BRIDGES — where growth meets open number theory
The object: the places where a growth world of GROWTH meets a question about primes, most of them configurations that exist when, or exactly when, a prime p exists with p − 1 of a prescribed multiplicative shape. A world grows its modulus by the least multiplier its demand admits: the transparency demand admits one that leaves λ fixed, the dynamics demand one that raises λ, the independence demand one coprime to the modulus; the transparency demand's walk dies at the wall, the largest modulus whose λ divides the current λ. The bridges concern the alphabet worlds, which fill to the wall and then push a prime of a fixed set S, the phoenix among them; the values the odd part of λ can take, λ(M) the exponent of the unit group of Z/M (the Carmichael function); and the tower read by the transparency demand. This page asks which of these bridges are exact equivalences, which run one way, and which rest on a proved theorem rather than an open one. Its verifier is bridges.py, which also checks instances of GROWTH's steering of a state onto a prime by a single multiplication (GROWTH: The lock-prime law): 200 states onto 2, and five states each onto 3, 5, 7, 13 and 17, each run watched for the 30 greedy picks after the multiplication.
The answer, in one line: two bridges are exact equivalences resting on open questions, the seating law on primes with p − 1 of a prescribed shape and the atoms (which odd h is the odd part of some λ(M), built from the odd parts of λ at odd prime powers) on primes h′·2^b + 1, h′ dividing h, whose least failure is, on the published record, the least prime Sierpiński number; two run one way, the lock into Dirichlet's theorem and so proved, the cascade closed wherever a known covering kills infinitely many dead levels and open past that (each argued on its own page, GROWTH and CASCADE); and one is a proved density theorem (TOWER: The density theorem), an upper bound on the rungs that move λ whose matching lower bound is open.
| bridge | growth side | number-theory side | direction | status |
|---|---|---|---|---|
| the lock | every prime is some seed's lock (GROWTH: The lock-prime law) | a prime ≡ 1 (mod B) and ≢ 1 modulo the odd prime locked onto, B the lcm the lock-prime law builds from the primes up to that prime, prime to it; 2 needs none | one way, into Dirichlet | proved |
| the seating law | an alphabet world (The seating law) seats infinitely many primes | infinitely many primes p with (p − 1)_S′ dividing F, n_S′ the part of n prime to S and F that part of lcm(λ(seed), q − 1 : q ∈ S); from seed 1 with every q − 1 S-smooth, infinitely many with p − 1 S-smooth | equivalence, per seed and alphabet | open: Fermat at {2}, Pierpont at {2, 3} |
| the atoms | an odd h is the odd part of some λ(M) | for each l^j ‖ h, a divisor h′ of h with v_l(h′) = j and h′·2^b + 1 prime, up to a finite correction | equivalence | the least h that is not: the least prime Sierpiński number, on the published record; open |
| the cascade | every ideal-world trajectory of a number ring locks (CASCADE: Dead levels recur at eleven characteristics) | infinitely many dead levels at one rank-1 characteristic, a prime with an unramified place over it whose norm is that prime | one way, sufficient | proved at a ring with a rank-1 characteristic among the eleven primes with a known covering of their levels' candidates, 2 by a covering of its own and 3, 5, 7, 11, 13, 17, 19, 23, 31 and 37 (odd characteristics below 1000 searched, covering primes below 2,000 modulo which the characteristic's order divides 5,040); open in general |
| the density | the transparency demand admits almost every move of independence's walk | TOWER: The density theorem | none needed | proved |
A one-way reduction into a theorem settles its growth side, and calling it an equivalence would add only that two true statements imply each other; an equivalence settles nothing until its number-theory side is settled. The lock and the cascade live on GROWTH and CASCADE; this page holds the other three.
The seating law
Tier: theorem; observation (the seat counts below 10⁶).
Verifier: proof; bridges.py::section_w.
An alphabet world starts at a seed s and repeats two phases: it fills, by the transparency demand's greedy walk, which dies at exactly the wall W(λ(M)) (GROWTH: The three fates), and then a push takes the least λ-raising move at one prime in a finite set S, its alphabet, under any schedule that pushes every letter, every prime of S, infinitely often. Its seats are the primes that ever divide its state. Write n_S′ for the part of n prime to S. Then
seats(s, S) = {p prime : (p − 1)_S′ divides F},
F = (lcm(λ(s), q − 1 : q ∈ S))_S′,
whatever the schedule. Fills leave λ fixed, and at the wall an odd prime r is seated iff (r − 1) | λ, at depth v_r(λ) + 1, its exponent in the wall. So a push of an unseated letter q seats it and moves λ's part prime to S to its lcm with (q − 1)_S′, once; a push of a seated letter q deepens it by one and adds one factor q and nothing prime to S, since (q − 1) | λ already; a push of 2 adds one factor 2. Once every letter has been pushed, λ's part prime to S is F forever and its part at every letter diverges, and the seats are the union of the walls' supports. An odd seat p outside S ends at depth v_p(F) + 1, and 2 outside S at v₂(F) + 2, v₂(F) being at least 1 since S then holds an odd letter. Nine alphabets from seed 1, and {2} and {2, 3} from each of the seeds 7, 11, 65 and 105, are run under a round-robin and a random schedule on λ alone, each fill read as its wall: at the 489 steps with λ ≤ 10⁶ the wall built from its primes carries that λ, and λ of it times its least move is the next step. The final λ's part prime to S is then F by construction, and the seats below 10⁶ after 25 pushes per letter match the predicate, a match that reads only that 25 pushes per letter carry every seat below 10⁶.
When every q − 1 with q ∈ S is S-smooth, F is λ(s)_S′, and from seed 1 the seats are exactly the primes p with p − 1 S-smooth: the world seats infinitely many primes iff there are infinitely many such primes. That is Fermat's question at S = {2} and Pierpont's at {2, 3}, one question per alphabet, each a Π₂ statement (for every bound, some seat past it). Below 10⁶ the seats number 6 at {2}, the count unchanged since 65537, against 42 at {2, 3}, 141 at {2, 3, 5} and 324 at {2, 3, 5, 7}. An alphabet without 2 bounds the power of 2 in p − 1: from seed 1, S = {3} has F = 2, so its seats are 2 and the primes 2·3^b + 1, 3 among them, each such p − 1 carrying exactly one factor 2; from seed 1, S = {2, 11} has F = 5, seating 41 and not 101. Another seed moves F: S = {3} from seed 5 has F = 4 and seats 13.
The phoenix
Tier: theorem; observation (the walls below 2·10⁷).
Verifier: proof; bridges.py::section_p.
The phoenix is the alphabet world S = {2}. At a wall every move raises λ, so the least move there is 2, which is the dynamics demand's own greedy pick: the phoenix is the single greedy law "the least move the transparency demand admits, else the least move", with no choice left to an operator. From seeds 1, 7 and 11 that law meets 15 walls in all below 2·10⁷, each equal to W(λ). Its F is odd(λ(s)), the odd part of λ(s), fixed at birth, so its seats are the spectrum of that odd number, where
spectrum(h) = {p prime : odd(p − 1) divides h},
and the spectrum only grows along divisibility, so every phoenix seats spectrum(1), which is 2 and the Fermat primes. The seed-1 phoenix seats 3, 5, 17, 257 and 65537 as v₂(λ) reaches 1, 2, 4, 8 and 16, and 2^j + 1 is composite for every other j with 1 ≤ j ≤ 40. Its seats being spectrum(1), it seats infinitely many primes iff there are infinitely many Fermat primes. From seed 1, the least move that restarts a transparency walk at its wall seats exactly 2 and the Fermat primes; from a seed with odd(λ(s)) = h it seats spectrum(h), which at h = 3 (seed 7) adds 7, 13, 97, 193, 769, …
The atom criterion
Tier: criterion (proved both ways; every odd h ≤ 999 realized by a
built modulus).
Verifier: proof; bridges.py::section_a, bridges.py::section_s,
bridges.py::section_c.
λ(M) is the lcm of λ(q^a) over q^a ‖ M, and the odd part of an lcm is the lcm of the odd parts, so odd(λ(M)) is the lcm of the atoms q^(a−1)·odd(q − 1) over the odd q^a ‖ M, the prime 2 contributing nothing. An odd h is realizable when it is odd(λ(M)) for some M, and it is realizable iff every prime power l^j ‖ h is served by an atom that divides h and carries l exactly to the power j. Only if: the lcm reaches l^j at some atom, and every atom divides h. If: keep each serving atom's q^a, the largest a per q, whose atom is a multiple of the atom at every smaller a and still divides h, so it carries each l^j those served; the product of those powers has odd(λ) = h.
An atom equals h iff h·2^b + 1 is prime for some b ≥ 1, or h = q^(a−1)·odd(q − 1) with a ≥ 2 and q | h, which is a finite check. So being an atom is that primality predicate with a finite correction, and realizability is a search over primes h′·2^b + 1 at the divisors h′ of h, with that correction. Every odd h ≤ 999 is realized; 47 needs b = 583 and 881 needs 1027, each the least b that works, while 383 has none at b ≤ 1000 and is realized at b = 6393, 383·2⁶³⁹³ + 1 certified prime by Proth's theorem. A covering set turns the search's failure into a proof. 78557 has the covering set {3, 5, 7, 13, 19, 37, 73} of period 36, so no prime q has odd(q − 1) = 78557, and neither 17^(a−1) nor 4621^(a−1)·1155 equals it, so it is the atom of no prime power, yet it is 17·4621 and two primes realize it: odd(λ(137·18927617)) = 78557, where 137 = 17·2³ + 1 and 18927617 = 4621·2¹² + 1.
The least unrealizable odd number
Tier: rule (proved; the search to b ≤ 1000 an observation; the other
survivors settled and the nine open primes, the published record's).
Verifier: bridges.py::section_a, bridges.py::section_s.
An odd h is a Sierpiński number when h·2^b + 1 is composite for every b ≥ 1. If l^j ‖ h is unserved, then l^j itself is no atom, and l^j is an atom iff l^j·2^b + 1 is prime for some b or l is a Fermat prime. So an unrealizable h has a prime-power divisor l^j ‖ h, with l not a Fermat prime, that is Sierpiński, and such an l^j is itself unrealizable, since the only divisor of l^j carrying l to the power j is l^j. The least odd number that is never the odd part of λ(M) is therefore the least Sierpiński prime power of non-Fermat base.
271129 is prime and Sierpiński, with the covering set {3, 5, 7, 13, 17, 241} of period 24, and 271128 = 2³·3·11·13·79 is no power of 2, so no M has odd(λ(M)) = 271129: it bounds the least unrealizable odd number. Below it lie 23,844 odd prime powers l^j of non-Fermat base; for 278 no l^j·2^b + 1 is prime at b ≤ 1000, the survivors below 1000 are exactly 383 and 881, and the one proper power among them, 143641 = 379², resolves at b = 1212, certified by Proth's theorem. The rest is the prime Sierpiński problem by another name: the published record of known primes k·2^b + 1 (W. Keller, The Sierpiński Problem: Definition and Status, prothsearch.com/sierp.html, updated 2026-07-29) settles every other prime survivor and leaves open, among the prime powers below 271129, exactly the nine primes 22699, 67607, 79309, 79817, 152267, 156511, 222113, 225931 and 237019, all survivors of the search here. So on that record the least odd number that is never the odd part of λ(M) IS the least prime Sierpiński number: one of the nine or 271129, which 271129 is exactly when the prime Sierpiński problem closes as conjectured.
Transparent rungs are transparent moves
Tier: property.
Verifier: proof; bridges.py::section_t.
λ(Mp) = lcm(λ(M), p − 1) for a prime p not dividing M, so rung k of the tower is transparent (TOWER: The transparency criterion) iff p_k is a move the transparency demand admits at p_(k−1)#, and the tower is the greedy independence walk from 1 (GROWTH: The three fates). The density theorem therefore says that along independence's walk the transparency demand admits almost every move: one walk, two demands, agreeing on a set of density 1. Up to k = 2000 the demand admits 1531 of the 1999 moves.