Erdős Problem #445

Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?

#445: [ErGr80,p.89]

number theory

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Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$.

This is discussed in this MathOverflow question.

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This page was last edited 27 December 2025.

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