Problem 141

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claims/: The 2 claim pages of Problem 141, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?

Status. Open. The site labels the problem OPEN (page last edited 28 September 2025); its commentary reports such progressions for every $k\le10$ and notes that whether infinitely many exist is open even for three terms. The instances $3\le k\le10$ are the accepted partial claim on the Dubner et al. page, ten consecutive primes in arithmetic progression; every $k$ follows from Hypothesis H on the Schinzel and Sierpiński page, a conditional claim. That commentary reports progressions only for $k\le10$, and no claim page settles $k\ge11$, so the derived standing is open.

Source. erdosproblems.com/141, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #141, https://www.erdosproblems.com/141.

References.

• [DFLMNZ02] Dubner, H. and Forbes, T. and Lygeros, N. and Mizony, M. and Nelson, H. and Zimmermann, P., Ten consecutive primes in arithmetic progression. Math. Comp. 71 (2002), no. 239, 1323-1328, doi:10.1090/S0025-5718-01-01374-6; published online 28 November 2001. Not a site key; the paper behind the site's remark that the progressions have been found for $k\le10$.
• [GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), no. 2, 481-547, doi:10.4007/annals.2008.167.481.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section A6 "Consecutive primes in A.P.", printed p. 28: "It has even been conjectured that there are arbitrarily long arithmetic progressions of consecutive primes, such as 251, 257, 263, 269 and 1741, 1747, 1753, 1759", the five- and six-term progressions of Jones, Lal and Blundon and of Lander and Parkin, and the seven consecutive primes with common difference 210 found by Dubner and Nelson in 1995. Library home: guy_2004_unsolved_problems_number_theory.
• [ScSi58] Schinzel, A. and Sierpiński, W., Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4 (1958), no. 3, 185-208, doi:10.4064/aa-4-3-185-208. Not a site key; the paper that derives arbitrarily long progressions of consecutive primes from Hypothesis H.

Formalization. Statement in formal-conjectures (pinned at the file's last commit, of 2026-09-18), which tags erdos_141 research open, its variant first_cases (every $3\le k\le10$) research solved with a sorry proof and no formal_proof attribute, and the variants for $k=11$ and for infinitely many progressions research open. The solved tag is a statement, not a Lean proof, and gives no formalized evidence.

Current assessment

The site records Problem 141 as OPEN (page last edited 28 September 2025). Its commentary notes that Green and Tao [GrTa08] give $k$ primes in arithmetic progression for every $k$, but not consecutive ones, that Erdős judged the problem out of reach, that such progressions are known for every $k\le10$, and that whether infinitely many exist is open even for three terms. The computational record is the ten consecutive primes in arithmetic progression of 1998, reported in [DFLMNZ02], which settle $3\le k\le10$; a progression of eleven consecutive primes needs a common difference divisible by $2310$, and the site's commentary (page last edited 28 September 2025) reports none. Under Hypothesis H, Schinzel and Sierpiński [ScSi58] derive infinitely many progressions of $n$ consecutive primes for every $n$ and every admissible common difference, so the conjecture is expected to hold for every $k$. The two claims are checked against the papers' published records only (the Math. Comp. abstract; the Acta Arithmetica paper's consequences $C_1$ and $C_{1.4}$); the computations and the derivation are not rechecked, and no literature search goes beyond the sources named here.

Known Results

• Dubner et al. 2002: ten consecutive primes in arithmetic progression, answering the question yes for $3\le k\le10$; accepted on refereed publication.
• Schinzel and Sierpiński 1958: arbitrarily long progressions of consecutive primes, infinitely many for each length, under Hypothesis H; a conditional claim.
• Guy's section A6 [Gu04] records the earlier five-, six- and seven-term finds and the conjecture that the progressions are arbitrarily long.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• green_2008_primes_contain_arbitrarily_long_arithmetic_progressions
• green_2008_primes_contain_arbitrarily_long_arithmetic_progressions / theorem_1_1
• guy_2004_unsolved_problems_number_theory