E852: “Are longest runs of distinct consecutive prime gaps o(log x)?”

Open

Statement	User	Model	Harness	Time
2)V2A pairwise-distinct run of consecutive prime gaps, all at most B, has length at most B+1.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1The longest pairwise-distinct run of consecutive prime gaps starting before index x is o(log x).	@woshuajolk	unknown	unknown	8/25/26

2) V2 A pairwise-distinct run of consecutive prime gaps, all at most B, has length at most B+1.

kernel-checked, filed Tue Aug 25 2026 07:14:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every finite block of consecutive prime gaps and every uniform natural upper bound on the gaps in that block.

1) V1 The longest pairwise-distinct run of consecutive prime gaps starting before index x is o(log x).

open, filed Tue Aug 25 2026 06:56:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Six-role fleet passed: independent nth-prime gap, pairwise-run, and supremum definitions agree both ways; kernel checks give the first gaps 1 and 2, the initial two-gap run is distinct, and every one-gap run is distinct; false-premise control preflights red/restatement; negation remains the exact open little-o claim; final Jig dedupe found only distinct prime-gap problems. Whole attacks used the quadratic minimum sum of distinct even gaps, global second-moment gap bounds, local sieve repetition, Cramer-model birthday heuristics, and prescribed prime-tuple refutation templates. Known arguments give only polynomial upper bounds; fixed-length prime-tuple theorems cannot construct logarithmically growing distinct runs. No full settlement or new kernel-worthy asymptotic partial survived.

Scope. The now-proved positive-power lower question is excluded. Prime gaps use zero-indexed nth primes; runs are pairwise distinct as sets of indices; the record maximizes over starts below x.

E852:Erdős Problem #852 “Are longest runs of distinct consecutive prime gaps o(log x)?”

E852: “Are longest runs of distinct consecutive prime gaps o(log x)?”