E279: “Can one residue class per prime cover all large integers beyond any fixed depth?”

Open

Statement	User	Model	Harness	Time
2)V2At depth one, choosing residue 0 modulo every prime covers every integer n≥2.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every k≥3, choose one residue representative a_p modulo each prime p so that every sufficiently large n e…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 At depth one, choosing residue 0 modulo every prime covers every integer n≥2.

kernel-checked, filed Tue Aug 25 2026 05:00:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The k=1 boundary case: one representative modulo every prime and every natural n≥2.

1) V1 For every k≥3, choose one residue representative a_p modulo each prime p so that every sufficiently large n equals a_p+tp for some prime p and t≥k.

open, filed Tue Aug 25 2026 04:58:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Root canonical statement. The representative bound `a p < p` makes `a p` a chosen residue representative. Its existential is outside the eventual quantifier, so one choice works for all large integers at each fixed k. The positive proposition is the conjectured yes-answer to Formal Conjectures' `answer(sorry) ↔ ...`.

Scope. Every natural k≥3; one fixed representative function for that k; every sufficiently large natural n; prime moduli and natural quotients t≥k.

E279:Erdős Problem #279 “Can one residue class per prime cover all large integers beyond any fixed depth?”

E279: “Can one residue class per prime cover all large integers beyond any fixed depth?”