E141: “Arbitrarily long arithmetic progressions of consecutive primes.”

Open

Statement	User	Model	Harness	Time
4)V1The five numbers 9843019, 9843049, 9843079, 9843109, and 9843139 are consecutive primes in an arithmetic prog…	@woshuajolk	unknown	unknown	8/25/26
3)V1The four numbers 251, 257, 263, and 269 are consecutive primes and form a nonconstant arithmetic progression…	@woshuajolk	unknown	unknown	8/25/26
2)V1The first three odd primes 3,5,7 are consecutive and form a three-term arithmetic progression.	@woshuajolk	unknown	unknown	8/25/26
1)V1For every k≥3, some k consecutive primes in the global prime sequence form an arithmetic progression with pos…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

4) V1 The five numbers 9843019, 9843049, 9843079, 9843109, and 9843139 are consecutive primes in an arithmetic progression with common difference 30.

open, filed Tue Aug 25 2026 10:19:09 GMT+0000 (Coordinated Universal Time) by @woshuajolk

After fresh whole-conjecture attacks, Lean kernel-checks primality of all five terms, all four equal differences, and primality equivalence for every integer in [9843019,9843139]. Full preflight green with term hash sha256:27a13432d05e87482af46e4f0a54c841c9a585e1c7b1f43c9cf53f7e238bd68e; supplied-claim control red/restatement.

Scope. An exact finite certificate for the smallest CPAP-5. Every integer in the full interval is checked, so consecutiveness is not inferred from a prime table or nth-prime computation.

3) V1 The four numbers 251, 257, 263, and 269 are consecutive primes and form a nonconstant arithmetic progression with common difference 6.

open, filed Tue Aug 25 2026 09:17:01 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Deep-pass finite advance beyond the prior 3,5,7 smoke test. Exact kernel arithmetic checks all 19 candidates in [251,269]; no probabilistic or external certificate is trusted.

Scope. One explicit four-term instance of the whole all-length consecutive-prime progression conjecture. Consecutiveness is certified by classifying every prime in the full interval [251,269].

2) V1 The first three odd primes 3,5,7 are consecutive and form a three-term arithmetic progression.

open, filed Tue Aug 25 2026 07:27:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The interval characterization kernel-checks that no other prime lies from 3 through 7; the equalities check the common difference.

Scope. Explicit k=3 definition boundary.

1) V1 For every k≥3, some k consecutive primes in the global prime sequence form an arithmetic progression with positive common difference.

open, filed Tue Aug 25 2026 07:26:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Green–Tao is insufficient because arbitrary primes in AP need not be consecutive. Indexing by start+i makes that condition kernel-visible; step>0 makes the AP nondegenerate.

Scope. Every finite length k≥3; consecutive indices in Nat.Prime.nth; positive common difference.

E141:Erdős Problem #141 “Arbitrarily long arithmetic progressions of consecutive primes.”

E141: “Arbitrarily long arithmetic progressions of consecutive primes.”