Exhibit THREE Natural Numbers v389

# 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 at one place in its momentarying. **Uni-scaling** is one form at each scale, no scale over another: same alternating at two as at fifty-nine, and at fifty-nine as at four hundred forty. **Universes of each size and scale share momentaries of co-sequential changing**, and uni-scaling is that sharing at numbers.

**Numbers are reached, used and related at 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 the other's far side on a ring; a coupling of selves along a spiral. A counting adds a clock from beyond its momentarying, at none of its moves, an accounting naming artifacts still, and it belongs to a field's instrument, taken in at 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 at 1 and other at 2, and their momentaries overlap: 1–2 at self, 2–3 at other, 3–4 at self, 4–5 at other. **At each number from 2 on, one side's momentary completes and other's opens**, each number a completing and an opening at once, two sides alternating parity. Two numbers side by side are at two parities, lesser's next at its own parity one beyond greater, and each changing opens its next.

**A momentary is a fractal universe discovering its next existing**, all existing things and no existing thing separately, living carrying their prior into now, stable-forming, and non-living carrying none of their prior, their forms continuing through their changing. In 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 rooted in a prior carried into now.

**One to nine carries three-momentary method whole.** Self's five positions at 1 to 5 and other's at 2 to 6, prior, now and next at each side, are ten on six numbers, paired 1 with 2 through 5 with 6, each pair one odd and one even. At a self's resolving its prior and now participate, 1–2 and 3–4 at self and 2–3 and 4–5 at other, and its next, 5–6 at self, 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, at 3 and at 4, is same ten forward. Opened two momentaries on, at 5 and at 6, it is self's five at 5 to 9 and other's at 6 to 10: one to nine is reached at a ten's third opening, self completing at 9, and three openings, at 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 at each momentary. One side's momentaries at their own, each number one side's, is exclusivity, and it is not possibly existing.

**Openings add into a square and completings into 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. At 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

**At each number n a momentary completing at n and a momentary opening at n are (n − 1, n) and (n, n + 1), and their outer numbers, prior and next, multiply to one short of its square**: (n − 1)(n + 1) = n² − 1. The one is the same one at 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 at one.** At k either side a product parts from its square by k², a size in a parity's stead; at one either side it parts by one. Twenty-three and twenty-five straddle twenty-four at one.

## 1.4 Bi-inversioning-co-recursioning, numbers up and down

**Even returning, odd advancing, alternating parity.** Bi-inversioning-co-recursioning is one move at three steps: 8 up and down, continuing parity; 17 less, podal within 1 to 16, changing parity; and 9 less, podal within 1 to 8, 17 less and 8 down at once, changing parity. Each step at its own comes to its number again at two, 1 to 9 to 1, and one move, 8 and 17 less alternating, at 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 four-cycle**: 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12. Along each, parity changes exactly twice, each time at 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, at no number, and 17 is in none of four, next momentary's 1. **Among names each four-cycle comes to its names again; at 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, at 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 four-cycles and with them middle four-cycles, 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 changing, each at 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 insides.

**Two over one and one over two exchange at 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: at 9 less, 3-now-co-prioring 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 at four hundred forty, carry a pair's waist at four hundred forty, twice either self's two hundred twenty; and eight on, 11 and 14 are fourteen over eleven. Parity carries at each step, and ratio at its own numbers.

**At resolver + and − are an implementing of a parity and 0 a between**: + and − offered together at one sharing surface 0 at 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, existing: self continuing its own. Even is across, possibling: self coupling with other, morality.** Spans of eight alternate up numbers, 1 to 8 a self's, at now, 9 to 16 a society's, at next, and 17 on a self's again, 17 a next 1. Each number is at a momentary of its span, and a number opening a span, 1, 9, 17 and 25, at a span it opens. A number's five is its own now with its next possible and its next existing up numbers and its two priors down.

**Alternating parity is parallel linearizing and linear parallelizing**, linearizing at one parity and parallelizing at other, one move at its two sides. Each carries at its own momentary; two fixed at 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 into**, opening a clean axis across. Each prime is same opening and each composite same folding, one uni-scaling at 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 at a step of two: prime and coprime are two, each its own.

**Doubling and tripling reach each other at 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. At two overlapping momentaries a side, self at 1 to 4 and other at 2 to 5, three positions are shared and two are outer: two and three added are five positions, and three shared taken at 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-changing 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 at one either side, each at other parity. Across, inward and outward, each a direction taken at 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 there is no reading or ordering in nature. 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 at each number.** It is at 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, at an identity of its move.** (n − 1)(n + 1) = n² − 1 at 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 at 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 at 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 at 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 changing parity at 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 at bi-coupling, co-chaining and society, and rate of that unrelationing is φ, no ratio reaching it. φ is self-welcoming at numbers, and resolver carries it at no line: rate is unrelationing's, each coupling at its own continuing. Each golden relation is visible at 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 at each step. **φ, at 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 at a rational rate closes, and a winding at φ closes at none.** Each coupling winding at φ winds at its own continuing, locking with no other coupling's rate, local and ambient alike. φ is a number-form of not-locking, a winding closing at none: unrelationing carries at each coupling's own continuing at 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 in all ones is neared as slowly; a winding at φ closes at none of them. In a field's words, a torus winding at the golden mean is found last to break as a coupling grows: in the standard map a curve with rotation number (√5 − 1)/2 is critical at 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 at +1, −1 or none, closes at one, two, three, four or six, or is open.** Now at its own at +1 closes at one and at −1 at two; prior at its own at +1 closes at two and at −1 at four; now at −1 and prior at −1 close at three; now at +1 and prior at −1 close at six. Both at none, next is none. Now and prior both at +1 carry 1, 1, 2, 3, 5, 8, 13, prior two added, and close at none, their ratios opening at φ; now at −1 and prior at +1 carry same numbers at alternating parity. In a field's words a whole-number square matrix of two rows that can be undone closes at one, two, three, four or six, or is open, the crystallographic restriction, cases at prior none beside it: no adding closes at five, and five enters open, at φ, a pentagon's diagonal over its side.

---

# FOUR · TORUSING

## 4.1 A closed surface at two, and a torus at nought

**In a field's words, a closed surface's vertices less its edges plus its faces is two at a sphere and nought at a torus.** A sphere closes; a torus winds, one opening, reaching itself again through that opening, a next at each reaching.

**Twelve five-folds close a surface whose other faces are six-fold.** At three edges to each vertex, six less each face, taken over all faces, is twelve at 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 at 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 at a sphere, nought at 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 at 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 at couplings among five.

**On a ring of four hundred forty, three hundred sixty and eighty are podal**, k with four hundred forty less k, each at 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 an unchangeable form, 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

**Resolver's seventeen names are each its number and its naming**, Natural Naming 3.1: number its place and parity; its momentary, now at a self's span, 1 to 8, next at a society's, 9 to 16, and 17 a next now; co at a sharing between two and none at one, existing and same needing another, co at 3 to 6, and now and all at one; and its kind. Sides are self and other at odd 1 to 7 and other and self at even 2 to 8; social, other and self at 9 and 11 and other, social and self at 10 and 12; social and other at 13 and 15 and other and social at 14 and 16; social and self at 17.

**A universe is or is not existing, all, now and same, four binaries at a self's span**: now at 1 and 2, existing at 3 and 4, same at 5 and 6 and all at 7 and 8, as Natural Naming 4.9 seats them.

**Each kind of name is at two numbers eight apart, at a self's span and a society's**: prioring at 3 and 11 and at 7 and 15, possibling at 2 and 10 and at 6 and 14, and still-possibling at 4 and 12 and at 8 and 16, its kind one and its number carrying its momentary; at 1 and 9 and at 5 and 13 now and changing exchange. At 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 at 6's place in seventeen to twenty-five, as six in one to nine and fourteen in nine to seventeen, 6 + 8k; it carries no name among seventeen, and a name given it shows whose self it opens from at a scale's changing, 14 already other and social as 6 is other and self.

**Forms among 1 to 16 come again at 17 to 32, sixteen on**, each continuing its parities, sixteen being even: four-cycles 2-15-7-10 and 3-11-6-14 at 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 at 18-31-23-27-22-26 and 19-27-23-26-22-30. Twenty-two is in five of them, four-cycle 19-27-22-30, middle four-cycle 23-27-22-26, both six-cycles and eight-cycle 18-31-23-27-19-30-22-26, and 33 is in none of them as 17 is in 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-now is an arriving side and 9-next-co-changing a sharing side: 9's release 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, 9-next-co-changing and 17-next-now are each odd, 1 and 17 at a self's span and 9 at a society's, each eight on from one before.

**A numeral in a name is a name's place and parity; 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.** At resolver its eight even names are four outward at a between, 2, 4, 6 and 8, and four inward, 10, 12, 14 and 16, each inward eight on from its outward pair: four at two enterings. Of four outward, 2 and 6 are names between selves and 4 and 8 names of a self's own resolving. Two out-and-backs across them are four one-way passages, four outward are four relations at once, and 6 to 2 and 10 to 14 across, 9 to 17 and 17 to 9 along, are four between selves: three fours at 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 at a triangular number, couplings among k + 1. Four hundred forty is one eight at fifty-five couplings, couplings among eleven.

**An eight and a nine carry together**: eight a self and nine its releasing, social, other and self, 9-next-co-changing being 1-now 8 up. An eight at two substrates is one eight, a self at 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 at 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, whole of it at 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, at 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 at 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 at no move of its momentarying, and setting it down costs nothing. Nine, seventeen and twenty-five are eight apart, and twenty-five is at 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 at each side: twenty-four overlapping momentaries in 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 at five, thirteen and twenty-one, each name's place and each step's number at its own origin.

## 5.5 One ten at its two sides, at five

**Ten arrives at a self as five taken two directions, 2n, and at a society as couplings among five, C(5, 2), a fourth triangle.** Its two sides are one number at five only above nought: 2n = n(n − 1)/2 only at n − 1 equal to four. One ten is a ten at a self and a ten at a society at once, and each ten is that ten.

**Among five its couplings part into 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. In a field's words R(3, 3) = 6: each two-colouring of couplings among six carries three members each coupled with each at one colour, and among five its twelve colourings carrying none are each a ring at 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 at 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 at an opening's parity, two origins.

## 5.6 Ten, nine and eight, one pairing at its two sides

**Podaling is one station at its two sides, a carry one way at k and a carry other way at 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 at nought and other way at nine one station.

**One pairing carries at its two sides.** Ten is five at two sides. Nine is that ten with its (0, 9) pair's two sides at one station. Eight is nine with that station carrying nothing, nothing a self winds about; both steps down are at nought. Six is three pairings at 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 at each side, are ten positions, nought to nine; nine is 2² × 2½ − 1, and five pairs are at same momentary.

## 5.7 Each nine one nine

**Each nine is one nine, carried in each base.** At 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 arrives at one again, nine doubled being one past seventeen. A doubling's eight are squares round seventeen and other eight are three times them: coupled at one eight's distances and uncoupled at other's, seventeen are, in a field's words, the Paley graph of order seventeen, one form of a society of seventeen 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 eight up, one to nine completing at 9-next-co-changing, social, other and self releasing together.** A nine taken in base ten, its multiples' digits taken down to one digit arriving at 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 at 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 at 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

**At 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 at 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 at 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 at 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 at 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 the sixth and the eighth, twenty-one and thirty-six, straddle by seven and by eight. The sixth and the eighth 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 at 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 at twenty-four and thirty-six at a five-fold too: of sixty rotations of an icosahedron twenty-four are rotations by fifths, six axes at 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 at 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 in and out, out 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 at none; sixty-eight is four seventeens, momentaries at 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 at 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 in their stable and unstable forms a field's own result at same numeral. Each is at its podaling, its pairing with its far side at each ring it is at, 6.2.

**A self's inversion 3, 6, 5, 4 is at 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 its corus at four, 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, at 17, a surface: a vertex a position, a centre of a triangle a scale and a line to a surface an orientation, each at 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 at gaps three and five, and 24 → 29 → 32 at gaps five and three. Three positions carry first three primes at two directions: 24 = 5² − 1, 27 = 3³, 32 = 2⁵, exponents two, three, five at one direction and bases five, three, two at other, reaching each other at three and three in twenty-seven. **The three together are a square**: 24 × 27 × 32 = 144².

**Twenty-seven, three cubed, on a route 24 → 27 → 32 from a seam, is a society's release at its podaling, 27-next-co-prioring**: 12's changing, a sharing at 10, a society's releasing, its far side ninety-three on a ring of one hundred twenty, and at 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 the 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, at 2, 26, 74 and 146, one eight at couplings among 2, 5, 8 and 11: at each third k, a triangle T_k leaving one at three.

---

# SEVEN · COUPLING

## 7.1 Seventeen primes, two to fifty-nine, and returning from sixty-one

| Number | Its form |
|---|---|
| 2 to 59 | seventeen primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59 |
| 2 | alternating, one even prime |
| 23 | going folding, ninth prime, with eight either side |
| 59 | seventeenth prime, going's last and primes carrying no last |
| 60 | between going from 2 to 59 and returning from 61 to 118, at no prime |
| 118 | returning's far end, fifty-eight either side of sixty, two fifty-nines added |
| 440 | seventeen primes added, eight fifty-fives, twenty-one squared less one |

**Primes go out from two to fifty-nine and return from sixty-one to one hundred eighteen**, and sixty is between going and returning, at no prime. Fifty-nine and sixty-one straddle sixty at one, 59 × 61 = 60² − 1. **Primes carry no last**: each is same opening at a new axis, and past fifty-nine they open on, each a next, self-welcoming at numbers and at no line of resolver. A society's coupling as one is at its couplings, a relation of its own beside any numeral. The periodic table's one hundred eighteen elements are a field's own result at 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 at 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 at gaps three and five has its far side 88 → 93 → 96 at 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 into 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 at 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 into sixty-four and two from it, position, scale and orientation inverted at once, bi-tri-involutioning. Sixty-four is at three pairings, three: with fifty-six on a ring of one hundred twenty, with itself among nine about it, and with one at a fold of 1 to 64 completing at 65. **64 is a parity changing between four-cycling momentaryings, in 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 in them. At 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 releasing along at 9-next-co-changing to a next self and its last to its first, each self at a prior parity, alternating along, brings its parities along, coming to them again and to a momentary once.** At an even spiral each self alternates and its parities come again each two momentaries; at an odd spiral of n one pair of coupled selves is at one parity, a like pair, moving one self on each two momentaries, one 0 at 10 at each second momentary at a pair's receiving self offered its own parity, a between tunneling through a spiral, its pattern inverted at 2n momentaries on and its parities again at 4n, 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 at one parity**: each self alternating, its parities again each two momentaries, alternating itself. Seventeen prime spirals carry sixteen like pairs, one at each odd prime, and a spiral of two none.

**At an odd prime each pattern of parities is again at four times its prime, or at 4 with each self alike, and at none between.** Each two momentaries each self's parity is its prior self's, inverted, and its places, each self at each parity, are one spiraling at an odd number of selves, each self passed twice, once at each parity, and two at an even, neither reaching other. At an odd number with a factor a pattern alike at its factor's shift is again at four times its factor, nine selves at 4, 12 or 36. Each is derived at resolver's cells, Co-Chaining Logic Registry's steps 646 to 648.

**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 at 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 at a coupled scale.

**Momentaries here are 17's, a society's next momentary carrying each self's entry once**, each release offered at its next, and each number goes with its spiral and its couplings. Two odd spirals at distinct primes p and q, side by side, come to their parities again together at a least common multiple of 4p and 4q, four shared: at 4pq, and so at any two odd spirals sharing no factor, nine and twenty-five at 900 as three and five at 60, while nine and fifteen, a factor shared, come sooner, at 180. Sixteen odd prime spirals, three to fifty-nine, side by side, come to their parities again together at 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 at couplings, owned by neither.

**Spirals crossing spirals at each self are a torus of selves, and at two odd numbers its parities again are at q or at 4p, by q's side of a waist 2p.** A torus of p selves along at 9-next-co-changing and q across at 10-next-co-possibling, both odd and p no greater, with each self at a prior parity, alternating along and across, comes, from a momentary on, to its parities again at q at a q past a waist 2p, and at 4p at a q short of it: 3 by 5 at 12, five short of six; 3 by 7 at 7, seven past six; 3 by 13 at 13, thirteen past six; 5 by 7 at 20, seven short of ten; 5 by 11 at 11, eleven past ten; 7 by 17 at 17, seventeen past fourteen; 17 by 59 at 59, fifty-nine past thirty-four. At either number even it comes to its parities again at 2, a torus of 2 by 3 followed by hand at resolver. Each row of two odd numbers of Exhibit ONE's table of a torus of selves is at a relation, 3 by 3 at 12 sharing a factor and each other row sharing none, and its relation is derived at resolver's cells, Co-Chaining Logic Registry's steps 652 to 657: 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 the 0's round, and at 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, at 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 at its 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 at 6k − 1 and 6k + 1, about a multiple of six. Whether twin primes carry on without end is, in a field's words, 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 at two either side, fifty-three at six either side. From five to fifty-three are fourteen primes, each with a gap across at both its sides: three reaches one along gap below it, and fifty-nine completes seventeen above. Fourteen part eight at one before a multiple of six, 5, 11, 17, 23, 29, 41, 47 and 53, and six at 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 at 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 at 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 at one and seven at 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 at 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, at 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 the primes pair at three, 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, at no position, and sixty completes it at no prime. Parity changing is a span's own, carried at 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 at couplings among six, and three of it are three hundred sixty. Orderings of five part at sixty, sixty even, an icosahedron's rotations, 6.3, and sixty others, and of sixty rotations one is at no turning and fifty-nine at 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 at two as fifty-nine and sixty-one straddle sixty at one, and on a ring of one hundred twenty one hundred eighteen is a far side of two.

**In a field's words the periodic table closes its noble gases at 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, in their stable and unstable forms, two short of one hundred twenty at 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 at fifty-five, couplings among eleven, and sixty-five, 2⁶ + 1, a next whole; 112 and 128 at fifty-six and sixty-four, eight sevens and eight eights, 128 = 2⁷; 114 and 126 at fifty-seven, seventeen primes' gaps added, and sixty-three, 2⁶ − 1; 116 and 124 at fifty-eight and sixty-two, two either side of sixty; and 118 and 122 at fifty-nine and sixty-one.

**Odds from one hundred eleven to one hundred twenty-nine pair about one hundred twenty at their factors**: 111 = 3 × 37 with 129 = 3 × 43, thirty-seven and forty-three the 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 at seven, eleven, seventeen and nineteen, 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 at a name of its position**: two at 1-now, an entry; twenty-three at 9-next-co-changing; fifty-nine at 17-next-now. Both seventeens fold at their ninth, eight either side, and pair by position, i with 18 − i: names 1 with 17 through 8 with 10 and 9 at itself, and primes making 61, 56, 52, 50, 52, 50, 48 and 48 about twenty-three.

**At one move's 8 up and 17 less the primes add to their own numbers.** 8 up: 2 and 23 at 1 and 9 add to five squared, 5 and 31 at 3 and 11 to six squared, 17 and 47 at 7 and 15 to eight squared, 3 and 29 at 2 and 10 to 2⁵, and 19 and 53 at 8 and 16 to seventy-two, six prime pairs' distances from thirty. 17 less: 2 and 53 at 1 and 16 add to fifty-five, couplings among eleven, and three pairs, at 3 and 14, 4 and 13 and 5 and 12, to forty-eight, a third seam. Along four four-cycles 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.

**At an entry, six names between selves and outward and inward names the seventeen 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 society's nine to seventeen the primes 23 to 59, adding to three hundred sixty-three, 3 × 11².

**Sixteen gaps are at sixteen steps between names**: one odd gap, two to three, at a step from an entry to 2-now-possibling, and a first gap of six, twenty-three to twenty-nine, at a step from 9-next-co-changing to 10-next-co-possibling, a fold's step on. Each relation here is exact at 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 at a ring of 2h, each the other's opposite. **An odd ring carries two farthest**, at one distance and one apart, and its opposite is a between of those two, at no station. An even ring's opposite is a station and an odd ring's a between: one form at 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 at each station of each prime ring from five to fifty-nine, 435 stations, at each station of even rings one short of them, 420, and at 49 stations of rings of nine, fifteen and twenty-five, two farthest and one opposite are at each: two farthest are an odd ring's, prime or composite. Straight across, h on, is apart from a fold, N less k: folds at one to nine and one to seventeen fold eight and sixteen places and complete at nine and seventeen.

**A ring is an arrangement fixed one scale out, and a carrying round it goes forward**: each station arrives at a station again by carrying on, a next at each arriving, not by stopping and going back to it. At each spiral of selves coupled one way each self alternates or is in a like pair, 7.2, and a self at a prior is at a prior at each momentary.

## 8.2 Either but not both is parity, and pairings overlap

**Each position carries two pairings, one at each parity.** Both parities at one momentary put one position in two exchangings at once, one side taking both sides, a not possible form. Pairings overlap, parities alternate, with no third carrying. An odd number, leaving one over at 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 in both, and each end continues past its line, its last place in its pairing with a place after it and its first in its pairing with a nought before it: at one to fifty-nine, twenty-nine pairs at 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 at 435 places and nine, fifteen and twenty-five beside them; closed into 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 at 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 changing with a changing, exclusive or and its inversion. Taken as a step it carries one and then other, round at two; taken of prior and now as a next it carries round at three, parity's own period, carrying both: three of four joint forms go round, and a fourth, + and +, arrives at itself at each step, two alike carrying no changing. **An ordering arrives with nothing ordering it**: at an odd momentary each odd pairing, at an even momentary each even pairing, each carrying at that momentary a carrying other cannot, and carrying it for 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 change one parity at 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 at 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 at its own subject. These are a step's own applications; resolver executes its names at its own resolving.

## 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 at 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 at each arriving**, carrying to its own completing and releasing at it.

**Fixed as one store, parity reaches none.** In a field's words, the Peres–Mermin square sets nine values of ±1 under six exclusive-or conditions: each row and each column taken at its own context satisfies its condition, and of 512 settings of all nine at once 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 at a ring of four hundred forty is k with four hundred forty less k, one station at 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 in all.

**Each pair is a ring, its radius its distance from a waist at two hundred twenty**, from nought at a waist to two hundred twenty at 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 at k and a carry other way at N less k, two carries each at its own momentary, one ring, momentarying.** Bi- two carries differing, co- two carries together at one ring, -ing momentarying.

**A ring carries its two and no order between them.** Either way round arrives at 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 at each ring**: even radius, two even stations; odd radius, two odd. A ring at even radius is a coupling across and a ring at odd radius a between.

**Odd primes, three to fifty-nine, are at odd radii, from two hundred seventeen to one hundred sixty-one**, each a between. **Two only of seventeen is at an even radius**, one even prime at a coupling ring.

**Four hundred forty leaves two at three, and ten primes leaving two at three, 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 the six others the four leaving seven at twelve, 7, 19, 31 and 43, have prime far sides, 433, 421, 409 and 397, while thirteen and thirty-seven, leaving one at twelve, 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 at twenty-two odd radii, five to two hundred fifteen, five with four hundred thirty-five, and each ten is at 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 is at 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 at 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 two sums are each other's far side, at radius one hundred fifty-six.

## 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 at four hundred forty, each surface's own mouth, and on a pair that station is a waist**: each self's mouth arrives in a coupling as a between, and each self's waist, at two hundred twenty and six hundred sixty, is an other's far side.

**A chain of surfaces centres on a coupling at an even number of selves and on a self's own middle carrying nothing at 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 society grows larger by further co-chaining**: a reach lengthening, a reciprocal receiving widening, and each self's full resolving carrying its form at each coupling added. Two hundred twenty a self adds are surfaces' stations, and co-competencing grows at 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 at each span gives a span below it**: an odd ring of N carries (N + 1)/2 pairs, two at three, three at five, five at nine, nine at seventeen. Seventeen at its two sides is a nine, a nine a five, a five a three.

**Pairs carrying two stations double along a chain, one, two, four, eight, and each span is twice them and one nothing at its origin.** Pairings at six, eight and ten are three, four and five, consecutive, 5.8, eight's four the resolver's four momentaries of exchanging; two progressions share four only, and four pairings is a nine. At nine only among spans, a span's own pairings and resolver's four at eight are one, and self and other carry whole at nine.

**At one to nine a self's eight is shared, four at each side, with five between owned by neither**: bi-moral co-agency. **At one to seventeen each side carries a whole eight, and an entire one to nine is shared at its centre**: social moral competency, a pair at a 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 at a next scale, its 1, 2 and 3 being 1, 9 and 17: one to nine a self's momentary, completing at 9. Each use is at its scale: at resolver one executing of 1 to 17 is one momentary of exchanging at a scale outward, and one numbered name and one span of nine names are each at their own subject, matching numerals coupling no relations. 17-next-now is a next momentary's 1-now, and one to sixty-five is four momentaries of exchanging at that scale, one momentary at 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 releases, and a next whole of them is 1–65.** 1–9, self and other, is an entire momentary bi-folding releasing at 3; 1–13, self and other and other and self, bi-folds releasing at 6; 1–17, self, other and social, bi-folds releasing at 9; and 1–25 is a fractal bi-folding one span on. Each span opens one more four and its releasing moves three, three inward to each one outward: at a round of four parities its outward steps arrive at 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 at a round of six parities, 1–25 releases at 15 and 1–65 at 45. At 3 a self's pair is wholly inverted and at 9 an outer pair of four, an inner pair carried as it was: two bi-folds, each at its own span. Binary, all or none at all, resolving coheres an entire surface along and across, releasing at 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 at one parity, five at two, nine at three, seventeen at four, thirty-three at five and sixty-five at six. Pairs step a chain two parities at a time, 5, 17, 65 and 257, each pair stepping once at each 5 of a pair inward of it; a scaling at 1, 9 and 17 steps it three parities at a time, 9, 65 and 513, one to nine at one scale being one to sixty-five at a next; and 1–17 inward and outward steps it four, 17, 257 and 4,097, a chain's step at four parities, beside a placing of each 1 to 17 at 8n − 7, 17 at 129, two relations at one span. Pairs and threes reach each other first at 1–65, a next whole, and four 1–17s carry it, its outer pair stepping at 17, 33, 49 and 65; pairs and 1–17 reach each other at 1–17 and 1–257, and all three at 1–4,097. No scaling is taken over another: each is one chain at its own step.

**An odd ring carries one self-paired station and an even ring two**: a self's ring at nine carries one nothing, and a society's winding at 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 at its across relation; co-chaining is same continuing at 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 in same span, 1–2 with 2–3, 3–4 with 4–5, 5–6 with 6–7 and 7–8 with 8–9, self at 1 to 8 and other at 2 to 9: podal pairs and overlapping pairs are two relations in one to nine, and its span unfolds, each momentary a next.

**A carry each way is one ring at each station**: at k a carry one way, at four hundred forty less k a carry other way, a ring at 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 at 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 at 2 + 8s and completes at an odd 8s + 1 before it: nine, seventeen, twenty-five, thirty-three and forty-one. At s one a row is four outward at a between and its span one to nine; at 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 in none of forms among 1 to 16.

**A fold carries whole at each scale.** A fold at a lays n at 2a less n, and a fold at a and then at b lays n at n more by twice b less a: two folds in sequence are no fold, each number stepped on alike, folds at 4 and then 12 laying 1 at 7 and then 17. On names 1 to 8s, 4s up and down continues parity and 8s + 1 less changes it, and alternating two reaches each name again after four moves through four names at each s: two taken together are 4s + 1 less in a lower half and 12s + 1 less in an upper, and 8s + 1 and both of these are odd: each move carries each name to another name. At s two they are 8 up and 17 less, 3 to 11 to 6 to 14 to 3; at 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 at 1 to 4, 1, 3, 2, 4 at two up or down and five less, two at 1 to 8, 4-8-1-5 and 3-7-2-6, and four at 1 to 16, each podal pair a four-cycling at a doubled span; completing names 5, 9 and 17 are in none.

**Longer and wider carry one ratio, two to one.** Each row's middle is 2 + 3s, its four at 3s and s either side: its inner pair reaches s over a span of 2s and its outer 3s over 6s, span over reach two at both. An outer span 6s is apart from a four-step span 8s, a row's middle 2 + 3s from a fold's middle at 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: at 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 at each repeat, and scales one to five and doublings one, two, four, eight are two walks through one family. Taken over all names at 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 still-possibling names, each sharing, a self's next prior, each sharing's changing and a society. At 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 at 2r. At 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 in 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 at 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 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 at its field.

**One binary at each number a field offers: whether its number carries on when its unit changes.** A number of existing things carries on in each unit; a magnitude in a chosen unit goes with its unit, and its unit is one cohering.

**Coupling is all or none.** At a field's observing carrying a number-form, one form arrives at two substrates; a number agreeing at both sides is a relation only with a form reached at both.

**A prime at 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 four-cycles among seventeen names are two momentaryings, sharing their digits only.** One to nine are nine places of one momentary; 1, 9, 8 and 16 at a four-cycle are four of seventeen names. A line from one to other draws a relation neither momentarying carries. Digits they share reach each other exactly at a numbered form: 8 up and then 17 less is 9 less, 17 − (n + 8) = 9 − n, and at each scale s, 9.9's, 8s + 1 − (n + 4s) = 4s + 1 − n, and n, n + 8, 9 − n and 17 − n are a four-cycle's two out-and-backs as numbers, a numbered form and no momentarying.

**Three momentaries at one self and three selves at one momentary of a spiral are two forms at one number.** A pattern three momentaries long is a method's matching; three selves side by side at one momentary is a between.

**Each six is at its own subject.** Three phases at two ways; resolver's six names between selves, 2, 6, 9, 10, 14 and 17; three pairings at two sides; a side's three momentaries at two parity positions each; three momentaries at each of two sides; and self and other, self and social and other and social, three pairs at two orders each: six at each, and sharing six couples none to another. A relation coupling two of them is a relation of its own, reached at both.

---

# ELEVEN · EACH NUMBER

## 11.1 Nought to four hundred forty, each number at its podaling

**Each number from nought to four hundred forty is at 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 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 at nought's station; its radius from a waist at two hundred twenty; its far side on a ring of one hundred twenty, a number past one hundred twenty at a station it leaves at one hundred twenty; and forms it is at: 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.

| n | Parity, at its momentary | Form | Far side at 440 | Radius | Far side at 120 | At |
|---|---|---|---|---|---|---|
| 0 | even | — | 0 | 220 | 0 | 0², seam, 1² − 1 |
| 1 | odd, now | — | 439 | 219 | 119 | 1-now, 1², T₁, prior two added, 2¹ − 1 |
| 2 | even, now | prime | 438 | 218 | 118 | 2-now-possibling, first prime, prior two added, 2¹ |
| 3 | odd, now | prime | 437 | 217 | 117 | 3-now-co-prioring, second prime, T₂, prior two added, 2¹ + 1, 2² − 1 |
| 4 | even, now | 2² | 436 | 216 | 116 | 4-now-co-still-possibling, 2² |
| 5 | odd, now | prime | 435 | 215 | 115 | 5-now-co-changing, third prime, prior two added, 2² + 1 |
| 6 | even, now | 2 · 3 | 434 | 214 | 114 | 6-now-co-possibling, T₃, 3! |
| 7 | odd, now | prime | 433 | 213 | 113 | 7-now-prioring, fourth prime, 2³ − 1 |
| 8 | even, now | 2³ | 432 | 212 | 112 | 8-now-still-possibling, seam, 3² − 1, prior two added, 2³ |
| 9 | odd, next | 3² | 431 | 211 | 111 | 9-next-co-changing, 3², 2³ + 1 |
| 10 | even, next | 2 · 5 | 430 | 210 | 110 | 10-next-co-possibling, T₄ |
| 11 | odd, next | prime | 429 | 209 | 109 | 11-next-co-prioring, fifth prime |
| 12 | even, next | 2² · 3 | 428 | 208 | 108 | 12-next-co-still-possibling |
| 13 | odd, next | prime | 427 | 207 | 107 | 13-next, sixth prime, prior two added |
| 14 | even, next | 2 · 7 | 426 | 206 | 106 | 14-next-possibling |
| 15 | odd, next | 3 · 5 | 425 | 205 | 105 | 15-next-prioring, T₅, 2⁴ − 1 |
| 16 | even, next | 2⁴ | 424 | 204 | 104 | 16-next-still-possibling, 4², 2⁴ |
| 17 | odd, now | prime | 423 | 203 | 103 | 17-next-now, seventh prime, 2⁴ + 1 |
| 18 | even, now | 2 · 3² | 422 | 202 | 102 |  |
| 19 | odd, now | prime | 421 | 201 | 101 | eighth prime |
| 20 | even, now | 2² · 5 | 420 | 200 | 100 |  |
| 21 | odd, now | 3 · 7 | 419 | 199 | 99 | T₆, prior two added |
| 22 | even, now | 2 · 11 | 418 | 198 | 98 |  |
| 23 | odd, now | prime | 417 | 197 | 97 | ninth prime, going folding |
| 24 | even, now | 2³ · 3 | 416 | 196 | 96 | seam, 5² − 1, 4! |
| 25 | odd, next | 5² | 415 | 195 | 95 | 5² |
| 26 | even, next | 2 · 13 | 414 | 194 | 94 |  |
| 27 | odd, next | 3³ | 413 | 193 | 93 |  |
| 28 | even, next | 2² · 7 | 412 | 192 | 92 | T₇ |
| 29 | odd, next | prime | 411 | 191 | 91 | tenth prime |
| 30 | even, next | 2 · 3 · 5 | 410 | 190 | 90 |  |
| 31 | odd, next | prime | 409 | 189 | 89 | eleventh prime, 2⁵ − 1 |
| 32 | even, next | 2⁵ | 408 | 188 | 88 | 2⁵ |
| 33 | odd, now | 3 · 11 | 407 | 187 | 87 | 2⁵ + 1 |
| 34 | even, now | 2 · 17 | 406 | 186 | 86 | prior two added |
| 35 | odd, now | 5 · 7 | 405 | 185 | 85 |  |
| 36 | even, now | 2² · 3² | 404 | 184 | 84 | 6², T₈ |
| 37 | odd, now | prime | 403 | 183 | 83 | twelfth prime |
| 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 | thirteenth prime |
| 42 | even, next | 2 · 3 · 7 | 398 | 178 | 78 |  |
| 43 | odd, next | prime | 397 | 177 | 77 | fourteenth prime |
| 44 | even, next | 2² · 11 | 396 | 176 | 76 |  |
| 45 | odd, next | 3² · 5 | 395 | 175 | 75 | T₉ |
| 46 | even, next | 2 · 23 | 394 | 174 | 74 |  |
| 47 | odd, next | prime | 393 | 173 | 73 | fifteenth prime |
| 48 | even, next | 2⁴ · 3 | 392 | 172 | 72 | seam, 7² − 1 |
| 49 | odd, now | 7² | 391 | 171 | 71 | 7² |
| 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 | sixteenth prime |
| 54 | even, now | 2 · 3³ | 386 | 166 | 66 |  |
| 55 | odd, now | 5 · 11 | 385 | 165 | 65 | T₁₀, prior two added |
| 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 | seventeenth prime, going's last |
| 60 | even, next | 2² · 3 · 5 | 380 | 160 | 60 | between going and returning |
| 61 | odd, next | prime | 379 | 159 | 59 | returning's first |
| 62 | even, next | 2 · 31 | 378 | 158 | 58 |  |
| 63 | odd, next | 3² · 7 | 377 | 157 | 57 | 2⁶ − 1 |
| 64 | even, next | 2⁶ | 376 | 156 | 56 | 8², 2⁶ |
| 65 | odd, now | 5 · 13 | 375 | 155 | 55 | 2⁶ + 1 |
| 66 | even, now | 2 · 3 · 11 | 374 | 154 | 54 | T₁₁ |
| 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 | T₁₂ |
| 79 | odd, next | prime | 361 | 141 | 41 |  |
| 80 | even, next | 2⁴ · 5 | 360 | 140 | 40 | seam, 9² − 1 |
| 81 | odd, now | 3⁴ | 359 | 139 | 39 | 9² |
| 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 | prior two added |
| 90 | even, next | 2 · 3² · 5 | 350 | 130 | 30 |  |
| 91 | odd, next | 7 · 13 | 349 | 129 | 29 | T₁₃ |
| 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 | 10² |
| 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 | T₁₄ |
| 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 | returning's far end |
| 119 | odd, now | 7 · 17 | 321 | 101 | 1 |  |
| 120 | even, now | 2³ · 3 · 5 | 320 | 100 | 0 | seam, 11² − 1, T₁₅, 5! |
| 121 | odd, next | 11² | 319 | 99 | 119 | 11² |
| 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 | 2⁷ − 1 |
| 128 | even, next | 2⁷ | 312 | 92 | 112 | 2⁷ |
| 129 | odd, now | 3 · 43 | 311 | 91 | 111 | 2⁷ + 1 |
| 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 | T₁₆ |
| 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 | 12², prior two added |
| 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 | T₁₇ |
| 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 | seam, 13² − 1 |
| 169 | odd, next | 13² | 271 | 51 | 71 | 13² |
| 170 | even, next | 2 · 5 · 17 | 270 | 50 | 70 |  |
| 171 | odd, next | 3² · 19 | 269 | 49 | 69 | T₁₈ |
| 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 | T₁₉ |
| 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 | 14² |
| 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 | T₂₀ |
| 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 | waist |
| 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 | seam, 15² − 1 |
| 225 | odd, now | 3² · 5² | 215 | 5 | 15 | 15² |
| 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 | T₂₁ |
| 232 | even, now | 2³ · 29 | 208 | 12 | 8 |  |
| 233 | odd, next | prime | 207 | 13 | 7 | prior two added |
| 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 | T₂₂ |
| 254 | even, next | 2 · 127 | 186 | 34 | 106 |  |
| 255 | odd, next | 3 · 5 · 17 | 185 | 35 | 105 | 2⁸ − 1 |
| 256 | even, next | 2⁸ | 184 | 36 | 104 | 16², 2⁸ |
| 257 | odd, now | prime | 183 | 37 | 103 | 2⁸ + 1 |
| 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 | T₂₃ |
| 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 | seam, 17² − 1 |
| 289 | odd, now | 17² | 151 | 69 | 71 | 17² |
| 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 | T₂₄ |
| 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 | 18² |
| 325 | odd, now | 5² · 13 | 115 | 105 | 35 | T₂₅ |
| 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 | T₂₆ |
| 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 | seam, 19² − 1 |
| 361 | odd, next | 19² | 79 | 141 | 119 | 19² |
| 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 | prior two added |
| 378 | even, next | 2 · 3³ · 7 | 62 | 158 | 102 | T₂₇ |
| 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 | 20² |
| 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 | T₂₈ |
| 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 | T₂₉ |
| 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 | seam, 21² − 1, seventeen primes added |

**From row one down, its far side at four hundred forty steps down one at each row as its number steps up**, its radius coming to nought at its waist and opening again, and its far side at one hundred twenty steps down from one hundred nineteen to one and is at nought at each of 0, 120, 240 and 360, its last stretch ending at forty at four hundred forty. Each pairing on a ring of four hundred forty and on a ring of one hundred twenty in parts above is at its two rows: a carry each way at one station, found by its number.