On the divisibility of Fermat quotients
For a prime \(p\) and an integer \(a\) the Fermat quotient is defined as \[ q_p(a)=\frac{a^{p-1}-1}{p}. \] Let \(l_p\) be the smallest value of \(a\) for which \(q_p(a)\not\equiv 0\pmod p\). The authors prove that \[ l_p\leq (\log p)^{\frac{463}{252}+o(1)} \] as \(p\to\infty\). As a corollary, for every \(\varepsilon>0\) and a sufficiently large integer \(n\), if \(a^{n-1}\equiv 1\pmod n\) for every positive integer \(a\leq (\log n)^{\frac{463}{252}+\varepsilon}\) then \(n\) is squarefree. Furthermore, for every \(\varepsilon>0\), there is \(\delta>0\) such that for all but one prime \(Q^{1-\delta} <p\leq Q\), we have \(l_p\leq (\log p)^{\frac{59}{35}+\varepsilon}\). Finally, for every \(\varepsilon>0\), there is \(\delta>0\) such that for all but \(O(Q^{1-\delta})\) primes \(p\leq Q\), we have \(l_p\leq (\log p)^{\frac{5}{3}+\varepsilon}\).
scientific article; zbMATH DE number 5792255
scientific article