Problem Statement
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 $ab\equiv 1\pmod{p}$?
Categories:
Number Theory
Progress
Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$.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.
Source: erdosproblems.com/445 | Last verified: January 15, 2026