Problem 9

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Statement. Let $A$ be the set of all odd integers $\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?

Status. Open.

Source. erdosproblems.com/9, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #9, https://www.erdosproblems.com/9.

References.

• [Cr71] Crocker, Roger, On the sum of a prime and of two powers of two. Pacific J. Math. (1971), 103-107.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section A19 "Values of $n$ making $n-2^k$ prime. Odd numbers not of the form $\pm p^a\pm2^b$.", p. 67: Crocker's infinitely many odd integers not of the form $2^k+2^l+p$ and the question whether there are $cx$ of them below $x$. Library home: guy_2004_unsolved_problems_number_theory.
• [Pa11] Pan, Hao, On the integers not of the form $p+2^a+2^b$. Acta Arith. (2011), 55-61.

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.

• crocker_1971_sum_prime_two_powers_two
• crocker_1971_sum_prime_two_powers_two / lemma_ii
• crocker_1971_sum_prime_two_powers_two / theorem_i
• pan_2011_integers_not_form
• guy_2004_unsolved_problems_number_theory