Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 6

claims/: The 1 claim page of Problem 6, one per claimant's result; the problem's standing derives from them.

Statement. Let $d_n=p_{n+1}-p_n$ . Are there infinitely many $n$ such that $d_n<d_{n+1}<d_{n+2}$ ?

Status. PROVED (LEAN), the site's label. The accepted claim is the 2013 theorem of Banks, Freiberg and Turnage-Butterbaugh, refereed and credited by the site's curator. The site links no Lean proof; the one found, a third-party formalization, is recorded on the claim page and in the Formalization paragraph below, and this corpus has built none.

Source. erdosproblems.com/6, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #6, https://www.erdosproblems.com/6.

References.

[BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.
[Er85c] Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.
[ErTu48] Erdős, P. and Turán, P., On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc. 54 (1948), 371-378.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A11 "Increasing and decreasing gaps", printed p. 43: Erdős and Turán's results on $d_n>d_{n+1}$ , "but it is not known if there are infinitely many decreasing or increasing sets of three consecutive values of $d_n$ ", with a prize offer. Library home: guy_2004_unsolved_problems_number_theory.
[Ma15] Maynard, James, Small gaps between primes. Ann. of Math. (2) 181 (2015), 383-413.

Formalization. Statement in formal-conjectures, at the linked commit: a statement with sorry marked research solved, with the general increasing and decreasing runs as variants. Boris Alexeev's lean-proofs repository holds a file declaring itself a formalization of the Banks, Freiberg and Turnage-Butterbaugh proof, recorded on the claim page; this corpus has not built it.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

Linked from (15)

Banks, Freiberg and Turnage-Butterbaugh: increasing runs of prime gaps problems/primes/E0006/claims Primes integer_sequences/banks_2014_consecutive_primes_tuples Corollary 1 (p. 2): infinitely many increasing and decreasing runs of m consecutive prime gaps Theorem 1 (p. 2): an admissible tuple of shifts infinitely often holds m consecutive primes number_theory/guy_2004_unsolved_problems_number_theory Primes primes/erdos_1948_new_questions_distribution_prime_numbers Lemma (p. 372): infinitely often a small prime gap is followed by a larger one, and infinitely often a larger gap by a small one Closing question (1) (p. 378): for every fixed k, are there infinitely many runs of k consecutive strictly increasing prime gaps?Theorem 1 (p. 372): for every t the t-th power mean of p_{n-1} and p_{n+1} exceeds p_n infinitely often and falls below it infinitely often primes/maynard_2015_small_gaps_between_primes Proposition 4.2: level of distribution theta gives ceil(theta M_k / 2) primes among n + h_i infinitely often Theorem 1.1: liminf (p_{n+m} - p_n) << m^3 e^{4m} for every m

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