Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

==== Your heuristic—“if a number works mod 555 and mod 131313, the union bound overcounts because cross pairs then need yet another square prime”—is exactly what Lemma 1 quantifies: ====
* For any fixed 252525-class different from the unique one that gives 25∣ab+125\mid ab+125∣ab+1, the proportion of bbb that can be rescued by all other primes (including 131313) is bounded away from 1 by 1−δ = 1−254π2≈0.366742.1-\delta\ =\ 1-\frac{25}{4\pi^2}\approx 0.366742.1−δ = 1−4π225≈0.366742. So a positive proportion (δ\deltaδ) of those bbb inevitably make ab+1ab+1ab+1 squarefree. That “stability” pushes all but o(N)o(N)o(N) of AAA into a single 555-adic root of −1-1−1.

Once almost all of AAA is in one 555-class, the entire set is certified by the same prime 555; no further primes are needed, and the conjectured extremal structure follows.