The proof for $$a^p \equiv a \pmod p\;,$$
where $p$ is a prime number, is pretty straightforward. But as was characteristic of Fermat, he never provided a proof (not that I know of), from what I know, Euler was the first to publish the proof. Even though Leibniz had a proof long before him, which went unpublished. Does anyone know what this was, even though it's basically the same proof? Maybe a link to a document of it?
Many authors (e.g. Weil) refer to D. Mahnke (1912) for Lebniz's unpublished work on number theory. Purportedly an unpublished manuscript of Leibniz from 12 September 1680 (and also Leibniz 1697) contains the first proof of Fermat's little theorem; see Mahnke (1912/13), page 38, Notes 4.4, Vacca (1894), Tropfke (1902), page 62, and Dickson (1919), Chapter III, pages 59-60. Mahnke considers it likely that Leibniz found the statement of the theorem himself, but it cannot be completely ruled out that he had already read Fermat's Varia Opera, published in 1679. [from von zur Gathen, Modern computer algebra]
D. Mahnke, Leibniz auf der Suche nach einer allgemeinen Primzahlgleichung,
Bibliotheca mathematica. Zeitschrift für Geschichte der Mathematischen Wissenschaften, 13, 1912, 29-61
Giovanni Vacca. Intorno alia prima dimostrazione di un teorema di Fermat.
Bibliotheca Mathematica, Serie 2, 8, 1894, 46-48.
Johannes Tropfke. Geschichte der Elementar-mathematik in systematischer Darstellung (1902).