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Wieferich’s criterion and the abc-conjecture. (English) Zbl 0654.10019

The following result is proved. The so called “abc-conjecture” of Masser and Oesterlé implies that the number of primes less than \(X\) for which \(\alpha^{p-1}\not\equiv 1 \pmod{p^2}\) where \(\alpha\) is a fixed rational number and \(\alpha \neq \pm 1,0\), is at least \(O(\log X)\). An analogous result is also proved for points of infinite order on elliptic curves having certain \(j\)-invariants. The proofs base on several skillful lemmas.

MSC:

11D41 Higher degree equations; Fermat’s equation
14H52 Elliptic curves
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Online Encyclopedia of Integer Sequences:

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

References:

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