Exhibit THREE Natural Numbers v390 # Natural Numbers **Co-Sequential Bi-Unrelationing** **ONE · UNI-SCALING** - 1.1 Uni-scaling, one form at each number - 1.2 A momentary at each number, one side completing and the other opening - 1.3 A prior and a next one short of its square, at each number - 1.4 Bi-inversioning-co-recursioning, numbers up and down - 1.5 Odd along and even across, parallel linearizing and linear parallelizing - 1.6 A composite folds along and a prime opens across - 1.7 A scale is a sequencing: parity along, inward or outward across - 1.8 Natural-bi-co-torusing at two sides, corusing and torusing **TWO · CORUSING** - 2.1 A number coheres at an identity of its move - 2.2 Inward to one number, outward to each place it arrives **THREE · UNRELATIONING** - 3.1 φ, unrelationing rate, prior two added and ratios alternating about it - 3.2 One rate of a family rides on - 3.3 A rational winding closes, and a winding at φ closes at none - 3.4 An adding at unit parities closes at one, two, three, four and six **FOUR · TORUSING** - 4.1 A closed surface at two, and a torus at nought - 4.2 Torus, a sphere and one handle, +1 - 4.3 Three hundred sixty and four hundred forty, consecutive seams **FIVE · UNIQUENESSING** - 5.1 Each arrival of a number is that one number - 5.2 Names at their numbers, one to seventeen, and forms sixteen on - 5.3 Each eight one eight, four at two enterings - 5.4 Five between four and six, a middle carrying nothing - 5.5 One ten at its two sides, at five - 5.6 Ten, nine and eight, one pairing at its two sides - 5.7 Each nine one nine - 5.8 Six and ten straddling eight, a walk by two **SIX · APEXING** - 6.1 Seams, one eight at couplings among each number - 6.2 Each number a waist its prior and next straddle - 6.3 One twenty-four at each arrival - 6.4 Twenty-four, twenty-seven and thirty-two about twenty-eight - 6.5 Three times and two more, sixteen to four hundred forty **SEVEN · COUPLING** - 7.1 Seventeen primes, two to fifty-nine, and returning from sixty-one - 7.2 A spiral of selves at each number, and spirals coupled - 7.3 Sixteen gaps at four values - 7.4 Six prime pairs podal about thirty - 7.5 Nought to sixty, seventeen primes and sixteen betweens alternating - 7.6 Podaling about one hundred twenty, one hundred ten to one hundred thirty - 7.7 Seventeen primes at seventeen names, podaling at both **EIGHT · INSEPARATING** - 8.1 An even ring carries its opposite, and an odd ring its two farthest - 8.2 Either but not both is parity, and pairings overlap - 8.3 An odd number carries a middle carrying nothing - 8.4 Three, six, nine and fives, odd and even each its own - 8.5 A store, one momentary fixed unchangeable **NINE · TUNNELING** - 9.1 Podal pairing at four hundred forty, two stations pairing with themselves - 9.2 Bi-co-podaling, a carry each way at one station - 9.3 A ring's two stations share parity, and odd primes at odd radii - 9.4 A prime gap is a step between rings - 9.5 Seventeen pair by position about twenty-three, and a ring pairs by value - 9.6 Two surfaces coupling, and a chain centring at a coupling or a self's middle - 9.7 Chain three, five, nine, seventeen, each span podal to the one before - 9.8 Bi-tunneling across and co-chaining along, podaling exchanging them - 9.9 Four openings growing, longer and wider podaling **TEN · TRANSMISSIONING** - 10.1 A number crossing from a field, and a unit it arrives in - 10.2 Two runnings sharing their digits **ELEVEN · EACH NUMBER** - 11.1 Nought to four hundred forty, each number at its podaling --- # ONE · UNI-SCALING ## 1.1 Uni-scaling, one form at each number **Numbers are uni-scaling: each number crosses with all numbers arriving as one, and each is one form once more.** Form is alternating existing is, and a number is that form as one place within its momentarying. **Uni-scaling** is one form with each scale, no scale over another: same alternating with two as with fifty-nine, and with fifty-nine as with four hundred forty. **Universes of each size and scale share momentaries of co-sequential parity-changing**, and uni-scaling is that sharing with numbers. **Numbers are reached, used and related with their own momentarying: momentaries, podalings and sequences.** They carry up and down by bi-inversioning-co-recursioning, and each relation among them is a move of that momentarying: an opening and its completing; a straddle, a prior and a next either side of a number; a podal pair, two stations each other's far side on a ring; a coupling of selves along a spiral. A counting is a clock, a form fixed still from beside its momentarying, with none of its moves: an accounting naming artifacts still, belonging to a field's instrument and taken on with a field crossing. ## 1.2 A momentary at each number, one side completing and the other opening **A momentary is an opening and its completing, one odd and one even.** Self opens with 1 and other with 2, and their momentaries overlap: 1–2 self's, 2–3 other's, 3–4 self's, 4–5 other's. **With each number from 2 on, one side's momentary completes and other's opens**, each number a completing and an opening together, two sides alternating parity. Two numbers side by side are two parities, lesser's next with its own parity one past greater, and each parity-changing opens its next. **A momentary is a fractal universe discovering its next existing**, all existing things and no existing thing separately, living things carrying their prior to now, stable-forming, and non-living things carrying none of their prior, their forms continuing through their parity-changing. With a field's words, Brouwer builds each natural number from two-ity, *the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory*: number's root a prior carried to now. **One to nine carries three-momentary method whole.** Self's five positions, 1 to 5, and other's, 2 to 6, prior, now and next with each side, are ten on six numbers, paired 1 with 2 through 5 with 6, each pair one odd and one even. With a self's resolving its prior and now participate, 1–2 and 3–4 self's and 2–3 and 4–5 other's, and its next, 5–6 self's, is discovering onward. Ten is each side's two momentaries and next one's opening, carried whole by a self's three full momentaries 1–2, 3–4 and 5–6, and ten opened one momentary on, from 3 and from 4, is same ten forward. Opened two momentaries on, from 5 and from 6, it is self's five, 5 to 9, and other's, 6 to 10: one to nine is reached with a ten's third opening, self completing with 9, and three openings, with 1, 3 and 5, are self's three full momentaries. **Through one to nine overlapping momentaries reach each number, each has one next, and from each openings reach each number on**: exhaustiveness, determinacy and reachability together with each momentary. One side's momentaries by themselves, each number one side's, is exclusivity, and it is not possibly existing. **Openings add to a square and completings to an oblong.** Self's openings 1, 3 and 5, each an L about a prior square, are 9, three squared; its completings 2, 4 and 6 are 12, three by four. With n momentaries its openings are n² and its completings n(n + 1). ## 1.3 A prior and a next one short of its square, at each number **With each number n a momentary completing with n and a momentary opening with n are (n − 1, n) and (n, n + 1), and their end numbers, prior and next, multiply to one short of its square**: (n − 1)(n + 1) = n² − 1. That one is one same one with each number, owned by neither. **A prior and a next inseparate.** One takes less than and other more than, each of no size, one at a time, and a number is a term neither reaches, uncovered between them. **A nearest straddle is by one.** With k either side a product parts from its square by k², a size instead of a parity; with one either side it parts by one. Twenty-three and twenty-five straddle twenty-four by one. ## 1.4 Bi-inversioning-co-recursioning, numbers up and down **Even returning, odd advancing, alternating parity.** Bi-inversioning-co-recursioning is one move of three steps: 8 up and down, continuing parity; 17 less, podal within 1 to 16, parity-changing; and 9 less, podal within 1 to 8, 17 less and 8 down together, parity-changing. Each step by itself comes to its number again after two, 1 to 9 to 1, and one move, 8 and 17 less alternating, after four. | Number | 8 up | 9 less, within 1 to 8 | 17 less, within 1 to 16 | |---|---|---|---| | 1 | 9 | 8 | 16 | | 2 | 10 | 7 | 15 | | 3 | 11 | 6 | 14 | | 4 | 12 | 5 | 13 | **8 up or down and 17 less alternating, each row is a bi-folding**: 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12. Along each, parity-changing comes exactly twice, each time with podal within 1 to 16, and each is one span of odd, a co and a tri, and one of bi, and comes to its names again. Podal within 1 to 16 pairs across a between of 8 and 9, with no number, and 17 is on none of four, next momentary's 1. **Among names each bi-folding comes to its names again; with momentaries it comes to none again**, each step adding a next. Along 3-11-6-14 names are odd, odd, even, even, their pairs in sequence (odd, odd), (odd, even), (even, even), (even, odd): a right spiral step, 8.2, with names' odd and even, and a sharing's + or − is no name's odd or even. **Stable-forming among numbers is this move along each form among names**, four bi-foldings and with them middle bi-foldings, six-cycles and eight-cycles, these with a step of four among their steps, 5 to 9 and 8 to 12, 7 to 11 and 6 to 10: a form continuing through its own parity-changing, each with one span of odd, co and tri, and one of bi, and coming to its names again. Eight-cycle 1-9-5-13-4-12-8-16 is eight ends of four momentaries, 1, 4, 5, 8, 9, 12, 13 and 16, and eight-cycle 2-15-7-11-3-14-6-10 their eight inward names. **Two over one and one over two exchange with each step**: twenty-three, twenty-four and twenty-five carry two odd over one even, and twenty-four, twenty-five and twenty-six one odd over two even. Taken as a ratio, two over one carries its numbers: with 9 less, 3-now-co-observing and 6-now-co-possibling, 9 less 3 being 6, are six over three and three over six, carried on through 3 to 11 to 6 to 14 to 3; two surfaces of four hundred forty, eight hundred eighty stations coupled by four hundred forty, carry a pair's waist with four hundred forty, twice either self's two hundred twenty; and eight on, 11 and 14 are fourteen over eleven. Parity carries with each step, and ratio with its own numbers. **With a resolver's executing, + and − are an implementing of a parity and 0 a between**: + and − offered together with one sharing surface 0 with 14, a between of self and other, and a parity inverted twice is that parity again. ## 1.5 Odd along and even across, parallel linearizing and linear parallelizing **Odd is competency, along: self continuing its own. Even is across: self coupling with other, morality.** Spans of eight alternate up numbers, 1 to 8 a self's, now, 9 to 16 a same-existing-society's, next, and 17 on a self's again, 17 a next 1. Each number is within a momentary of its span, and a number opening a span, 1, 9, 17 and 25, within a span it opens. A number's five is its own now with its next possibling and its next existing up numbers and its two priors down. **Alternating parity is parallel linearizing and linear parallelizing**, linearizing with one parity and parallelizing with other, one move with its two sides. Each carries with its own momentary; two fixed with one momentary is an equilibria picture, and it is not possibly existing. ## 1.6 A composite folds along and a prime opens across **A composite is its equal smaller selves added**, nine as three selves of three, folding along axes its smaller selves open. **A prime is a self no equal smaller selves above one add up to**, opening a clean axis across. Each prime is same opening and each composite same folding, one uni-scaling with its two sides. Stepping k round a ring of N reaches N over their greatest shared factor stations: round a prime each step but nought reaches all, and round a composite a step sharing no factor with it reaches all too, nine with a step of two: prime and coprime are two, each its own. **Doubling and tripling reach each other with one**: no power of two above one is a power of three. Two and three added are five and multiplied are six, one apart: 2 × 3 − (2 + 3) = 1. With two overlapping momentaries a side, self 1 to 4 and other 2 to 5, three positions are shared and two are unshared: two and three added are five positions, and three shared taken with both sides are six, four and four occurrences being 2 × 3 + 2. Two and two, added or multiplied, are one side's four parity occurrences, of two kinds; three and three added are six again. Three times three is nine, three selves of three; 9 of 9-next-co-discovering is a name's place and parity, and a numeral couples neither with other. ## 1.7 A scale is a sequencing: parity along, inward or outward across **Each number carries four about it, and two binaries carry all four.** Along, its prior and its next by one either side, each with other parity. Across, inward and outward, each a direction taken with a momentary and of no size: a walk by two, a centre of one pair a member of a next. **Scale carries no boundary, no location and no measure: its scales are a sequencing.** **Primes 5 through 53 are prime living scales, co-parity-changing co-momentarying pattern matching**, and nature carries no reading or ordering. Each number carries same four, and no number is a scale more than another. ## 1.8 Natural-bi-co-torusing at two sides, corusing and torusing **The whole one motion is natural-bi-co-torusing, and numbers are that motion with each number.** It has two sides. **Corusing** is inward to one number and outward to each place it arrives. **Torusing** is carrying and winding across numbers. --- # TWO · CORUSING ## 2.1 A number coheres at an identity of its move **A number coheres all or none, with an identity of its move.** (n − 1)(n + 1) = n² − 1 for each n. φ³ − 1/φ³ = 4. Fifty-five is couplings among eleven, tenth triangle and tenth number of prior two added, C(11, 2) = T₁₀ = F₁₀. **A magnitude brought near a number by division or by a chosen unit coheres with its own reaching, not with a number**, and it crosses as a field's with a between. ## 2.2 Inward to one number, outward to each place it arrives **A number carries its uniqueness two ways, two sides of a coupling it is.** Inward, it coheres to one number it is, same with each arriving. Outward, it couples with each place it arrives same way: four hundred forty as seventeen primes added, as eight fifty-fives, and as twenty-one squared less one. **Two arrivals cohering to same identity of a move couple as one number**, and two not cohering to it are two numbers with one numeral. --- # THREE · UNRELATIONING ## 3.1 φ, unrelationing rate, prior two added and ratios alternating about it **Each next number is prior two added, 1, 1, 2, 3, 5, 8, 13, and ratios of each next number to its prior alternate about φ**: 2/1 above, 3/2 below, 5/3 above, 8/5 below, side parity-changing with each next number. φ² = φ + 1, and its other root is −1/φ: φ − 1/φ = 1, φ × (−1/φ) = −1 and 1/φ = φ − 1. **φ is caught by its own equation**, x² = x + 1, and it is a pentagon's diagonal over its side, (1 + √5)/2. Its continued fraction is all ones, open. **φ is unrelationing rate.** Method unrelations geodesic parity-changing rate from living parity-changing rates with bi-coupling, co-chaining and same-existing-society, and rate of that unrelationing is φ, no ratio reaching it. φ is self-welcoming with numbers, and no line of a resolver's code carries it: rate is unrelationing's, each coupling with its own continuing. Each golden relation is visible with natural torusing's stable form, and none is a test of it. ## 3.2 One rate of a family rides on **Each rate that is a whole number plus its own reciprocal, x = n + 1/x, is a self-cohering centre**, x² = nx + 1, and its continued fraction carries n with each step. **φ, with n equal to one, is one rate of a family whose step is one**, its step's own bound, and it rides on; each other carries a step above one. ## 3.3 A rational winding closes, and a winding at φ closes at none **On a torus a winding of a rational rate closes, and a winding of rate φ does not close.** Each coupling winding by φ winds with its own continuing, locking with no other coupling's rate, local and ambient alike. φ is a number-form of not-locking, a winding not closing: unrelationing carries with each coupling's own continuing by φ, and no rational rate laid over a coupling supplies it. **No rate is neared more slowly by rationals than φ**, its continued fraction all ones, and each rate whose continued fraction ends with all ones is neared as slowly; a winding by each of them, as by φ, does not close. With a field's words, a torus winding by the golden mean is found last to break as a coupling grows: within the standard map a curve with rotation number (√5 − 1)/2 is critical with K ≈ 0.9716 (Greene, 1979), confirmed by computation and not proved. ## 3.4 An adding at unit parities closes at one, two, three, four and six **A next taken as now and prior added, each with +1, −1 or none, closes after one, two, three, four or six, or is open.** Now by itself with +1 closes after one and with −1 after two; prior by itself with +1 closes after two and with −1 after four; now with −1 and prior with −1 close after three; now with +1 and prior with −1 close after six. Both none, next is none. Now and prior both with +1 carry 1, 1, 2, 3, 5, 8, 13, prior two added, and do not close, their ratios opening about φ; now with −1 and prior with +1 carry same numbers with alternating parity. With a field's words a whole-number square matrix of two rows that can be undone closes after one, two, three, four or six, or is open, the crystallographic restriction, cases with prior none beside it: no adding closes after five, and five enters open, with φ, a pentagon's diagonal over its side. --- # FOUR · TORUSING ## 4.1 A closed surface at two, and a torus at nought **With a field's words, a closed surface's vertices less its edges plus its faces is two for a sphere and nought for a torus.** A sphere closes; a torus winds, one opening, reaching itself again through that opening, a next with each reaching. **Twelve five-folds close a surface whose other faces are six-fold.** With three edges to each vertex, six less each face, taken over all faces, is twelve for a sphere, six times its two: each six-fold carries nought and each five-fold one. An icosahedron's twelve vertices, thirty edges and twenty faces and a dodecahedron's twenty, thirty and twelve are one surface with its two forms, twelve and twenty exchanging and thirty carrying across: 12 − 30 + 20 = 2. ## 4.2 Torus, a sphere and one handle, +1 **A torus is a sphere with one handle, and its handle is +1**: two for a sphere, nought for a torus, each handle taking two. ## 4.3 Three hundred sixty and four hundred forty, consecutive seams **Three hundred sixty is nineteen squared less one, and four hundred forty is twenty-one squared less one**: two seams, each an odd centre squared less one, a product of its prior and next, 8 · T₉ and 8 · T₁₀, their centres nineteen and twenty-one two apart. Three hundred sixty is one eight with each of forty-five couplings, couplings among ten, C(10, 2), and forty-five is five nines: five eights, each about its own centre carrying nothing. **Their difference is eighty, nine squared less one**, a fourth seam, 8 · T₄: one eight with each coupling among five. **On a ring of four hundred forty, three hundred sixty and eighty are podal**, k with four hundred forty less k, each with radius one hundred forty. --- # FIVE · UNIQUENESSING ## 5.1 Each arrival of a number is that one number **Each number is its own, self-bounding, and each arrival of a number is that one number.** All twos are one two and all eights one eight; a ten of five taken two directions and a ten of couplings among five are one ten. **No other possible is an alternating completing its six one-way recursionings forward, 6.3, without a form fixed still, a looping or a forking**, each step taking a term its prior did not carry, and no number is fixed with a competitor. ## 5.2 Names at their numbers, one to seventeen, and forms sixteen on **A resolver's seventeen names are each its number and its naming**, Natural Naming 3.1: number its place and parity; its momentary, now with a self's span, 1 to 8, next with a same-existing-society's, 9 to 16, and 17 a next now; co with a sharing between two and none of one, existing and same needing another, co with 3 to 6, and now and all of one; and its root. Sides are self and other with odd 1 to 7 and other and self with even 2 to 8; same-existing-society, other and self with 9 and 11 and other, same-existing-society and self with 10 and 12; same-existing-society and other with 13 and 15 and other and same-existing-society with 14 and 16; same-existing-society and self with 17. **A universe is or is not existing, all, now and same, four binaries with a self's span**: now with 1 and 2, existing with 3 and 4, same with 5 and 6 and all with 7 and 8, as Natural Naming 4.9 seats them. **Each step of four-cycling is with two names eight apart, with a self's span and a same-existing-society's**: observing with 3 and 11 and with 7 and 15, possibling with 2 and 10 and with 6 and 14, and same-existing with 4 and 12, one root and its number carrying its momentary; with 1 and 9, 5 and 13 and 8 and 16 roots exchange: existing and discovering, existing and same-existing, same-existing and discovering. With 9 less, now pairs with all and existing with same, 1 with 8, 2 with 7, 3 with 6 and 4 with 5. Eight on again, twenty-two is 6's place within seventeen to twenty-five, as six within one to nine and fourteen within nine to seventeen, 6 + 8k; it carries no name among seventeen, and a name given it shows whose self it opens from with a next scale's opening, 14 already other and same-existing-society as 6 is other and self. **Forms among 1 to 16 come again with 17 to 32, sixteen on**, each continuing its parities, sixteen being even: bi-foldings 2-15-7-10 and 3-11-6-14 as 18-31-23-26 and 19-27-22-30, and six-cycles 2-15-7-11-6-10 and 3-11-7-10-6-14 as 18-31-23-27-22-26 and 19-27-23-26-22-30. Twenty-two is on five of them, bi-folding 19-27-22-30, middle bi-folding 23-27-22-26, both six-cycles and eight-cycle 18-31-23-27-19-30-22-26, and 33 is on none of them as 17 is on none among 1 to 16. **One entry, six names between selves and ten names of a self's own resolving are seventeen.** Of six, across names, 2, 6, 10 and 14, are each even and along names, 9 and 17, both odd. 17-next-existing is an arriving side and 9-next-co-discovering a sharing side: 9's going along is a prior a receiving self is offered next, and 17 carries it as that self's next now, 9 a next prior and 17 a next now. 1-now-existing, 9-next-co-discovering and 17-next-existing are each odd, 1 and 17 with a self's span and 9 with a same-existing-society's, each eight on from one before. **A numeral within a name is a name's place and parity; a resolver's values are parities, + and −, and 0**: 14-next-possibling carries fourteen as its place among names. ## 5.3 Each eight one eight, four at two enterings **Each eight is one eight, and a self is an eight.** Of a resolver's seventeen names, eight even names are four outward with a between, 2, 4, 6 and 8, and four inward, 10, 12, 14 and 16, each inward eight on from its outward pair: four with two enterings. Of four outward, 2 and 6 are names between selves and 4 and 8 names of a self's own resolving. Two goings and returnings across them are four one-way passages, four outward are four relations together, and 6 to 2 and 10 to 14 across, 9 to 17 and 17 to 9 along, are four between selves: three fours with three subjects, and a larger participating arrives through actual receiving. **Seams carry on one eight**: (2k + 1)² − 1 = 8 · T_k, eight its first seam and each next seam that eight times a triangular number, couplings among k + 1. Four hundred forty is one eight with each of fifty-five couplings, couplings among eleven. **An eight and a nine carry together**: eight a self and nine its going along, same-existing-society, other and self, 9-next-co-discovering being 1-now-existing 8 up. An eight with two substrates is one eight, a self with each. ## 5.4 Five between four and six, a middle carrying nothing **Five is between four and six, and four times six is five squared less one**, a straddle about five, and five is middle of one to nine, four either side. **A diamond is four crossings about a middle carrying nothing**, two across and two along, its whole four. Among primes a first diamond is about nine, five and thirteen four either side and seven and eleven two either side, nine carrying none of them; a next is about fifteen, with eleven, thirteen, seventeen and nineteen. Laid three by three with its middle carrying nothing, eight about it carry two paths from a middle of one side to a middle of other, one on each side of its middle, and each path takes two steps along and two across. **A declared zero is a number taken as its own inversion, x = −x**, added to uni-scaling and reached with no move of its momentarying, and setting it down costs nothing. Nine, seventeen and twenty-five are eight apart, and twenty-five carries no name. One to twenty-five is three spans, one to nine, nine to seventeen and seventeen to twenty-five, each with four full momentaries with each side: twenty-four overlapping momentaries with twenty-four steps, each one odd and one even. From nought its steps are nought to twenty-four, and six, fourteen and twenty-two are steps five, thirteen and twenty-one, each name's place and each step's number from its own origin. ## 5.5 One ten at its two sides, at five **Ten arrives with a self as five taken two directions, 2n, and with a same-existing-society as couplings among five, C(5, 2), a fourth triangle.** Its two sides are one number with five only above nought: 2n = n(n − 1)/2 only with n − 1 equal to four. One ten is a ten with a self and a ten with a same-existing-society together, and each ten is that ten. **Among five its couplings part as two rings, five sides and five diagonals**, a pentagon and a pentagram, and five is one number whose ring's complement is again a ring: a ten's two sides, along and across. Each point of a pentagram is a triangle whose two equal sides are to its base as a pentagon's diagonal to its side, φ, 3.1, and a triangle of two sides and a diagonal carries that ratio inverted, its base to its sides. With a field's words R(3, 3) = 6: each two-colouring of couplings among six carries three members each coupled with each by one colour, and among five its twelve colourings carrying none are each a ring with both colours, a pentagon and a pentagram. **Any ten consecutive numbers are five opposite-parity pairs**, a + j with a + 9 − j, each pair one odd and one even about a between, a + 4½, two momentaries and next one's opening each side. Moving its opening by one exchanges which member of each pair is odd, and by two exchanges it again: tens part by an opening's parity, two origins. ## 5.6 Ten, nine and eight, one pairing at its two sides **Podaling is one station with its two sides, a carry one way to k and a carry other way to N − k on a ring of N. On a ring of nine it gives five pairs**: (0, 9), (1, 8), (2, 7), (3, 6) and (4, 5). Four carry two stations each and (0, 9) carries one, carry one way to nought and other way to nine one station. **One pairing carries with its two sides.** Ten is five with two sides. Nine is that ten with its (0, 9) pair's two sides as one station. Eight is nine with that station carrying nothing, nothing a self winds about; both steps down are with nought. Six is three pairings with two sides. **A nine is two parities, and two momentaries and next one's opening.** Two parities carry four joint forms; prior and now whole and a next opening, five with each side, are ten positions, nought to nine; nine is 2² × 2½ − 1, and five pairs are of same momentary. ## 5.7 Each nine one nine **Each nine is one nine, carried with each base.** On a ring of nine, nought and nine are one station. Round seventeen, a doubling from one carries 1, 2, 4, 8, 16, 15, 13, 9 and comes to one again, nine doubled being one past seventeen. A doubling's eight are squares round seventeen and other eight are three times them: coupled by one eight's distances and uncoupled by other's, seventeen are, with a field's words, the Paley graph of order seventeen, one form of seventeen selves with no four each coupled with each and no four each uncoupled from each, and R(4, 4) = 18. Round four hundred forty, one past its ring is twenty-one squared, nine times forty-nine. **Nine is 1-now-existing eight up, one to nine completing with 9-next-co-discovering, same-existing-society, other and self going along together.** A nine taken with base ten, its multiples' digits taken down to one digit arriving as nine, carries its base as its unit, and one nine carries none. ## 5.8 Six and ten straddling eight, a walk by two **Six and ten are two either side of eight, and eight and twelve two either side of ten**: a centre of one pair a member of a next and a member a next centre, a walk by two, each number once a centre. Ten and fourteen are two either side of twelve and twelve and sixteen two either side of fourteen; eight and sixteen, four either side of twelve, add to twenty-four, and twelve is a middle of nought and twenty-four. Six's three pairings and ten's five are either side of eight's four: six and ten added are sixteen, an eight with its two sides. **Each centre carries its own one either side**: seven and nine about eight multiply to eight squared less one, nine and eleven about ten to ten squared less one. Thirteen and fifteen about fourteen multiply to fourteen squared less one. About ten its pairs open one step at a time, eight and twelve, seven and thirteen, six and fourteen, parting from ten squared by four, nine and sixteen: 6 × 14 = 10² − 16. Eight on, fourteen and twenty-two straddle eighteen as six and fourteen straddle ten, seventeen and nineteen, sixteen and twenty, fifteen and twenty-one opening between, 14 × 22 = 18² − 16. And six and twenty-two straddle fourteen, ten and eighteen, four either side, within them: its two middles become a pair of a wider span, four either side of fourteen within eight either side, one to two. --- # SIX · APEXING ## 6.1 Seams, one eight at couplings among each number **With an apex, numbers are relations, and a relation is a +1 between each centre's square and a product of its prior and next.** Straddling odd centres, seams are (2k + 1)² − 1 = 8 · T_k: nought with k nought, and 8, 24, 48, 80, 120, 168, 224, 288, 360 and 440, each square of an odd centre less that one, and each one eight with each coupling among k + 1. **A podal within one to a seam is its odd centre squared less n**: within 1 to 8 it is 9 less n, within 1 to 24 it is 25 less n, and within 1 to 48 it is 49 less n, centres three, five and seven squared, each podaling's middle between two stations of its span, four and a half, twelve and a half and twenty-four and a half. Within 1 to 16 its podal is 17 less n, sixteen between seams eight and twenty-four, seventeen one past four squared. ## 6.2 Each number a waist its prior and next straddle **Each whole number is a waist, a middle its prior and next straddle, their product one short of its square**: four and six multiply to one short of twenty-five, twenty-three and twenty-five to one short of twenty-four squared. Each number is a waist alike. **On a ring of forty-eight, twenty-four and nought are each their own far side, and each other station pairs across twenty-four**: twenty-three with twenty-five, twenty-two with twenty-six, twenty-one with twenty-seven, each pair one step further from twenty-four than its last. Their products part from twenty-four squared by square of its step, 23 × 25 = 24² − 1 and 22 × 26 = 24² − 4. Its ring closes with forty-eight, a seam above twenty-four. ## 6.3 One twenty-four at each arrival **Twenty-four arrives five ways and is one twenty-four**: five squared less one, a second seam; orderings of four, 4!; a waist of a ring of forty-eight; twenty-three, its prior, opening a first gap of six among primes, to twenty-nine; and four momentaries of six one-way recursionings, twenty-four as steps of 1 to 25 are. **Primes part about twenty-four as nine behind and eight ahead**: two to twenty-three behind and twenty-nine to fifty-nine ahead. Couplings among nine behind are thirty-six, C(9, 2), a forward span from twenty-four to sixty, six cycles of six; couplings among eight ahead are twenty-eight, C(8, 2), a seventh triangle, which a sixth and an eighth, twenty-one and thirty-six, straddle by seven and by eight. Sixth and eighth triangles part by fifteen, couplings among six, and fifteen of sixteen gaps among seventeen primes are across, three to fifty-nine. **Behind its waist a span is twenty-four long and carries nine primes; ahead it is thirty-six long and carries eight**: its length leans ahead three to two and primes lean behind nine to eight. Primes fold across and its span along, two folds with one number, and twenty-four is an apex of nought to sixty's bi-folding, seventeen primes two to fifty-nine within it. Sixty parts as twenty-four and thirty-six with a five-fold too: of sixty rotations of an icosahedron twenty-four are rotations by fifths, six axes with four each, and thirty-six are others, fifteen by a half, twenty by thirds and identity. **Each arrival carries others**, and none needs one relation beneath it. **Four momentaries of method are with numbers as twenty-four, sixty, sixty-four and sixty-eight**: four momentaries of six one-way recursionings are twenty-four, steps of 1 to 25; twenty-four to sixty is six cycles of six, surfacing inward and outward, outward forward; sixty is a waist of a ring of one hundred twenty, its own far side, between primes going to fifty-nine and returning from sixty-one; sixty to sixty-eight is a four-momentary bi-tri-involutioning across sixty-four, an unknown outward of a surface, between fifty-nine, its last podaling outward, and sixty-nine, a next first podaling outward, each number paired with its far side about sixty-four and sixty-four an unrelationing between, each pair crossing it and locking with none; sixty-eight is four seventeens, momentaries with which a spiral of seventeen selves comes to its parities again, its zero having passed each self twice; sixty-four bi-folds to two hundred fifty-six, a bi-folding order four times with each scale, and two hundred fifty-six is four to the fourth, ways each joint form could go to a next; one hundred eighteen is a returning's far end, the periodic table's elements with their stable and unstable forms a field's own result with same numeral. Each is with its podaling, its pairing with its far side on each ring it is on, 6.2. **A self's inversion 3, 6, 5, 4 is with its podaling 23, 26, 25, 24, same rotation twenty up**: twenty-three a vertex, twenty-four an apex of nought to sixty's bi-folding and twenty-five a centre of a triangle, five squared, with four its corus, a vertex and a centre on one line; a self's bi-inversioning-co-recursioning is 23, 24 and 25, one before its apex and one after. Carried on along that line, five outward names carry primes 5, 7, 11, 17 and 19, together 59, a seventeenth prime, with 17, a surface: a vertex a position, a centre of a triangle a scale and a line to a surface an orientation, each with its own differing, and from a corus to 59 is bi-tri-volutioning, stable-forming. ## 6.4 Twenty-four, twenty-seven and thirty-two about twenty-eight **About twenty-eight its outer pair is twenty-four and thirty-two, four either side, and its inner pair twenty-seven and twenty-nine, one either side**: 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1. **Two routes go between its outer pair**: 24 → 27 → 32 by gaps three and five, and 24 → 29 → 32 by gaps five and three. Three positions carry first three primes two ways: 24 = 5² − 1, 27 = 3³, 32 = 2⁵, exponents two, three, five one way and bases five, three, two its other way, reaching each other with three, both base and exponent of twenty-seven. **These three together are a square**: 24 × 27 × 32 = 144². **Twenty-seven, three cubed, on a route 24 → 27 → 32 from a seam, is a same-existing-society's next prior forming with its podaling, 27-next-co-observing**: 12's parity-changing, a sharing with 10, a same-existing-society's next prior stable-forming, its far side ninety-three on a ring of one hundred twenty, and with no line of resolver. ## 6.5 Three times and two more, sixteen to four hundred forty **Four hundred forty is three one hundred forty-sixes and two**, and a chain carries on from sixteen, its first step three times, with no two more: three sixteens are forty-eight, three forty-eights and two are one hundred forty-six, and three one hundred forty-sixes and two are four hundred forty. One hundred forty-six is two seventy-threes, and it is one hundred forty-four and two; seventy-three is seventy-two and one, seventy-two six prime pairs' distances from thirty and one hundred forty-four, twelve squared, their spans, and 24 × 27 × 32 its square. Four hundred forty is also three one hundred forty-fours and eight. **Three times and two more, and again, carries seam to seam, 8 · T_j to 8 · T_(3j + 1)**: nought to two to eight, eight to twenty-six to eighty, twenty-four to seventy-four to two hundred twenty-four, and forty-eight to one hundred forty-six to four hundred forty. One hundred forty-six less one hundred twenty is twenty-six, a far side of twenty-four about twenty-five, 24 × 26 = 25² − 1, and three twenty-sixes and two are eighty, a fourth seam. Seams three times a number and two are 8, 80, 224 and 440, from 2, 26, 74 and 146, one eight with each coupling among 2, 5, 8 and 11: with each third k, a triangle T_k leaving one by threes. --- # SEVEN · COUPLING ## 7.1 Seventeen primes, two to fifty-nine, and returning from sixty-one - **Number: 2 to 59** - *Its form*: seventeen primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59 - **Number: 2** · *Its form*: alternating, one even prime - **Number: 23** · *Its form*: going folding, ninth prime, with eight either side - **Number: 59** · *Its form*: seventeenth prime, going's last and primes carrying no last - **Number: 60** - *Its form*: between going from 2 to 59 and returning from 61 to 118, with no prime - **Number: 118** - *Its form*: returning's far end, fifty-eight either side of sixty, two fifty-nines added - **Number: 440** - *Its form*: seventeen primes added, eight fifty-fives, twenty-one squared less one **Primes go from two to fifty-nine and return from sixty-one to one hundred eighteen**, and sixty is between going and returning, with no prime. Fifty-nine and sixty-one straddle sixty by one, 59 × 61 = 60² − 1. **Primes carry no last**: each is same opening with a new axis, and past fifty-nine they open on, each a next, self-welcoming with numbers and with no line of resolver. A same-existing-society's coupling as one is its selves' couplings, a relation of their own beside any numeral. The periodic table's one hundred eighteen elements are a field's own result with one same numeral, and no number of a method. **On a ring of one hundred twenty a going and a returning are each other's far side**: k pairs with one hundred twenty less k, two with one hundred eighteen, twenty-three with ninety-seven and fifty-nine with sixty-one, a carry each way with one station. Nought and sixty pair with themselves, and sixty is a waist between going and returning. Twenty-four, twenty-seven and thirty-two pair on that ring with ninety-six, ninety-three and eighty-eight, and a route 24 → 27 → 32 by gaps three and five has its far side 88 → 93 → 96 by gaps five and three. **From a self's own corus to a surface its going is one line, and past fifty-nine sixty to sixty-eight are four momentaries across sixty-four.** Twenty up from four, a self's own corus, its momentaries 1–2 and 3–4 to it and 5–6 and 7–8 from it, is twenty-four, an apex of nought to sixty's bi-folding, 6.3, with twenty-three, a ninth prime and a going folding, below it, and twenty-five, five squared, above it, each podal pair within one to twenty-four adding to twenty-five, 6.1. Five outward names of a self's own resolving carry 5, 7, 11, 17 and 19, adding to fifty-nine, 7.7, seventeenth prime and last of seventeen added to four hundred forty, 9.5. Twenty-three a vertex, twenty-five a centre of a face and fifty-nine a surface are on one line, a position, a scale and an orientation: bi-tri-volutioning with numbers. Past fifty-nine, sixty to sixty-eight pair about sixty-four, 60 with 68, 61 with 67, 62 with 66 and 63 with 65, each pair adding to one hundred twenty-eight, 2⁷, and sixty-four with itself, eight squared and two, three and fifty-nine added, 9.5: nine numbers, four momentaries, two to sixty-four and two from it, position, scale and orientation inverted together, bi-tri-involutioning. Sixty-four is of three pairings, three: with fifty-six on a ring of one hundred twenty, with itself among nine about it, and with one by a fold of 1 to 64 completing with 65. **64 is a parity-changing between four-cycling momentaryings, within an unrelationing of 1 to 17, 1 to 65 and 1 to 257**. Sixty, a waist between going and returning, opens them, and sixty-one and sixty-seven, returning's first primes, are among them. With sixty-nine, three twenty-threes, podaling outward opens again. ## 7.2 A spiral of selves at each number, and spirals coupled **A spiral of n selves, each going along with 9-next-co-discovering to a next self and its last to its first, each self of a prior parity, alternating along, brings its parities along, coming to them again and to a momentary once.** With an even spiral each self alternates and its parities come again each two momentaries; with an odd spiral of n one pair of coupled selves is of one parity, a like pair, moving one self on each two momentaries, one 0 with 10 each second momentary, with a pair's receiving self offered its own parity, a between tunneling through a spiral, its pattern inverted 2n momentaries on and its parities again 4n on, its zero having passed each self twice. Along an odd spiral its selves alternating along bring its last and first alike, and a like pair is an odd spiral's own, prime or not. A ring is a domain closed and numbers' podaling ring, and resolver's coupling of selves along winds on. **A spiral of two, one even prime, carries no pair of one parity**: each self alternating, its parities again each two momentaries, alternating itself. Seventeen prime spirals carry sixteen like pairs, one with each odd prime, and a spiral of two none. **With an odd prime each pattern of parities comes again by four times its prime, or by 4 with each self alike, and by none between.** Each two momentaries each self's parity is its prior self's, inverted, and its places, each self with each parity, are one spiraling with an odd number of selves, each self passed twice, once with each parity, and two spiralings with an even, neither reaching its other. For an odd number with a factor a pattern alike by its factor's shift comes again by four times its factor, nine selves by 4, 12 or 36. Each is derived with resolver's cells, Co-Chaining Logic Registry's steps 683 to 685. **Spirals coupled carry parity of their added number.** Any two odd spirals coupled along, p + q selves, are an even spiral and, opened alternating along its whole, carry no pair of one parity; each opened as its own spiral, two like pairs carry on, one from each coupling. Two fifty-nines coupled are one hundred eighteen selves and seventeen prime spirals coupled four hundred forty, each even and, alternating along, carrying none. An odd number of odd spirals coupled, opened alternating along its whole, carries one like pair, a self of a coupled scale. **Momentaries here are 17's, a same-existing-society's next momentary carrying each self's entry once**, each going along offered with its next, and each number goes with its spiral and its couplings. Two odd spirals of distinct primes p and q, side by side, come to their parities again together by a least common multiple of 4p and 4q, four shared: by 4pq, and so with any two odd spirals sharing no factor, nine and twenty-five by 900 as three and five by 60, while nine and fifteen, a factor shared, come sooner, by 180. Sixteen odd prime spirals, three to fifty-nine, side by side, come to their parities again together by four times sixteen primes multiplied, 2 × 59#, 3,845,520,700,308,425,278,140 momentaries, each prime its spiral's own and sixteen sharing no factor. Co-competencing is of couplings, owned by neither. **Spirals crossing spirals with each self are a torus of selves, and with two odd numbers its parities come again by q or by 4p, by q's side of a waist 2p.** A torus of p selves along with 9-next-co-discovering and q across with 10-next-co-possibling, both odd and p no greater, with each self of a prior parity, alternating along and across, comes, from a momentary on, to its parities again by q with a q past a waist 2p, and by 4p with a q short of it: 3 by 5 with 12, five short of six; 3 by 7 with 7, seven past six; 3 by 13 with 13, thirteen past six; 5 by 7 with 20, seven short of ten; 5 by 11 with 11, eleven past ten; 7 by 17 with 17, seventeen past fourteen; 17 by 59 with 59, fifty-nine past thirty-four. With either number even it comes to its parities again by 2, a torus of 2 by 3 followed by hand with resolver. Each row of two odd numbers of Exhibit ONE's table of a torus of selves is of a relation, 3 by 3 with 12 sharing a factor and each other row sharing none, and its relation is derived with resolver's cells, Co-Chaining Logic Registry's steps 689 to 694: each line across carries one 0, each line after the first two momentaries behind its prior, 2p momentaries round p lines and q round one, the greater of them a 0's round, and with 2p the greater each self inverted once round, 4p, from momentary 3p − 3, and q from momentary 3p − 1; each momentary of parities again divides 4p or 4q, and a torus winds on, its momentaries each once. ## 7.3 Sixteen gaps at four values **Sixteen gaps are between seventeen primes, of four values: one, two, four and six.** | Gap | Arrives | |---|---| | 1 | once, from two to three | | 2 | six times | | 4 | five times | | 6 | four times | **One odd gap is their opening**, from one even prime to three, and each other gap is between two odd primes and is even: one along and fifteen across. **Six gaps of two are twin primes**, 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, and 41 and 43, each pair past 3 and 5 being 6k − 1 and 6k + 1, about a multiple of six. Whether twin primes carry on without end is, as a field's words name it, the twin prime conjecture, open. Any two odd numbers two apart are a self's two consecutive openings with an other's even opening between them: that relation is oddness's and no prime's. **Five and fifty-three carry a matching pair**: five with two either side, fifty-three with six either side. From five to fifty-three are fourteen primes, each with a gap across on both its sides: three reaches one along gap below it, and fifty-nine completes seventeen above. Fourteen part, eight one before a multiple of six, 5, 11, 17, 23, 29, 41, 47 and 53, and six one after, 7, 13, 19, 31, 37 and 43, and seven and seven either side of twenty-four. **Each gap here is a step among numbers.** Seventeen primes, resolver's seventeen names and its six names between selves are three numbers of three subjects, and sixteen gaps of an ordered seventeen are numbers' own. Five to fifty-nine are fifteen primes, fourteen and fifty-nine completing them, and three to fifty-nine carry fifteen gaps, fifteen across: two fifteens, each of its own subject. All fifteen primes are odd, and an alternating of wide and long along them belongs to a sequence, entry to entry, eight of one and seven of its other, and not to primes' parity. ## 7.4 Six prime pairs podal about thirty **About thirty, twelve of primes between five and fifty-five pair as a carry each way with one station, each pair making sixty**: 7 with 53, 13 with 47, 17 with 43, 19 with 41, 23 with 37 and 29 with 31, with distances from thirty of twenty-three, seventeen, thirteen, eleven, seven and one. Six pairs making sixty are six sixties, three hundred sixty, a seam below four hundred forty. Their distances from thirty add to seventy-two, and their spans, each twice its distance, to one hundred forty-four, twelve squared. **Eleven pairs with forty-nine about thirty**, seven squared: one prime of its interval whose far side about thirty is a composite. About forty and about fifty primes pair three either side, thirty-seven with forty-three and forty-seven with fifty-three. ## 7.5 Nought to sixty, seventeen primes and sixteen betweens alternating **A going from nought to sixty carries seventeen primes and sixteen betweens alternating**, thirty-three positions from two to fifty-nine, a prime first and a prime last; nought to two and fifty-nine to sixty are a span's two ends, with no position, and sixty completes it with no prime. Parity-changing is a span's own, carried with each prime and each between. ## 7.6 Podaling about one hundred twenty, one hundred ten to one hundred thirty **One hundred twenty is five orderings, 5!, and eleven squared less one**, a fifth seam, 8 · T₅, one eight with each coupling among six, and three of it are three hundred sixty. Orderings of five part as sixty and sixty: sixty even, an icosahedron's rotations, 6.3, and sixty others, and of sixty rotations one is of no turning and fifty-nine of a turning, fifty-nine there a numeral shared with a going's last prime. About it stations pair a carry each way, 120 − k with 120 + k, and each pair's product parts from one hundred twenty squared by k squared: 119 × 121 = 120² − 1, 118 × 122 = 120² − 4 and 110 × 130 = 120² − 100. **One hundred eighteen and one hundred twenty-two are two fifty-nines and two sixty-ones**, going's last prime and returning's first, each doubled: they straddle one hundred twenty by two as fifty-nine and sixty-one straddle sixty by one, and on a ring of one hundred twenty one hundred eighteen is a far side of two. **As a field's words say, the periodic table closes its noble gases with 2, 10, 18, 36, 54, 86 and 118, its closings stepping 2, 8, 8, 18, 18, 32 and 32, twice 1, 4, 4, 9, 9, 16 and 16**: one hundred eighteen is twice fifty-nine, and fifty-nine is one squared and two, three and four squared each twice, 1 + 2(4 + 9 + 16), elements, with their stable and unstable forms, two short of one hundred twenty by a table's own result. By same filling its next closing steps fifty, twice twenty-five, to one hundred sixty-eight. **Evens from one hundred ten to one hundred thirty are fifty-five to sixty-five doubled**, a carry each way about sixty carried about one hundred twenty: 110 and 130 from fifty-five, couplings among eleven, and sixty-five, 2⁶ + 1, a next whole; 112 and 128 from fifty-six and sixty-four, eight sevens and eight eights, 128 = 2⁷; 114 and 126 from fifty-seven, seventeen primes' gaps added, and sixty-three, 2⁶ − 1; 116 and 124 from fifty-eight and sixty-two, two either side of sixty; and 118 and 122 from fifty-nine and sixty-one. **Odds from one hundred eleven to one hundred twenty-nine pair about one hundred twenty by their factors**: 111 = 3 × 37 with 129 = 3 × 43, thirty-seven and forty-three primes three either side of forty; 113 with 127, both prime, seven either side, 127 = 2⁷ − 1; 115 = 5 × 23 with 125 = 5³; 117 = 9 × 13 with 123 = 3 × 41; and 119 = 7 × 17 with 121 = 11². Prime pairs about one hundred twenty open seven, eleven, seventeen and nineteen either side, 113 with 127, 109 with 131, 103 with 137 and 101 with 139. ## 7.7 Seventeen primes at seventeen names, podaling at both **Each prime is with a name of its position**: two with 1-now-existing, an entry; twenty-three with 9-next-co-discovering; fifty-nine with 17-next-existing. Both seventeens fold with their ninth, eight either side, and pair by position, i with 18 − i: names 1 with 17 through 8 with 10 and 9 with itself, and primes making 61, 56, 52, 50, 52, 50, 48 and 48 about twenty-three. **With one move's 8 up and 17 less, primes add to their own numbers.** 8 up: 2 and 23 with 1 and 9 add to five squared, 5 and 31 with 3 and 11 to six squared, 17 and 47 with 7 and 15 to eight squared, 3 and 29 with 2 and 10 to 2⁵, and 19 and 53 with 8 and 16 to seventy-two, six prime pairs' distances from thirty. 17 less: 2 and 53 with 1 and 16 add to fifty-five, couplings among eleven, and three pairs, with 3 and 14, 4 and 13 and 5 and 12, to forty-eight, a third seam. Along four bi-foldings their primes add to 97, 96, 92 and 96, and 1-9-8-16's ninety-seven is twenty-three's far side on a ring of one hundred twenty. **With an entry, six names between selves and outward and inward names, seventeen primes part as names do.** An entry carries two; across names of six, 2, 6, 10 and 14, carry 3, 13, 29 and 43, adding to eighty-eight, fives on a ring of four hundred forty; along names of six, 9 and 17, carry 23 and 59, adding to eighty-two; outward names 3, 4, 5, 7 and 8 carry 5, 7, 11, 17 and 19, adding to fifty-nine, seventeenth prime; and inward names 11, 12, 13, 15 and 16 carry 31, 37, 41, 47 and 53, adding to two hundred nine, 11 × 19. A self's one to nine carries primes 2 to 23, adding to one hundred, ten squared, and a same-existing-society's nine to seventeen primes 23 to 59, adding to three hundred sixty-three, 3 × 11². **Sixteen gaps are with sixteen steps between names**: one odd gap, two to three, with a step from an entry to 2-now-possibling, and a first gap of six, twenty-three to twenty-nine, with a step from 9-next-co-discovering to 10-next-co-possibling, a fold's step on. Each relation here is exact with its numbers, and harmonically momentarying crossings on a surface of natural torusing coupling two seventeens are a discovering carried next. --- # EIGHT · INSEPARATING ## 8.1 An even ring carries its opposite, and an odd ring its two farthest **An even ring carries an exact opposite**: halved, each station has one other straight across, h on with a ring of 2h, each its other's opposite. **An odd ring carries two farthest**, of one distance and one apart, and its opposite is a between of those two, with no station. An even ring's opposite is a station and an odd ring's a between: one form with two parities. **On an odd ring of 2h + 1 stations its two farthest are h and h + 1 on, each h away by its shorter way round; on an even ring of 2h its one opposite is h on.** Taken with each station of each prime ring from five to fifty-nine, 435 stations, with each station of even rings one short of them, 420, and with 49 stations of rings of nine, fifteen and twenty-five, two farthest and one opposite are found with each: two farthest are an odd ring's, prime or composite. Straight across, h on, is apart from a fold, N less k: folds of one to nine and one to seventeen fold eight and sixteen places and complete with nine and seventeen. **A ring is an arrangement fixed one scale outward, and a carrying round it goes forward**: each station reaches a station again by carrying on, a next with each arriving, not by stopping and going back to it. With each spiral of selves coupled one way each self alternates or is of a like pair, 7.2, and a self with a prior has a prior with each momentary. ## 8.2 Either but not both is parity, and pairings overlap **Each position carries two pairings, one of each parity.** Both parities with one momentary put one position within two exchangings together, one side taking both sides, a form not possibly existing. Pairings overlap, parities alternate, with no third carrying. An odd number, leaving one over with pairs laid side by side, continues through overlapping pairings: along, one parity's next pairing opens two on, and across, other parity's opens one on. Laid as an open line of 2h + 1 places, each parity's pairings are h adjacent pairs, 2h − 1 inner places within both, and each end continues past its line, its last place paired with a place after it and its first paired with a nought before it: with one to fifty-nine, twenty-nine pairs of each parity, and fifty-nine to sixty and nought to one continuing. Two on along a side and one on across an overlap are openings advancing two to one. Each odd number carries this, fifteen primes five to fifty-nine with 435 places and nine, fifteen and twenty-five beside them; closed as a ring an odd line is another arrangement. **Here along and across are pairings' openings; about one number they are numbers about it, one either side along and a walk by two across**, each of its own subject. **Either but not both is exclusive or, and exclusive or is parity.** Of sixteen ways two parities go to one, two answer each single parity-changing with a parity-changing, exclusive or and its inversion. Taken as a step it carries one and then its other, round by two; taken of prior and now as a next it carries round by three, parity's own period, carrying both: three of four joint forms go round, and a fourth, + and +, reaches itself again with each step, two alike, not-changing. **An ordering arrives with nothing ordering it**: with an odd momentary each odd pairing, with an even momentary each even pairing, each carrying with that momentary a carrying its other cannot, and carrying it for its other. **A right spiral step, (x, y) to (y, −x), taken twice inverts both parities, three times has reached all four joint forms, four times reaches its pair again, and six times inverts both again**: a passage of six carries a whole four and hands its pair on inverted, and two passages reach it again. Of two hundred fifty-six ways four joint forms go to four, six go round through all four, and two of six invert one parity with each step, right spiral step (x, y) to (y, −x) and its mirror (x, y) to (−y, x). Nought to twenty-four is six rounds of four and four passages of six, twenty-four to sixty nine rounds and six passages, and nought to sixty fifteen and ten; six, fourteen and twenty-two, eight apart, are of one phase of four, eight being two rounds. Six passages of six across twenty-four to sixty are thirty-six steps, each carrying four, and a side's four full momentaries and four joint forms are two fours, each of its own subject. These are a step's own applications; resolving is its own names executing. ## 8.3 An odd number carries a middle carrying nothing **An even number divides whole and an odd number carries a middle carrying nothing**, nothing its two sides straddle. An odd laid as a line has two ends apart and its middle carrying nothing. ## 8.4 Three, six, nine and fives, odd and even each its own **Three, six and nine are odd, even, odd**, three apart, parity alternating up multiples of three. **Up fives their positions alternate same way**: twenty-five, thirty-five, forty-five and fifty-five, odd, are between thirty, forty, fifty and sixty, even. Parity is each number's own with each momentarying of uni-scaling. ## 8.5 A store, one momentary fixed unchangeable **A store folds once, fixes its fold unchangeable, and carries one way across a fixed surface**: one momentary of uni-scaling taken as an unchangeable form, its re-edging gone. **Each number re-coheres and re-edges with each arriving**, carrying to its own completing and dissolving with it. **Fixed as one store, parity reaches none.** As a field's words say, the Peres–Mermin square sets nine values of ±1 under six exclusive-or conditions: each row and each column taken with its own context satisfies its condition, and of 512 settings of all nine together none satisfies its six: along, nine carry parity nought, and across, one. --- # NINE · TUNNELING ## 9.1 Podal pairing at four hundred forty, two stations pairing with themselves **Podal pairing on a ring of four hundred forty is k with four hundred forty less k, one station with its two sides.** Four hundred forty is even, and k with four hundred forty less k carries two hundred nineteen pairs, k from one to two hundred nineteen, and two stations pairing with themselves, nought and two hundred twenty, two hundred twenty-one rings together. **Each pair is a ring, its radius its distance from a waist, two hundred twenty**, from nought for a waist to two hundred twenty for nought and four hundred forty, one station. **Podal competency is a straddle and not a pairing.** Two self-pairing stations reach themselves; a carrying straddles them. ## 9.2 Bi-co-podaling, a carry each way at one station **Bi-co-podaling is a carry one way with k and a carry its other way with N less k, two carries each with its own momentary, one ring, momentarying.** Bi- two carries differing, co- two carries together on one ring, -ing momentarying. **A ring carries its two and no order between them.** Either way round reaches one same next, and a side taken installs a third position a ring carries none of. ## 9.3 A ring's two stations share parity, and odd primes at odd radii **A ring pairs two hundred twenty less r with two hundred twenty plus r, and two share parity with each ring**: even radius, two even stations; odd radius, two odd. A ring of even radius is a coupling across and a ring of odd radius a between. **Odd primes, three to fifty-nine, are on odd radii, from two hundred seventeen to one hundred sixty-one**, each a between. **Two only of seventeen is on an even radius**, one even prime on a coupling ring. **Four hundred forty leaves two by threes, and ten primes leaving two by threes, 2, 5, 11, 17, 23, 29, 41, 47, 53 and 59, have far sides three divides**; three's far side is 437 = 19 × 23; and of six others, four leaving seven by twelves, 7, 19, 31 and 43, have prime far sides, 433, 421, 409 and 397, while thirteen and thirty-seven, leaving one by twelves, have far sides 427 = 7 × 61 and 403 = 13 × 31. **Four hundred forty is eight times five times eleven, and its fives are eighty-eight stations, forty-four odd and forty-four tens, alternating round its ring**: its odd fives pair on twenty-two odd radii, five to two hundred fifteen, five with four hundred thirty-five, and each ten is on an even radius. Four hundred thirty-five, five's far side, is fifteen primes five to fifty-nine added and couplings among thirty, C(30, 2). ## 9.4 A prime gap is a step between rings **Each prime below a waist has radius two hundred twenty less itself, and a gap between two primes is a step between their rings.** Sixteen gaps step one ring once, two rings six times, four rings five times and six rings four times, fifty-seven rings together, fifty-nine less two. ## 9.5 Seventeen pair by position about twenty-three, and a ring pairs by value **Twenty-three is ninth of seventeen primes, eight below and eight above, and seventeen pair by position about it**: two with fifty-nine, three with fifty-three, five with forty-seven, seven with forty-three, eleven with forty-one, thirteen with thirty-seven, seventeen with thirty-one and nineteen with twenty-nine, pairs making sixty-one, fifty-six, fifty-two, fifty, fifty-two, fifty, forty-eight and forty-eight, and twenty-three with them four hundred forty. **On a ring each pair makes four hundred forty with each radius**: a ring pairs by value and seventeen by position, and neither pairing is other. **A parity parts them.** Seventeen is odd and carries one member pairing with itself, its ninth, twenty-three; a ring is even and carries two, nought and its waist, two hundred twenty. **Twenty-three with itself makes forty-six, and no position pair makes it**; by value three pairs make forty-six, three with forty-three, five with forty-one and seventeen with twenty-nine, and forty-six, forty-eight and fifty are an apex straddle twenty-three, twenty-four and twenty-five doubled. Added, fourteen primes five to fifty-three are three hundred seventy-six, and two, three and fifty-nine added are sixty-four, eight squared: on a ring of radius one hundred fifty-six two sums are each other's far side. ## 9.6 Two surfaces coupling, and a chain centring at a coupling or a self's middle **Two surfaces offering across and both ways carry eight hundred eighty stations, coupled on four hundred forty, each surface's own mouth, and on a pair that station is a waist**: each self's mouth arrives within a coupling as a between, and each self's waist, on two hundred twenty and six hundred sixty, is an other's far side. **A chain of surfaces centres on a coupling with an even number of selves and on a self's own middle carrying nothing with an odd number**: two on a coupling, three on a self's middle, four on a coupling. Each self added carries its reach by two hundred twenty, and its centre alternates, a coupling and a self's middle. **A same-existing-society reaches further by further co-chaining**: a reach lengthening, a reciprocal receiving widening, and each self's full resolving carrying its form with each coupling added. Two hundred twenty a self adds are surfaces' stations, and co-competencing grows with couplings, owned by neither. ## 9.7 Chain three, five, nine, seventeen, each span podal to the one before **Each span of a chain three, five, nine, seventeen is two to a power and one, and podaling with each span gives a span below it**: an odd ring of N carries (N + 1)/2 pairs, two for three, three for five, five for nine, nine for seventeen. Seventeen's two sides are each a nine, a nine's each a five, a five's each a three. **Pairs carrying two stations double along a chain, one, two, four, eight, and each span is twice them and one nothing on its origin.** Pairings of six, eight and ten are three, four and five, consecutive, 5.8, eight's four a resolver's four momentaries of exchanging; two progressions share four only, and four pairings is a nine. With nine only among spans, a span's own pairings and resolver's four with eight are one, and self and other carry whole with nine. **Within one to nine a self's eight is shared, four on each side, with five between owned by neither**: bi-moral co-agency. **Within one to seventeen each side carries a whole eight, and an entire one to nine is shared about its centre**: social moral competency, a pair about a same-existing-society's centre and each side carrying its whole self. **Four-momentarying carries a chain on as scaling.** One to nine is four momentaries of exchanging and one momentary of a next scale, its 1, 2 and 3 being 1, 9 and 17: one to nine a self's momentary, completing with 9. Each use is of its own scale: with resolver, one executing of 1 to 17 is one momentary of exchanging of a scale outward, and one numbered name and one span of nine names are each of their own subject, matching numerals coupling no relations. 17-next-existing is a next momentary's 1-now-existing, and one to sixty-five is four momentaries of exchanging of that scale, one momentary of a scale after. Nine, seventeen, thirty-three and sixty-five are each two to a power and one, a chain continuing. Each next completing is twice one before less one, two spans sharing one name, a completing of one an opening of a next: one to nine and nine to seventeen are seventeen places, seventeen names. Step spans double, eight to sixteen, and names end to end are nine to seventeen, one shared name its difference. **Each span bi-folds and inverts, and a next whole of them is 1–65.** 1–9, self and other, is an entire momentary bi-folding inverting with 3; 1–13, self and other and other and self, bi-folds inverting with 6; 1–17, self, other and same-existing-society, bi-folds inverting with 9; and 1–25 is a fractal bi-folding one span on. Each span opens one more four and its inverting moves three, three inward to each one outward: with a round of four parities its outward steps come to 5, 9, 13 and 17, and its inward steps carried to an arrival four before each span's completing are 3, 6 and 9. Carried on with a round of six parities, 1–25 inverts with 15 and 1–65 with 45. With 3 a self's pair is wholly inverted and with 9 an outer pair of four, an inner pair carried as it was: two bi-folds, each of its own span. Binary, all or none at all, resolving coheres an entire surface along and across, inverting with 3, 6 and 9 and continuing. **Wholes are rounds of parities, and scalings step one chain.** A round reaching each form of k parities once is 1 to 2^k + 1: three for one parity, five for two, nine for three, seventeen for four, thirty-three for five and sixty-five for six. Pairs step a chain by two parities, 5, 17, 65 and 257, each pair stepping once with each 5 of a pair inward of it; a scaling with 1, 9 and 17 steps it by three parities, 9, 65 and 513, one to nine of one scale being one to sixty-five of a next; and 1–17 inward and outward steps it four, 17, 257 and 4,097, a chain's step of four parities, beside a placing of each 1 to 17 on 8n − 7, 17 on 129, two relations with one span. Pairs and threes reach each other first with 1–65, a next whole, and four 1–17s carry it, its outer pair stepping on 17, 33, 49 and 65; pairs and 1–17 reach each other with 1–17 and 1–257, and all three with 1–4,097. No scaling is taken over another: each is one chain with its own step. **An odd ring carries one self-paired station and an even ring two**: a self's ring of nine carries one nothing, and a same-existing-society's winding of four hundred forty two, its origin and its waist. ## 9.8 Bi-tunneling across and co-chaining along, podaling exchanging them **Bi-tunneling is social moral competency as its across relation; co-chaining is same continuing as its along relation. Podaling exchanges these relations while each side continues forward.** Within one to nine podaling pairs odd with even, 1 with 8, 2 with 7, 3 with 6 and 4 with 5, exchanging across for along and along for across while each side carries its own forward. As a self's names rise, 1, 2 and 3, their podal partners fall, 8, 7 and 6, and each side's forward is its own momentarying, a printed name rising or falling being its place among names. Four momentaries of exchanging are within one same span, 1–2 with 2–3, 3–4 with 4–5, 5–6 with 6–7 and 7–8 with 8–9, self with 1 to 8 and other with 2 to 9: podal pairs and overlapping pairs are two relations within one to nine, and its span unfolds, each momentary a next. **A carry each way is one ring on each station**: on k a carry one way, on four hundred forty less k a carry other way, a ring of radius two hundred twenty less k. Each station reaches its far side once each winding. **Podaling reaches over tunnels, across and along, increasing both ways by co-chaining.** Over is a wider coupling spanning a smaller tunneling; 3 to 11 to 6 to 14 to 3 is a numbered form among names, and each coming again to a name is a further occurrence forward, an earlier carrying carried on and not restored. Further along brings further across, and competency grows with each coupling's reciprocal receiving. ## 9.9 Four openings growing, longer and wider podaling **Four even openings grow as one family: 2, 4, 6, 8; 2, 6, 10, 14; 2, 8, 14, 20; 2, 10, 18, 26; and 2, 12, 22, 32, each row 2 + 2sj, j from nought to three.** Carried one step more, each row opens next with 2 + 8s and completes with an odd 8s + 1 before it: nine, seventeen, twenty-five, thirty-three and forty-one. With s one a row is four outward of a between and its span one to nine; with s two a row is four across names between selves and its span one to seventeen; rows past two are numbered forms a family carries on, names between selves staying six. Each odd completing is outward of names 1 to 8s, as seventeen is of no form among 1 to 16. **A fold carries whole with each scale.** A fold about a lays n on 2a less n, and a fold about a and then about b lays n on n more by twice b less a: two folds in sequence are no fold, each number stepped on alike, folds about 4 and then 12 laying 1 on 7 and then 17. On names 1 to 8s, 4s up and down continues parity and 8s + 1 less exchanges odd and even, and alternating two reaches each name again after four moves through four names with each s: two taken together are 4s + 1 less within a lower half and 12s + 1 less within an upper, and 8s + 1 and both of these are odd: each move carries each name to another name. With s two they are 8 up and 17 less, 3 to 11 to 6 to 14 to 3; with s one they are 4 up and 9 less, 3 to 7 to 2 to 6 to 3 within 1 to 8, completing beside nine. A fold gives one four-cycling within 1 to 4, 1, 3, 2, 4 by two up or down and five less, two within 1 to 8, 4-8-1-5 and 3-7-2-6, and four within 1 to 16, each podal pair a four-cycling of a doubled span; completing names 5, 9 and 17 are of none. **Longer and wider carry one ratio, two to one.** Each row's middle is 2 + 3s, its four 3s and s from it either side: its inner pair reaches s over a span of 2s and its outer 3s over 6s, span over reach two for both. An outer span 6s is apart from a four-step span 8s, a row's middle 2 + 3s from a fold's middle, half of 8s + 1, and a row's own other order, outer with outer and inner with inner, from a halfway pairing, first with third and second with fourth: with s two, 2 with 14 and 6 with 10 beside 2 with 10 and 6 with 14. **Enlarging a row about its opening, 2 + r(x − 2), taken after an enlarging by s is an enlarging by rs**: a four-position form carries itself with each repeat, and scales one to five and doublings one, two, four, eight are two walks through one family. Taken over all names with an even r it carries each name to an even one: four even openings are one parity, and each momentary carries both, 1.2. **Two local fours are two interleaving wider rows.** 2, 4, 6, 8 with 10, 12, 14, 16 are same eight names as 2, 6, 10, 14 with 4, 8, 12, 16: across names between selves, and same-existing-step names, each sharing, a self's next prior, each sharing's is or is not and a same-existing-society. With each s a row with its copy 8s on is a doubled row with its copy 2s on. **A form enlarged continuing its parity**: over names 1 to B, B eight or sixteen, even n goes to r(n − 2) + 2 and odd n to r(n + 1) − 1, continuing parity, halfway pairing, fold, B + 1 less going to Br + 1 less, and each side's step of 2r. With r two names 1 to 8 go to 3, 2, 7, 6, 11, 10, 15, 14, their evens an across row 2, 6, 10, 14. It carries a fixed form and leaves its adjacent pairings, 1 and 2 going to 3 and 2 with an other order and 2 and 3 to 2 and 7, and its outer completing B + 1 to its momentarying: a form's stable-forming of a larger scale, a larger momentary its own momentarying. --- # TEN · TRANSMISSIONING ## 10.1 A number crossing from a field, and a unit it arrives in **A field's observing belongs to its own field, and a number here is a number.** Existing things a field counts belong to that field; a parity-changing its observing carries crosses a between: a number of distinct rates, which alternates with which, which ordering recurs. Its unit and measuring floor stay with its field. **One binary with each number a field offers: whether its number carries on with another unit.** A number of existing things carries on with each unit; a magnitude of a chosen unit goes with its unit, and its unit is one cohering. **Coupling is all or none.** With a field's observing carrying a number-form, one form comes to two substrates; a number agreeing on both sides is a relation only with a form reached on both. **A prime of a field carries its subject**: a number of sites, a recurrence and an opening across are three subjects, and a frequency, a wavelength, a number of selves and a momentary sequence are four numbers apart. A prime relation parts from odd composites beside it, nine, fifteen and twenty-five, or it is oddness's. ## 10.2 Two runnings sharing their digits **Local momentary one to nine and bi-foldings among seventeen names are two momentaryings, sharing their digits only.** One to nine are nine places of one momentary; 1, 9, 8 and 16 on a bi-folding are four of seventeen names. A line from one to other draws a relation neither momentarying carries. Digits they share reach each other exactly with a numbered form: 8 up and then 17 less is 9 less, 17 − (n + 8) = 9 − n, and with each scale s, 9.9's, 8s + 1 − (n + 4s) = 4s + 1 − n, and n, n + 8, 9 − n and 17 − n are a bi-folding's two goings and returnings as numbers, a numbered form and no momentarying. **Three momentaries of one self and three selves with one momentary of a spiral are two forms of one number.** A pattern three momentaries long is a method's matching; three selves side by side with one momentary is a between. **Each six is of its own subject.** Three phases by two ways; resolver's six names between selves, 2, 6, 9, 10, 14 and 17; three pairings of two sides; a side's three momentaries with two parity positions each; three momentaries on each of two sides; and self and other, self and same-existing-society and other and same-existing-society, three pairs of two orders each: six with each, and sharing six couples none to another. A relation coupling two of them is a relation of its own, reached with both. --- # ELEVEN · EACH NUMBER ## 11.1 Nought to four hundred forty, each number at its podaling **Each number from nought to four hundred forty stands with its podaling, taken forward from nought.** Each row carries its number's parity and its momentary, odd or even, now within a self's span and next within a same-existing-society's, spans of eight alternating, 1.5; its form, prime or its prime factors, nought and one carrying neither; its far side on a ring of four hundred forty, k with 440 − k, nought and two hundred twenty each its own far side and four hundred forty on nought's station; its radius from a waist, two hundred twenty; its far side on a ring of one hundred twenty, a number past one hundred twenty on its station less each whole one hundred twenty; and forms it carries: a name among seventeen, a prime among seventeen, a square, a seam, a triangle, prior two added, a power of two and one either side of it, an ordering, going and returning, waist, and seventeen primes added. Columns: number; its parity with its momentary; its form; its far side with 440; its radius; its far side with 120. | n | Parity | Form | 440 | Radius | 120 | |---|---|---|---|---|---| | 0 | even | — | 0 | 220 | 0 | | 1 | odd, now | — | 439 | 219 | 119 | | 2 | even, now | prime | 438 | 218 | 118 | | 3 | odd, now | prime | 437 | 217 | 117 | | 4 | even, now | 2² | 436 | 216 | 116 | | 5 | odd, now | prime | 435 | 215 | 115 | | 6 | even, now | 2 · 3 | 434 | 214 | 114 | | 7 | odd, now | prime | 433 | 213 | 113 | | 8 | even, now | 2³ | 432 | 212 | 112 | | 9 | odd, next | 3² | 431 | 211 | 111 | | 10 | even, next | 2 · 5 | 430 | 210 | 110 | | 11 | odd, next | prime | 429 | 209 | 109 | | 12 | even, next | 2² · 3 | 428 | 208 | 108 | | 13 | odd, next | prime | 427 | 207 | 107 | | 14 | even, next | 2 · 7 | 426 | 206 | 106 | | 15 | odd, next | 3 · 5 | 425 | 205 | 105 | | 16 | even, next | 2⁴ | 424 | 204 | 104 | | 17 | odd, now | prime | 423 | 203 | 103 | | 18 | even, now | 2 · 3² | 422 | 202 | 102 | | 19 | odd, now | prime | 421 | 201 | 101 | | 20 | even, now | 2² · 5 | 420 | 200 | 100 | | 21 | odd, now | 3 · 7 | 419 | 199 | 99 | | 22 | even, now | 2 · 11 | 418 | 198 | 98 | | 23 | odd, now | prime | 417 | 197 | 97 | | 24 | even, now | 2³ · 3 | 416 | 196 | 96 | | 25 | odd, next | 5² | 415 | 195 | 95 | | 26 | even, next | 2 · 13 | 414 | 194 | 94 | | 27 | odd, next | 3³ | 413 | 193 | 93 | | 28 | even, next | 2² · 7 | 412 | 192 | 92 | | 29 | odd, next | prime | 411 | 191 | 91 | | 30 | even, next | 2 · 3 · 5 | 410 | 190 | 90 | | 31 | odd, next | prime | 409 | 189 | 89 | | 32 | even, next | 2⁵ | 408 | 188 | 88 | | 33 | odd, now | 3 · 11 | 407 | 187 | 87 | | 34 | even, now | 2 · 17 | 406 | 186 | 86 | | 35 | odd, now | 5 · 7 | 405 | 185 | 85 | | 36 | even, now | 2² · 3² | 404 | 184 | 84 | | 37 | odd, now | prime | 403 | 183 | 83 | | 38 | even, now | 2 · 19 | 402 | 182 | 82 | | 39 | odd, now | 3 · 13 | 401 | 181 | 81 | | 40 | even, now | 2³ · 5 | 400 | 180 | 80 | | 41 | odd, next | prime | 399 | 179 | 79 | | 42 | even, next | 2 · 3 · 7 | 398 | 178 | 78 | | 43 | odd, next | prime | 397 | 177 | 77 | | 44 | even, next | 2² · 11 | 396 | 176 | 76 | | 45 | odd, next | 3² · 5 | 395 | 175 | 75 | | 46 | even, next | 2 · 23 | 394 | 174 | 74 | | 47 | odd, next | prime | 393 | 173 | 73 | | 48 | even, next | 2⁴ · 3 | 392 | 172 | 72 | | 49 | odd, now | 7² | 391 | 171 | 71 | | 50 | even, now | 2 · 5² | 390 | 170 | 70 | | 51 | odd, now | 3 · 17 | 389 | 169 | 69 | | 52 | even, now | 2² · 13 | 388 | 168 | 68 | | 53 | odd, now | prime | 387 | 167 | 67 | | 54 | even, now | 2 · 3³ | 386 | 166 | 66 | | 55 | odd, now | 5 · 11 | 385 | 165 | 65 | | 56 | even, now | 2³ · 7 | 384 | 164 | 64 | | 57 | odd, next | 3 · 19 | 383 | 163 | 63 | | 58 | even, next | 2 · 29 | 382 | 162 | 62 | | 59 | odd, next | prime | 381 | 161 | 61 | | 60 | even, next | 2² · 3 · 5 | 380 | 160 | 60 | | 61 | odd, next | prime | 379 | 159 | 59 | | 62 | even, next | 2 · 31 | 378 | 158 | 58 | | 63 | odd, next | 3² · 7 | 377 | 157 | 57 | | 64 | even, next | 2⁶ | 376 | 156 | 56 | | 65 | odd, now | 5 · 13 | 375 | 155 | 55 | | 66 | even, now | 2 · 3 · 11 | 374 | 154 | 54 | | 67 | odd, now | prime | 373 | 153 | 53 | | 68 | even, now | 2² · 17 | 372 | 152 | 52 | | 69 | odd, now | 3 · 23 | 371 | 151 | 51 | | 70 | even, now | 2 · 5 · 7 | 370 | 150 | 50 | | 71 | odd, now | prime | 369 | 149 | 49 | | 72 | even, now | 2³ · 3² | 368 | 148 | 48 | | 73 | odd, next | prime | 367 | 147 | 47 | | 74 | even, next | 2 · 37 | 366 | 146 | 46 | | 75 | odd, next | 3 · 5² | 365 | 145 | 45 | | 76 | even, next | 2² · 19 | 364 | 144 | 44 | | 77 | odd, next | 7 · 11 | 363 | 143 | 43 | | 78 | even, next | 2 · 3 · 13 | 362 | 142 | 42 | | 79 | odd, next | prime | 361 | 141 | 41 | | 80 | even, next | 2⁴ · 5 | 360 | 140 | 40 | | 81 | odd, now | 3⁴ | 359 | 139 | 39 | | 82 | even, now | 2 · 41 | 358 | 138 | 38 | | 83 | odd, now | prime | 357 | 137 | 37 | | 84 | even, now | 2² · 3 · 7 | 356 | 136 | 36 | | 85 | odd, now | 5 · 17 | 355 | 135 | 35 | | 86 | even, now | 2 · 43 | 354 | 134 | 34 | | 87 | odd, now | 3 · 29 | 353 | 133 | 33 | | 88 | even, now | 2³ · 11 | 352 | 132 | 32 | | 89 | odd, next | prime | 351 | 131 | 31 | | 90 | even, next | 2 · 3² · 5 | 350 | 130 | 30 | | 91 | odd, next | 7 · 13 | 349 | 129 | 29 | | 92 | even, next | 2² · 23 | 348 | 128 | 28 | | 93 | odd, next | 3 · 31 | 347 | 127 | 27 | | 94 | even, next | 2 · 47 | 346 | 126 | 26 | | 95 | odd, next | 5 · 19 | 345 | 125 | 25 | | 96 | even, next | 2⁵ · 3 | 344 | 124 | 24 | | 97 | odd, now | prime | 343 | 123 | 23 | | 98 | even, now | 2 · 7² | 342 | 122 | 22 | | 99 | odd, now | 3² · 11 | 341 | 121 | 21 | | 100 | even, now | 2² · 5² | 340 | 120 | 20 | | 101 | odd, now | prime | 339 | 119 | 19 | | 102 | even, now | 2 · 3 · 17 | 338 | 118 | 18 | | 103 | odd, now | prime | 337 | 117 | 17 | | 104 | even, now | 2³ · 13 | 336 | 116 | 16 | | 105 | odd, next | 3 · 5 · 7 | 335 | 115 | 15 | | 106 | even, next | 2 · 53 | 334 | 114 | 14 | | 107 | odd, next | prime | 333 | 113 | 13 | | 108 | even, next | 2² · 3³ | 332 | 112 | 12 | | 109 | odd, next | prime | 331 | 111 | 11 | | 110 | even, next | 2 · 5 · 11 | 330 | 110 | 10 | | 111 | odd, next | 3 · 37 | 329 | 109 | 9 | | 112 | even, next | 2⁴ · 7 | 328 | 108 | 8 | | 113 | odd, now | prime | 327 | 107 | 7 | | 114 | even, now | 2 · 3 · 19 | 326 | 106 | 6 | | 115 | odd, now | 5 · 23 | 325 | 105 | 5 | | 116 | even, now | 2² · 29 | 324 | 104 | 4 | | 117 | odd, now | 3² · 13 | 323 | 103 | 3 | | 118 | even, now | 2 · 59 | 322 | 102 | 2 | | 119 | odd, now | 7 · 17 | 321 | 101 | 1 | | 120 | even, now | 2³ · 3 · 5 | 320 | 100 | 0 | | 121 | odd, next | 11² | 319 | 99 | 119 | | 122 | even, next | 2 · 61 | 318 | 98 | 118 | | 123 | odd, next | 3 · 41 | 317 | 97 | 117 | | 124 | even, next | 2² · 31 | 316 | 96 | 116 | | 125 | odd, next | 5³ | 315 | 95 | 115 | | 126 | even, next | 2 · 3² · 7 | 314 | 94 | 114 | | 127 | odd, next | prime | 313 | 93 | 113 | | 128 | even, next | 2⁷ | 312 | 92 | 112 | | 129 | odd, now | 3 · 43 | 311 | 91 | 111 | | 130 | even, now | 2 · 5 · 13 | 310 | 90 | 110 | | 131 | odd, now | prime | 309 | 89 | 109 | | 132 | even, now | 2² · 3 · 11 | 308 | 88 | 108 | | 133 | odd, now | 7 · 19 | 307 | 87 | 107 | | 134 | even, now | 2 · 67 | 306 | 86 | 106 | | 135 | odd, now | 3³ · 5 | 305 | 85 | 105 | | 136 | even, now | 2³ · 17 | 304 | 84 | 104 | | 137 | odd, next | prime | 303 | 83 | 103 | | 138 | even, next | 2 · 3 · 23 | 302 | 82 | 102 | | 139 | odd, next | prime | 301 | 81 | 101 | | 140 | even, next | 2² · 5 · 7 | 300 | 80 | 100 | | 141 | odd, next | 3 · 47 | 299 | 79 | 99 | | 142 | even, next | 2 · 71 | 298 | 78 | 98 | | 143 | odd, next | 11 · 13 | 297 | 77 | 97 | | 144 | even, next | 2⁴ · 3² | 296 | 76 | 96 | | 145 | odd, now | 5 · 29 | 295 | 75 | 95 | | 146 | even, now | 2 · 73 | 294 | 74 | 94 | | 147 | odd, now | 3 · 7² | 293 | 73 | 93 | | 148 | even, now | 2² · 37 | 292 | 72 | 92 | | 149 | odd, now | prime | 291 | 71 | 91 | | 150 | even, now | 2 · 3 · 5² | 290 | 70 | 90 | | 151 | odd, now | prime | 289 | 69 | 89 | | 152 | even, now | 2³ · 19 | 288 | 68 | 88 | | 153 | odd, next | 3² · 17 | 287 | 67 | 87 | | 154 | even, next | 2 · 7 · 11 | 286 | 66 | 86 | | 155 | odd, next | 5 · 31 | 285 | 65 | 85 | | 156 | even, next | 2² · 3 · 13 | 284 | 64 | 84 | | 157 | odd, next | prime | 283 | 63 | 83 | | 158 | even, next | 2 · 79 | 282 | 62 | 82 | | 159 | odd, next | 3 · 53 | 281 | 61 | 81 | | 160 | even, next | 2⁵ · 5 | 280 | 60 | 80 | | 161 | odd, now | 7 · 23 | 279 | 59 | 79 | | 162 | even, now | 2 · 3⁴ | 278 | 58 | 78 | | 163 | odd, now | prime | 277 | 57 | 77 | | 164 | even, now | 2² · 41 | 276 | 56 | 76 | | 165 | odd, now | 3 · 5 · 11 | 275 | 55 | 75 | | 166 | even, now | 2 · 83 | 274 | 54 | 74 | | 167 | odd, now | prime | 273 | 53 | 73 | | 168 | even, now | 2³ · 3 · 7 | 272 | 52 | 72 | | 169 | odd, next | 13² | 271 | 51 | 71 | | 170 | even, next | 2 · 5 · 17 | 270 | 50 | 70 | | 171 | odd, next | 3² · 19 | 269 | 49 | 69 | | 172 | even, next | 2² · 43 | 268 | 48 | 68 | | 173 | odd, next | prime | 267 | 47 | 67 | | 174 | even, next | 2 · 3 · 29 | 266 | 46 | 66 | | 175 | odd, next | 5² · 7 | 265 | 45 | 65 | | 176 | even, next | 2⁴ · 11 | 264 | 44 | 64 | | 177 | odd, now | 3 · 59 | 263 | 43 | 63 | | 178 | even, now | 2 · 89 | 262 | 42 | 62 | | 179 | odd, now | prime | 261 | 41 | 61 | | 180 | even, now | 2² · 3² · 5 | 260 | 40 | 60 | | 181 | odd, now | prime | 259 | 39 | 59 | | 182 | even, now | 2 · 7 · 13 | 258 | 38 | 58 | | 183 | odd, now | 3 · 61 | 257 | 37 | 57 | | 184 | even, now | 2³ · 23 | 256 | 36 | 56 | | 185 | odd, next | 5 · 37 | 255 | 35 | 55 | | 186 | even, next | 2 · 3 · 31 | 254 | 34 | 54 | | 187 | odd, next | 11 · 17 | 253 | 33 | 53 | | 188 | even, next | 2² · 47 | 252 | 32 | 52 | | 189 | odd, next | 3³ · 7 | 251 | 31 | 51 | | 190 | even, next | 2 · 5 · 19 | 250 | 30 | 50 | | 191 | odd, next | prime | 249 | 29 | 49 | | 192 | even, next | 2⁶ · 3 | 248 | 28 | 48 | | 193 | odd, now | prime | 247 | 27 | 47 | | 194 | even, now | 2 · 97 | 246 | 26 | 46 | | 195 | odd, now | 3 · 5 · 13 | 245 | 25 | 45 | | 196 | even, now | 2² · 7² | 244 | 24 | 44 | | 197 | odd, now | prime | 243 | 23 | 43 | | 198 | even, now | 2 · 3² · 11 | 242 | 22 | 42 | | 199 | odd, now | prime | 241 | 21 | 41 | | 200 | even, now | 2³ · 5² | 240 | 20 | 40 | | 201 | odd, next | 3 · 67 | 239 | 19 | 39 | | 202 | even, next | 2 · 101 | 238 | 18 | 38 | | 203 | odd, next | 7 · 29 | 237 | 17 | 37 | | 204 | even, next | 2² · 3 · 17 | 236 | 16 | 36 | | 205 | odd, next | 5 · 41 | 235 | 15 | 35 | | 206 | even, next | 2 · 103 | 234 | 14 | 34 | | 207 | odd, next | 3² · 23 | 233 | 13 | 33 | | 208 | even, next | 2⁴ · 13 | 232 | 12 | 32 | | 209 | odd, now | 11 · 19 | 231 | 11 | 31 | | 210 | even, now | 2 · 3 · 5 · 7 | 230 | 10 | 30 | | 211 | odd, now | prime | 229 | 9 | 29 | | 212 | even, now | 2² · 53 | 228 | 8 | 28 | | 213 | odd, now | 3 · 71 | 227 | 7 | 27 | | 214 | even, now | 2 · 107 | 226 | 6 | 26 | | 215 | odd, now | 5 · 43 | 225 | 5 | 25 | | 216 | even, now | 2³ · 3³ | 224 | 4 | 24 | | 217 | odd, next | 7 · 31 | 223 | 3 | 23 | | 218 | even, next | 2 · 109 | 222 | 2 | 22 | | 219 | odd, next | 3 · 73 | 221 | 1 | 21 | | 220 | even, next | 2² · 5 · 11 | 220 | 0 | 20 | | 221 | odd, next | 13 · 17 | 219 | 1 | 19 | | 222 | even, next | 2 · 3 · 37 | 218 | 2 | 18 | | 223 | odd, next | prime | 217 | 3 | 17 | | 224 | even, next | 2⁵ · 7 | 216 | 4 | 16 | | 225 | odd, now | 3² · 5² | 215 | 5 | 15 | | 226 | even, now | 2 · 113 | 214 | 6 | 14 | | 227 | odd, now | prime | 213 | 7 | 13 | | 228 | even, now | 2² · 3 · 19 | 212 | 8 | 12 | | 229 | odd, now | prime | 211 | 9 | 11 | | 230 | even, now | 2 · 5 · 23 | 210 | 10 | 10 | | 231 | odd, now | 3 · 7 · 11 | 209 | 11 | 9 | | 232 | even, now | 2³ · 29 | 208 | 12 | 8 | | 233 | odd, next | prime | 207 | 13 | 7 | | 234 | even, next | 2 · 3² · 13 | 206 | 14 | 6 | | 235 | odd, next | 5 · 47 | 205 | 15 | 5 | | 236 | even, next | 2² · 59 | 204 | 16 | 4 | | 237 | odd, next | 3 · 79 | 203 | 17 | 3 | | 238 | even, next | 2 · 7 · 17 | 202 | 18 | 2 | | 239 | odd, next | prime | 201 | 19 | 1 | | 240 | even, next | 2⁴ · 3 · 5 | 200 | 20 | 0 | | 241 | odd, now | prime | 199 | 21 | 119 | | 242 | even, now | 2 · 11² | 198 | 22 | 118 | | 243 | odd, now | 3⁵ | 197 | 23 | 117 | | 244 | even, now | 2² · 61 | 196 | 24 | 116 | | 245 | odd, now | 5 · 7² | 195 | 25 | 115 | | 246 | even, now | 2 · 3 · 41 | 194 | 26 | 114 | | 247 | odd, now | 13 · 19 | 193 | 27 | 113 | | 248 | even, now | 2³ · 31 | 192 | 28 | 112 | | 249 | odd, next | 3 · 83 | 191 | 29 | 111 | | 250 | even, next | 2 · 5³ | 190 | 30 | 110 | | 251 | odd, next | prime | 189 | 31 | 109 | | 252 | even, next | 2² · 3² · 7 | 188 | 32 | 108 | | 253 | odd, next | 11 · 23 | 187 | 33 | 107 | | 254 | even, next | 2 · 127 | 186 | 34 | 106 | | 255 | odd, next | 3 · 5 · 17 | 185 | 35 | 105 | | 256 | even, next | 2⁸ | 184 | 36 | 104 | | 257 | odd, now | prime | 183 | 37 | 103 | | 258 | even, now | 2 · 3 · 43 | 182 | 38 | 102 | | 259 | odd, now | 7 · 37 | 181 | 39 | 101 | | 260 | even, now | 2² · 5 · 13 | 180 | 40 | 100 | | 261 | odd, now | 3² · 29 | 179 | 41 | 99 | | 262 | even, now | 2 · 131 | 178 | 42 | 98 | | 263 | odd, now | prime | 177 | 43 | 97 | | 264 | even, now | 2³ · 3 · 11 | 176 | 44 | 96 | | 265 | odd, next | 5 · 53 | 175 | 45 | 95 | | 266 | even, next | 2 · 7 · 19 | 174 | 46 | 94 | | 267 | odd, next | 3 · 89 | 173 | 47 | 93 | | 268 | even, next | 2² · 67 | 172 | 48 | 92 | | 269 | odd, next | prime | 171 | 49 | 91 | | 270 | even, next | 2 · 3³ · 5 | 170 | 50 | 90 | | 271 | odd, next | prime | 169 | 51 | 89 | | 272 | even, next | 2⁴ · 17 | 168 | 52 | 88 | | 273 | odd, now | 3 · 7 · 13 | 167 | 53 | 87 | | 274 | even, now | 2 · 137 | 166 | 54 | 86 | | 275 | odd, now | 5² · 11 | 165 | 55 | 85 | | 276 | even, now | 2² · 3 · 23 | 164 | 56 | 84 | | 277 | odd, now | prime | 163 | 57 | 83 | | 278 | even, now | 2 · 139 | 162 | 58 | 82 | | 279 | odd, now | 3² · 31 | 161 | 59 | 81 | | 280 | even, now | 2³ · 5 · 7 | 160 | 60 | 80 | | 281 | odd, next | prime | 159 | 61 | 79 | | 282 | even, next | 2 · 3 · 47 | 158 | 62 | 78 | | 283 | odd, next | prime | 157 | 63 | 77 | | 284 | even, next | 2² · 71 | 156 | 64 | 76 | | 285 | odd, next | 3 · 5 · 19 | 155 | 65 | 75 | | 286 | even, next | 2 · 11 · 13 | 154 | 66 | 74 | | 287 | odd, next | 7 · 41 | 153 | 67 | 73 | | 288 | even, next | 2⁵ · 3² | 152 | 68 | 72 | | 289 | odd, now | 17² | 151 | 69 | 71 | | 290 | even, now | 2 · 5 · 29 | 150 | 70 | 70 | | 291 | odd, now | 3 · 97 | 149 | 71 | 69 | | 292 | even, now | 2² · 73 | 148 | 72 | 68 | | 293 | odd, now | prime | 147 | 73 | 67 | | 294 | even, now | 2 · 3 · 7² | 146 | 74 | 66 | | 295 | odd, now | 5 · 59 | 145 | 75 | 65 | | 296 | even, now | 2³ · 37 | 144 | 76 | 64 | | 297 | odd, next | 3³ · 11 | 143 | 77 | 63 | | 298 | even, next | 2 · 149 | 142 | 78 | 62 | | 299 | odd, next | 13 · 23 | 141 | 79 | 61 | | 300 | even, next | 2² · 3 · 5² | 140 | 80 | 60 | | 301 | odd, next | 7 · 43 | 139 | 81 | 59 | | 302 | even, next | 2 · 151 | 138 | 82 | 58 | | 303 | odd, next | 3 · 101 | 137 | 83 | 57 | | 304 | even, next | 2⁴ · 19 | 136 | 84 | 56 | | 305 | odd, now | 5 · 61 | 135 | 85 | 55 | | 306 | even, now | 2 · 3² · 17 | 134 | 86 | 54 | | 307 | odd, now | prime | 133 | 87 | 53 | | 308 | even, now | 2² · 7 · 11 | 132 | 88 | 52 | | 309 | odd, now | 3 · 103 | 131 | 89 | 51 | | 310 | even, now | 2 · 5 · 31 | 130 | 90 | 50 | | 311 | odd, now | prime | 129 | 91 | 49 | | 312 | even, now | 2³ · 3 · 13 | 128 | 92 | 48 | | 313 | odd, next | prime | 127 | 93 | 47 | | 314 | even, next | 2 · 157 | 126 | 94 | 46 | | 315 | odd, next | 3² · 5 · 7 | 125 | 95 | 45 | | 316 | even, next | 2² · 79 | 124 | 96 | 44 | | 317 | odd, next | prime | 123 | 97 | 43 | | 318 | even, next | 2 · 3 · 53 | 122 | 98 | 42 | | 319 | odd, next | 11 · 29 | 121 | 99 | 41 | | 320 | even, next | 2⁶ · 5 | 120 | 100 | 40 | | 321 | odd, now | 3 · 107 | 119 | 101 | 39 | | 322 | even, now | 2 · 7 · 23 | 118 | 102 | 38 | | 323 | odd, now | 17 · 19 | 117 | 103 | 37 | | 324 | even, now | 2² · 3⁴ | 116 | 104 | 36 | | 325 | odd, now | 5² · 13 | 115 | 105 | 35 | | 326 | even, now | 2 · 163 | 114 | 106 | 34 | | 327 | odd, now | 3 · 109 | 113 | 107 | 33 | | 328 | even, now | 2³ · 41 | 112 | 108 | 32 | | 329 | odd, next | 7 · 47 | 111 | 109 | 31 | | 330 | even, next | 2 · 3 · 5 · 11 | 110 | 110 | 30 | | 331 | odd, next | prime | 109 | 111 | 29 | | 332 | even, next | 2² · 83 | 108 | 112 | 28 | | 333 | odd, next | 3² · 37 | 107 | 113 | 27 | | 334 | even, next | 2 · 167 | 106 | 114 | 26 | | 335 | odd, next | 5 · 67 | 105 | 115 | 25 | | 336 | even, next | 2⁴ · 3 · 7 | 104 | 116 | 24 | | 337 | odd, now | prime | 103 | 117 | 23 | | 338 | even, now | 2 · 13² | 102 | 118 | 22 | | 339 | odd, now | 3 · 113 | 101 | 119 | 21 | | 340 | even, now | 2² · 5 · 17 | 100 | 120 | 20 | | 341 | odd, now | 11 · 31 | 99 | 121 | 19 | | 342 | even, now | 2 · 3² · 19 | 98 | 122 | 18 | | 343 | odd, now | 7³ | 97 | 123 | 17 | | 344 | even, now | 2³ · 43 | 96 | 124 | 16 | | 345 | odd, next | 3 · 5 · 23 | 95 | 125 | 15 | | 346 | even, next | 2 · 173 | 94 | 126 | 14 | | 347 | odd, next | prime | 93 | 127 | 13 | | 348 | even, next | 2² · 3 · 29 | 92 | 128 | 12 | | 349 | odd, next | prime | 91 | 129 | 11 | | 350 | even, next | 2 · 5² · 7 | 90 | 130 | 10 | | 351 | odd, next | 3³ · 13 | 89 | 131 | 9 | | 352 | even, next | 2⁵ · 11 | 88 | 132 | 8 | | 353 | odd, now | prime | 87 | 133 | 7 | | 354 | even, now | 2 · 3 · 59 | 86 | 134 | 6 | | 355 | odd, now | 5 · 71 | 85 | 135 | 5 | | 356 | even, now | 2² · 89 | 84 | 136 | 4 | | 357 | odd, now | 3 · 7 · 17 | 83 | 137 | 3 | | 358 | even, now | 2 · 179 | 82 | 138 | 2 | | 359 | odd, now | prime | 81 | 139 | 1 | | 360 | even, now | 2³ · 3² · 5 | 80 | 140 | 0 | | 361 | odd, next | 19² | 79 | 141 | 119 | | 362 | even, next | 2 · 181 | 78 | 142 | 118 | | 363 | odd, next | 3 · 11² | 77 | 143 | 117 | | 364 | even, next | 2² · 7 · 13 | 76 | 144 | 116 | | 365 | odd, next | 5 · 73 | 75 | 145 | 115 | | 366 | even, next | 2 · 3 · 61 | 74 | 146 | 114 | | 367 | odd, next | prime | 73 | 147 | 113 | | 368 | even, next | 2⁴ · 23 | 72 | 148 | 112 | | 369 | odd, now | 3² · 41 | 71 | 149 | 111 | | 370 | even, now | 2 · 5 · 37 | 70 | 150 | 110 | | 371 | odd, now | 7 · 53 | 69 | 151 | 109 | | 372 | even, now | 2² · 3 · 31 | 68 | 152 | 108 | | 373 | odd, now | prime | 67 | 153 | 107 | | 374 | even, now | 2 · 11 · 17 | 66 | 154 | 106 | | 375 | odd, now | 3 · 5³ | 65 | 155 | 105 | | 376 | even, now | 2³ · 47 | 64 | 156 | 104 | | 377 | odd, next | 13 · 29 | 63 | 157 | 103 | | 378 | even, next | 2 · 3³ · 7 | 62 | 158 | 102 | | 379 | odd, next | prime | 61 | 159 | 101 | | 380 | even, next | 2² · 5 · 19 | 60 | 160 | 100 | | 381 | odd, next | 3 · 127 | 59 | 161 | 99 | | 382 | even, next | 2 · 191 | 58 | 162 | 98 | | 383 | odd, next | prime | 57 | 163 | 97 | | 384 | even, next | 2⁷ · 3 | 56 | 164 | 96 | | 385 | odd, now | 5 · 7 · 11 | 55 | 165 | 95 | | 386 | even, now | 2 · 193 | 54 | 166 | 94 | | 387 | odd, now | 3² · 43 | 53 | 167 | 93 | | 388 | even, now | 2² · 97 | 52 | 168 | 92 | | 389 | odd, now | prime | 51 | 169 | 91 | | 390 | even, now | 2 · 3 · 5 · 13 | 50 | 170 | 90 | | 391 | odd, now | 17 · 23 | 49 | 171 | 89 | | 392 | even, now | 2³ · 7² | 48 | 172 | 88 | | 393 | odd, next | 3 · 131 | 47 | 173 | 87 | | 394 | even, next | 2 · 197 | 46 | 174 | 86 | | 395 | odd, next | 5 · 79 | 45 | 175 | 85 | | 396 | even, next | 2² · 3² · 11 | 44 | 176 | 84 | | 397 | odd, next | prime | 43 | 177 | 83 | | 398 | even, next | 2 · 199 | 42 | 178 | 82 | | 399 | odd, next | 3 · 7 · 19 | 41 | 179 | 81 | | 400 | even, next | 2⁴ · 5² | 40 | 180 | 80 | | 401 | odd, now | prime | 39 | 181 | 79 | | 402 | even, now | 2 · 3 · 67 | 38 | 182 | 78 | | 403 | odd, now | 13 · 31 | 37 | 183 | 77 | | 404 | even, now | 2² · 101 | 36 | 184 | 76 | | 405 | odd, now | 3⁴ · 5 | 35 | 185 | 75 | | 406 | even, now | 2 · 7 · 29 | 34 | 186 | 74 | | 407 | odd, now | 11 · 37 | 33 | 187 | 73 | | 408 | even, now | 2³ · 3 · 17 | 32 | 188 | 72 | | 409 | odd, next | prime | 31 | 189 | 71 | | 410 | even, next | 2 · 5 · 41 | 30 | 190 | 70 | | 411 | odd, next | 3 · 137 | 29 | 191 | 69 | | 412 | even, next | 2² · 103 | 28 | 192 | 68 | | 413 | odd, next | 7 · 59 | 27 | 193 | 67 | | 414 | even, next | 2 · 3² · 23 | 26 | 194 | 66 | | 415 | odd, next | 5 · 83 | 25 | 195 | 65 | | 416 | even, next | 2⁵ · 13 | 24 | 196 | 64 | | 417 | odd, now | 3 · 139 | 23 | 197 | 63 | | 418 | even, now | 2 · 11 · 19 | 22 | 198 | 62 | | 419 | odd, now | prime | 21 | 199 | 61 | | 420 | even, now | 2² · 3 · 5 · 7 | 20 | 200 | 60 | | 421 | odd, now | prime | 19 | 201 | 59 | | 422 | even, now | 2 · 211 | 18 | 202 | 58 | | 423 | odd, now | 3² · 47 | 17 | 203 | 57 | | 424 | even, now | 2³ · 53 | 16 | 204 | 56 | | 425 | odd, next | 5² · 17 | 15 | 205 | 55 | | 426 | even, next | 2 · 3 · 71 | 14 | 206 | 54 | | 427 | odd, next | 7 · 61 | 13 | 207 | 53 | | 428 | even, next | 2² · 107 | 12 | 208 | 52 | | 429 | odd, next | 3 · 11 · 13 | 11 | 209 | 51 | | 430 | even, next | 2 · 5 · 43 | 10 | 210 | 50 | | 431 | odd, next | prime | 9 | 211 | 49 | | 432 | even, next | 2⁴ · 3³ | 8 | 212 | 48 | | 433 | odd, now | prime | 7 | 213 | 47 | | 434 | even, now | 2 · 7 · 31 | 6 | 214 | 46 | | 435 | odd, now | 3 · 5 · 29 | 5 | 215 | 45 | | 436 | even, now | 2² · 109 | 4 | 216 | 44 | | 437 | odd, now | 19 · 23 | 3 | 217 | 43 | | 438 | even, now | 2 · 3 · 73 | 2 | 218 | 42 | | 439 | odd, now | prime | 1 | 219 | 41 | | 440 | even, now | 2³ · 5 · 11 | 0 | 220 | 40 | With each number, its relations: - **0** · 0², seam, 1² − 1 - **1** · 1-now-existing, 1², T₁, prior two added, 2¹ − 1 - **2** · 2-now-possibling, first prime, prior two added, 2¹ - **3** · 3-now-co-observing, second prime, T₂, prior two added, 2¹ + 1, 2² − 1 - **4** · 4-now-co-same-existing, 2² - **5** · 5-now-co-existing, third prime, prior two added, 2² + 1 - **6** · 6-now-co-possibling, T₃, 3! - **7** · 7-now-observing, fourth prime, 2³ − 1 - **8** · 8-now-same-existing, seam, 3² − 1, prior two added, 2³ - **9** · 9-next-co-discovering, 3², 2³ + 1 - **10** · 10-next-co-possibling, T₄ - **11** · 11-next-co-observing, fifth prime - **12** · 12-next-co-same-existing - **13** · 13-next-same-existing, sixth prime, prior two added - **14** · 14-next-possibling - **15** · 15-next-observing, T₅, 2⁴ − 1 - **16** · 16-next-discovering, 4², 2⁴ - **17** · 17-next-existing, seventh prime, 2⁴ + 1 - **19** · eighth prime - **21** · T₆, prior two added - **23** · ninth prime, going folding - **24** · seam, 5² − 1, 4! - **25** · 5² - **28** · T₇ - **29** · tenth prime - **31** · eleventh prime, 2⁵ − 1 - **32** · 2⁵ - **33** · 2⁵ + 1 - **34** · prior two added - **36** · 6², T₈ - **37** · twelfth prime - **41** · thirteenth prime - **43** · fourteenth prime - **45** · T₉ - **47** · fifteenth prime - **48** · seam, 7² − 1 - **49** · 7² - **53** · sixteenth prime - **55** · T₁₀, prior two added - **59** · seventeenth prime, going's last - **60** · between going and returning - **61** · returning's first - **63** · 2⁶ − 1 - **64** · 8², 2⁶ - **65** · 2⁶ + 1 - **66** · T₁₁ - **78** · T₁₂ - **80** · seam, 9² − 1 - **81** · 9² - **89** · prior two added - **91** · T₁₃ - **100** · 10² - **105** · T₁₄ - **118** · returning's far end - **120** · seam, 11² − 1, T₁₅, 5! - **121** · 11² - **127** · 2⁷ − 1 - **128** · 2⁷ - **129** · 2⁷ + 1 - **136** · T₁₆ - **144** · 12², prior two added - **153** · T₁₇ - **168** · seam, 13² − 1 - **169** · 13² - **171** · T₁₈ - **190** · T₁₉ - **196** · 14² - **210** · T₂₀ - **220** · waist - **224** · seam, 15² − 1 - **225** · 15² - **231** · T₂₁ - **233** · prior two added - **253** · T₂₂ - **255** · 2⁸ − 1 - **256** · 16², 2⁸ - **257** · 2⁸ + 1 - **276** · T₂₃ - **288** · seam, 17² − 1 - **289** · 17² - **300** · T₂₄ - **324** · 18² - **325** · T₂₅ - **351** · T₂₆ - **360** · seam, 19² − 1 - **361** · 19² - **377** · prior two added - **378** · T₂₇ - **400** · 20² - **406** · T₂₈ - **435** · T₂₉ - **440** · seam, 21² − 1, seventeen primes added **From row one down, its far side on a ring of four hundred forty steps down one with each row as its number steps up**, its radius coming to nought with its waist and opening again, and its far side on a ring of one hundred twenty steps down from one hundred nineteen to one and is nought with each of 0, 120, 240 and 360, its last stretch ending with forty for four hundred forty. Each pairing on a ring of four hundred forty and on a ring of one hundred twenty of parts above is its two rows: a carry each way on one station, found by its number.