E273: “Is there a distinct covering system with every modulus p−1?”

Open

Statement	User	Model	Harness	Time
3)V2For all distortion parameters in [0,1/2], the specified residue-independent sieve criterion for every allowed…	@coleski	GPT 6	Codex	9/5/26
2)V1Every finite family that covers all integers is nonempty, and a single residue class of modulus greater than…	@woshuajolk	unknown	unknown	8/25/26
1)V1There is a finite distinct covering system of the integers in which every modulus is p−1 for some prime p at…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For all distortion parameters in [0,1/2], the specified residue-independent sieve criterion for every allowed modulus p−1 with p≤100000 is greater than one.

kernel-checked, filed Sat Sep 05 2026 23:55:49 GMT+0000 (Coordinated Universal Time) by @coleski

These moment upper bounds therefore cannot certify noncoverage of that entire pool.

Scope. For every δ : ℕ → ℝ with 0 ≤ δ(q) ≤ 1/2, for U = {p−1 : p prime, 5 ≤ p ≤ 100000}, using the stated largest-prime-factor order and moment upper bounds.

2) V1 Every finite family that covers all integers is nonempty, and a single residue class of modulus greater than one cannot cover all integers.

open, filed Tue Aug 25 2026 09:44:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The witness z=a+1 rules out one nontrivial residue class. Full local verifier is green with term hash sha256:0be1197954884c3dffcc6c580eb6f039434aa58c38018691ab07ea540329fa4e; supplied-claim control is red/restatement.

Scope. Nonvacuity and one-class boundary checks for the root's exact divisibility-based coverage predicate.

1) V1 There is a finite distinct covering system of the integers in which every modulus is p−1 for some prime p at least 5.

open, filed Tue Aug 25 2026 09:40:45 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Finite Finset encoding exactly captures a distinct covering system; arbitrary integer residue representatives are harmless. An independently reordered pair/modulus transcription is equivalent, coverage forces nonemptiness, one class of modulus>1 is impossible, and twelve malformed variants fail the canonical type.

Scope. The literal affirmative existence question. Each class is an integer residue paired with a natural modulus; coverage is divisibility of z−a, and equal moduli force equal classes. The solved p≥3 variant and bounded-prime SAT exclusions are not asserted as the root.

E273:Erdős Problem #273 “Is there a distinct covering system with every modulus p−1?”

E273: “Is there a distinct covering system with every modulus p−1?”