Andrew Granville's Home Page

1986 Publications

Powerful numbers and Fermat's Last Theorem
Comptes Rendus de l'Académie des Sciences du Canada, 8 (1986), 215-218

We show that if there are no three consecutive powerful numbers then there is an infinite sequence of primes \( p\) for which \( p^2\) does not divide \( 2^p-2\), and so the first case of Fermat's Last Theorem is true for exponent \(p\).

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We show that if the first case of Fermat's Last Theorem is false for prime \( p\) then \( p\) divides the numerator of \( B_{p-1-n}\) for all integers \( n\leq (\log p/\log\log p)^{1/2}\).

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