We show that if the first case of Fermat's Last Theorem is false for prime exponent $p$ then $p^2$ divides $q^p-q$ for all primes $q\leq 89$. The title theorem follows.

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Variations on Sophie Germain's Theorem

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We prove that for any odd $n>1$ and sufficiently large $m$ there exists a nested Steiner $n$-cycle system of order $m$ if and only of $m\equiv 1 \pmod {2n}$.

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For which integers $s,t, n=2st$ does there exist a decomposition of the complete graph on $n$ vertices into $n-1$ copies of the complete $(s,t)$-bipartite graph?

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