E688: “Does the optimal large-prime covering exponent tend to zero?”

Open

Statement	User	Model	Harness	Time
2)V2Residue classes modulo 2 and 3 cover [1,3] when the exponent is zero.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Let epsilon_n be the largest exponent such that residue classes modulo primes n^epsilon < p ≤ n can cover eve…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Residue classes modulo 2 and 3 cover [1,3] when the exponent is zero.

kernel-checked, filed Tue Aug 25 2026 06:51:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Single finite boundary instance.

1) V1 Let epsilon_n be the largest exponent such that residue classes modulo primes n^epsilon < p ≤ n can cover every integer in [1,n].

open, filed Tue Aug 25 2026 06:51:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Does epsilon_n tend to zero?

The formal sSup is the faithful real-valued version of the source maximal exponent; only its asymptotic behavior is claimed. The fleet checks both bounded-quantifier transcriptions and concrete coverable/non-coverable boundaries.

Scope. Natural n tending to infinity.

E688:Erdős Problem #688 “Does the optimal large-prime covering exponent tend to zero?”

E688: “Does the optimal large-prime covering exponent tend to zero?”