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The abc conjecture and non-Wieferich primes in arithmetic progressions. (English) Zbl 1272.11014

If a prime \(p\) satisfies the congruence \(2^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime and if it satisfies the congruence \(a^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime for base \(a\). As was noted by the authors it can be shown by elementary tools that under the abc conjecture there are infinitely many non-Wieferich primes. In this paper the authors show that under the abc conjecture the number of non-Wieferich primes is \(\ll \frac{\log x}{\log\log x}\).

MSC:

11A41 Primes
11N13 Primes in congruence classes

Citations:

Zbl 1269.11007
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Online Encyclopedia of Integer Sequences:

Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.