Problem #436: If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be...

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Progress

Asked by Lehmer and Lehmer [LeLe62], who note that for example $\Lambda(2,2)=9$ - indeed, $9$ is always a quadratic residue, and if $10$ isn't then either $2$ or $5$ is, and hence at least one of $1,2$ or $4,5$ or $9,10$ is a consecutive pair of quadratic residues (and similarly there are infinitely many $p$ for which there are no consecutive quadratic residues below $9,10$).

A similar argument of Dunton [Du65] proves $\Lambda(3,2)=77$, and Bierstedt and Mills [BiMi63] proved $\Lambda(4,2)=1224$. Lehmer and Lehmer proved that $\Lambda(k,3)=\infty$ for all even $k$ and $\Lambda(k,4)=\infty$ for all $k\leq 1048909$.

Lehmer, Lehmer, and Mills [LLM63] proved $\Lambda(5,2)=7888$ and $\Lambda(6,2)=202124$. Brillhart, Lehmer, and Lehmer [BLL64] proved $\Lambda(7,2)=1649375$. Lehmer, Lehmer, Mills, and Selfridge [LLMS62] proved that $\Lambda(3,3)=23532$.

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 $k$.