  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.
