Erdős Problem #979

Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to\[n=p_1^k+\cdots+p_k^k,\]where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n)=\infty$?

#979: [Er65b,p.224]

number theory

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Erdős [Er37b] proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished).

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This page was last edited 19 September 2025.

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Formalised statement? Yes
Related OEIS sequences: A385316 possible

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