Non-Wieferich primes in arithmetic progressions

Abstract:

Graves and Murty proved that for any integer $a\ge 2$ and any fixed integer $k\ge 2$, there are $\gg \log x/\log \log x$ primes $p\le x$ such that $a^{p-1}\not \equiv 1\pmod {p^2}$ and $p\equiv 1\pmod k$, under the assumption of the abc conjecture. In this paper, for any fixed $M$, the bound $\log x/\log \log x$ is improved to $(\log x/\log \log x) (\log \log \log x)^M$.