Erdős Problem #436

declared status 'open'. Formalized: no. If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let$$\Lambda(k,m)=\limsup_{p\to \infty} r(k,m,p).$$Is it true that $\Lambda(k,2)$ is finite for all $k$? Is $\Lambda(k,3)$ finite for all odd $k$? How large are they? Current best: Lehmer and Lehmer proved that $\Lambda(k,3)=\infty$ for all even $k$ and $\Lambda(k,4)=\infty$ for all $k\leq 1048909$. Graham [Gr64g] proved that $\Lambda(k,l)=\infty$ for all $k\geq 2$ and $l\geq 4$. Hildebrand [Hi91] resolved the first question, proving that $\Lambda(k,2)$ is finite for all $k$: in other words, for any $k\geq 2$, if $p$ is sufficiently large then there exists a pair of consecutive $k$th power residues modulo $p$ in $[1,O_k(1)]$. The remaining questions are to examine whether $\Lambda(k,3)$ is finite for all odd $k\geq 5$, and the growth rate of $\Lambda(k,2)$ and $\Lambda(k,3)$ as functions of $k$. Prize: no. OEIS: A000445. Tags: number theory.

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