Problem 1142

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Statement. Are there infinitely many $n$ (or any $n>105$) such that $n-2^k$ is prime for all $1<2^k<n$?

Status. Open.

Source. erdosproblems.com/1142, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1142, https://www.erdosproblems.com/1142.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; A19 "Values of $n$ making $n-2^k$ prime. Odd numbers not of the form $\pm p^a\pm2^b$.", printed p. 67: Erdős's conjecture that 4, 7, 15, 21, 45, 75 and 105 are the only such $n$, the verifications to $2^{44}$ ([MiWe69]) and $2^{77}$ (Uchiyama and Yorinaga), Vaughan's bound $x\exp(-(\ln x)^c)$ ([Va73]) and the conditional bounds $O(x^c)$, $c<1$, of Hooley and Narkiewicz. Library home: guy_2004_unsolved_problems_number_theory.
• [MiWe69] Mientka, Walter E. and Weitzenkamp, Roger C., On $f$-plentiful numbers. J. Combinatorial Theory (1969), 374-377.
• [Va73] Vaughan, R. C., Some applications of Montgomery's sieve. J. Number Theory (1973), 64-79.

Formalization. Statement in formal-conjectures.

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.

• guy_2004_unsolved_problems_number_theory
• erdos_1950_integers_form_related_problems
• erdos_1950_integers_form_related_problems / conjecture_p115