If $p$ is a prime and $k,m\ge 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\dots ,r+m-1$ are all $k$th power residues modulo $p$. Let$$\Lambda (k,m)=\underset{p\to \infty }{\mathrm{limsup}}r(k,m,p).$$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?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you call (\Lambda(k,m)) is exactly the function $A(k,m)$ introduced by D. H. Lehmer and Emma Lehmer [[nomath]](they define $r(k,m,p)$ for “sufficiently large” primes $p$ and then take $\limsup$)[[/nomath]]. ([UMD Computer Science][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.