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

Updated Oct 7

Problem 17

Statement. 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 $q_1,q_2\leq p$ ?

Status. Open.

Source. erdosproblems.com/17, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #17, https://www.erdosproblems.com/17.

References.

[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_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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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 (9)

Primes number_theory/guy_2004_unsolved_problems_number_theory Primes primes/blecksmith_1999_cluster_primes Conjecture (p. 45): a bound x / e^{α (log log x)^2} for the cluster-prime count Definition (p. 43): cluster primes, with the question whether there are infinitely many Theorem 1 (p. 44): fewer than x/(log x)^s cluster primes up to x, for each fixed s Theorem 2 (p. 45): the reciprocals of the cluster primes have a finite sum primes/elsholtz_2003_cluster_primes

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