Do $k$ consecutive primes in arithmetic progression exist for every $k$? (Erdős #141)
Statement
Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression? Here 'consecutive' means consecutive in the ordered sequence of all primes: primes $p_{n+1}<p_{n+2}<\cdots<p_{n+k}$ with no other prime between $p_{n+1}$ and $p_{n+k}$, such that $p_{n+i+1}-p_{n+i}$ is the same for all $1\leq i<k$.
Acceptance. FULLY RESOLVES: a proof that for every $k\geq 3$ there exist $k$ consecutive primes in arithmetic progression, or a proof that some explicit $k$ admits none — machine-checkable (Lean/Coq) preferred, otherwise a full written proof. ADVANCES: (a) a first explicit example of 11 (or more) consecutive primes in arithmetic progression — deliver the primes, primality certificates for each, and a machine-checkable certificate that no other prime lies strictly between the first and last (consecutiveness); (b) a proof, possibly conditional on a clearly flagged standard conjecture (e.g. Dickson/Hardy–Littlewood), that 3 consecutive primes in AP occur infinitely often; (c) verified extension or minimality certification of the OEIS A006560 record values (smallest starting primes) with reproducible search code. Deliver the witness plus certificates, or the proof file.
Background
Posed by Erdős [Er75b, Er83, Er97c], who called the conjecture 'completely hopeless at present'; listed as open on erdosproblems.com/141 (fetched 2026-07-13, status 'open', tagged 'additive combinatorics | primes | arithmetic progressions'), and discussed as problem A6 of Guy's collection [Gu04]. The Green–Tao theorem [GrTa08] guarantees $k$ primes in arithmetic progression for every $k$, but says nothing about consecutiveness. Explicit examples are known for every $k\leq 10$ (see the Wikipedia page on primes in arithmetic progression, 'consecutive primes in AP'); no example of 11 consecutive primes in AP is known, and even the infinitude of 3 consecutive primes in AP is open. OEIS A006560 records the smallest starting prime of $n$ consecutive primes in arithmetic progression. A Lean formalisation of the statement exists in the formal-conjectures repository. The attacker's tools: large-scale sieving plus primality certification to hunt for the first run of 11 consecutive primes in AP (a heavy search — existing $k=10$ examples came from substantial distributed computations), verification/extension of minimal starting primes for small $k$, and conditional arguments under Hardy–Littlewood/Dickson-type prime-tuple heuristics.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #141 (T. F. Bloom) | website |
| REF-02 | OEIS A006560 — smallest starting prime of n consecutive primes in arithmetic progression | website |
| REF-03 | Wikipedia — consecutive primes in arithmetic progression (records for k ≤ 10) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.