Misc,
Observations on a certain theorem of Fermat and on others concerning prime numbers
(April 2008)
Abstract
E26 in the Enestrom index. Translated from the Latin original, Öbservationes
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.
