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Updated Oct 7

Problem 236

Statement. Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$ . Is it true that $f(n)=o(\log n)$ ?

Status. Open.

Source. erdosproblems.com/236, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #236, https://www.erdosproblems.com/236.

References.

[Er50] Erdős, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (5)

Primes primes/chen_2023_conjecture_erdos_p_2_k primes/erdos_1950_integers_form_related_problems Conjectures, p. 115: f(n) = o(log n), 105 as the largest n with every n - 2^k prime, and a generalization of Theorem 1 Theorem 2 (p. 113): every moment of f(n), the number of representations n = 2^k + p, is bounded on average

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