Observations on a certain theorem of Fermat and on others concerning prime numbers

Details Observations on a certain theorem of Fermat and on others concerning prime numbers Observations on a certain theorem of Fermat and on others concerning prime numbers

2005-01-09

arxiv.org/abs/math/0501118v3

Mathematics

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4 pages

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E26 in the Enestrom index. Translated from the Latin original, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus" (1732). In this paper Euler gives a counterexample to Fermat's claim that all numbers of the form 2^{2^m}+1 are primes, by showing 2^{2^5}+1=4294967297 is divisible by 641. He also considers many cases in which we are guaranteed that a number is composite, but he notes clearly that it is not possible to have a full list of circumstances under which a number is composite. He then gives a theorem and several corollaries of it, but he says that he does not have a proof, although he is sure of the truth of them. The main theorem is that a^n-b^n is always able to be divided by n+1 if n+1 is a prime number and both a and b cannot be divided by it.

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