Problem 9

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?

Source: erdosproblems.com/9

Number theory
Additive bases
Primes

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Open

No claim settles this problem.

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.
Pa11: Pan, Hao, On the integers not of the form $p+2^a+2^b$ . Acta Arith. (2011), 55-61.

Problem 9

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