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Updated Oct 7

Problem 244

claims/: The 2 claim pages of Problem 244, one per claimant's result; the problem's standing derives from them.

Statement. Let $C>1$ . Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$ , have density $>0$ ?

Formulation. The question is read as Ding [Di25] reads Erdős's 1961 statement and as the formal-conjectures statement reads it: for every real $C>1$ , does the set have positive lower density? The site's commentary reads it the same way, since it counts Romanoff's lower-density theorem as a yes for integer $C$ . Romanoff's theorem and Ding's theorems are lower-density statements, and none of them shows that the natural density exists. Whether $k$ starts at $0$ or at $1$ does not matter, since the integers $p+1$ have density zero.

Status. Open, the site's label (OPEN; page last edited 28 October 2025). Romanoff's theorem [Ro34] gives positive lower density for every integer $C\ge2$ , recorded as the accepted partial claim on Romanoff's claim page; Ding's theorems [Di25], positive lower density for almost every real $C>1$ and for the golden ratio, are the pending partial claim on Ding's claim page. No claim covers every $C>1$ , so the problem stays open.

Source. erdosproblems.com/244, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #244, https://www.erdosproblems.com/244.

References.

[Di25] Y. Ding, On a Romanoff type problem of Erdős and Kalmár. arXiv:2503.22700 (2025); the record's current version, of 10 September 2026, is titled On two Romanoff type problems of Erdős. Library home: ding_2025_two_romanoff_type_problems_erdos.
[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678. Library home: romanoff_1934_uber_einige_satze_der_additiven.

Formalization. Statement in formal-conjectures.

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Linked library material

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Linked from (9)

Romanoff: positive lower density for every integer base problems/primes/E0244/claims Primes additive_bases/romanoff_1934_uber_einige_satze_der_additiven Satz II (p. 668): the integers that are a prime plus a power of a have positive lower density Primes primes/ding_2025_two_romanoff_type_problems_erdos primes/erdos_1950_integers_form_related_problems Theorem 4 (p. 114): for a divisor chain a_1 | a_2 | ..., the integers p + a_k have positive density exactly when (4) and (5) hold

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