Problem 141

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

Source: erdosproblems.com/141

Additive combinatorics
Primes
Arithmetic progressions

Wiki page

Status

Open

No claim settles this problem.

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.

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.
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.

Problem 141

Status

Proof claims

Discussion