is an integer, if x is an integer, and n + 1 is a prime integer which is not a factor of x.

This is not a neat proof of Fermat's theorem, but, as far as I know, it is a new proof, and it may have some little interest, as it seems very possible that the theorem may have been suggested to Fermat in this way. Fermat published the theorem without any demonstration and without indicating what had suggested it, and any proofs, that I have seen, give no indication why one should look for such a theorem, and would very possibly never have been given if the theorem had not been already known.