THEOREM OF THE DAY: Theorem (Fermat’s Little Theorem)

Theorem

If p is a prime number, then $$a^{p −1} ≡ 1 (mod p )$$ for any positive integer a not divisible by p.

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Fermat announced this result in 1640, in a letter to a fellow civil servant Frenicle de Bessy. As with his ‘Last Theorem’ he ´ claimed that he had a proof but that it was too long to supply. In this case, however, the challenge was more tractable: Leonhard Euler supplied a proof almost 100 years later which, as a matter of fact, echoed one in an unpublished manuscript of Gottfried Wilhelm von Leibniz, dating from around 1680