Problem #244: Let $C>1$. Does the set of integers of the form $p+\lfloor...

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 $>0$?

SOLVED

Problem 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 $>0$?

Categories: Number Theory Primes

Progress

Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.

Ding [Di25] has proved that this is true for almost all $C>1$.

Source: erdosproblems.com/244 | Last verified: January 14, 2026

Problem Statement

Progress

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