Problem 453

Is it true that, for all sufficiently large $n$ , there exists some $i<n$ such that

p_n^2 < p_{n+i}p_{n-i},

where $p_k$ is the $k$ th prime?

Source: erdosproblems.com/453

Number theory

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved, the site's label, with a Lean suffix that is a catalog label: the Lean file announced in the thread is linked from the claim page as a formalization of Pomerance's proof, neither built nor audited here, and this page records the statement file of formal-conjectures, which points at that Lean file. The disproof is Pomerance's 1979 corollary that infinitely many $n$ have $p_n^2>p_{n-i}p_{n+i}$ for all $0<i<n$ , recorded on the claim page Pomerance's infinitely many good primes (accepted; refereed in Mathematics of Computation, and the result the site's commentary names).

Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115.
Gu04: Guy, Richard K., Unsolved problems in number theory, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. A14 '"Good" primes and the prime number graph', printed p. 54: Erdős and Straus call $p_n$ good if $p_n^2>p_{n-i}p_{n+i}$ for all $1\le i\le n-1$ , and Pomerance used the prime number graph to show that there are infinitely many good primes, which is the negative answer; no proofs. Library home: Guy 2004.
Po79: Pomerance, Carl, The prime number graph. Math. Comp. 33 (1979), no. 145, 399--408, DOI 10.1090/S0025-5718-1979-0514836-7.

Problem 453

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