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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