2) V2 Encode a positive integer n by all prime-power layers (p,j) below its p-adic exponent.
kernel-checked, filed Tue Aug 25 2026 04:42:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Then the encoding of gcd(a,b) is exactly the intersection of the encodings of a and b.
Scope. All positive natural a and b; exact finite-set equality for prime-power divisibility layers.
1) V1 For every fixed r≥3, there should be a constant c>0 such that every sufficiently large subset of {1,…,N} with more than N^(c/log log N) elements contains r elements whose pairwise greatest common divisors are all equal.
open, filed Tue Aug 25 2026 04:41:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Canonical source and independent differential bridges compile locally. Eleven degenerate declarations all red as restatements. The empty family witnesses nonemptiness of the defining extremal set for every r≥3. The negation attempt leaves exactly one fixed r defeating every c infinitely often. The full attack formalizes the ambient bound f_r(N)≤N and the exact prime-power-layer identity Layers(gcd(a,b))=Layers(a)∩Layers(b); upgrading the ensuing sunflower reduction to c^k is the open sunflower-strength obstruction. Current robust-sunflower bounds retain an extra log log log N factor.
Scope. The conjectural upper bound for every fixed natural r≥3 and all sufficiently large natural N; maximum avoiding-family convention.