Problem 17

Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p$ ?

Source: erdosproblems.com/17

Number theory
Primes

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No claim settles this problem.

BES99: Blecksmith, Richard and Erdős, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43-48.
El03: Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281-284.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A8 "Gaps between primes. Twin primes.", printed p. 36: the cluster primes, 97 the smallest non-cluster prime, and Blecksmith and Selfridge's question whether there are infinitely many; the same question is the second definition of a good prime in A14 "“Good” primes and the prime number graph", printed p. 54. Library home: Guy 2004.

Problem 17

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