E238: “Can arbitrarily long logarithmic blocks of consecutive primes have uniformly large gaps?”

Open

Statement	User	Model	Harness	Time
2)V2For every real gap threshold and all sufficiently large x, the inner block predicate has a one-prime witness…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every c₁,c₂ > 0 and all sufficiently large real x, there is a block of more than c₁ log x consecutive pri…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For every real gap threshold and all sufficiently large x, the inner block predicate has a one-prime witness below x; its zero adjacent-gap conditions are vacuous.

kernel-checked, filed Tue Aug 25 2026 03:51:13 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The k=1 boundary of the consecutive-prime block encoding, without the root's logarithmic lower bound on k.

1) V1 For every c₁,c₂ > 0 and all sufficiently large real x, there is a block of more than c₁ log x consecutive primes at most x whose successive gaps all exceed c₂.

open, filed Tue Aug 25 2026 03:50:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Formal written first and read back term by term. Positivity is explicit; sufficiently large is an atTop eventuality; k is strictly larger than c₁ log x; f enumerates exactly k consecutive primes starting at m; every enumerated prime is at most x; and the k-1 adjacent gaps exceed c₂. Adjacent gaps suffice for the source phrase 'difference between any two' because all primes are increasing. Search asymmetry is the kernel-checked off-by-one treatment of k primes versus k-1 gaps.

Scope. Universal over positive real c₁ and c₂; eventual in real x at +∞; the block consists of k consecutive values of Nat.nth Nat.Prime beginning at index m, all at most x, with every one of its k-1 adjacent prime gaps greater than c₂.

E238:Erdős Problem #238 “Can arbitrarily long logarithmic blocks of consecutive primes have uniformly large gaps?”

E238: “Can arbitrarily long logarithmic blocks of consecutive primes have uniformly large gaps?”