Problem 6

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

Number theory
Primes

Wiki page

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SolvedProved

An accepted solution exists. The statement is true.

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.

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.
Ma15: Maynard, James, Small gaps between primes. Ann. of Math. (2) 181 (2015), 383-413.

Problem 6

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