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.
%0 Generic
%1 citeulike:3036301
%A Euler, Leonhard
%D 2008
%K Vor1750 available-in-tex-format mathematics number-theory pre1750
%T Observations on a certain theorem of Fermat and on others concerning prime numbers
%U http://arxiv.org/abs/math/0501118
%X 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.
@misc{citeulike:3036301,
abstract = {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.},
added-at = {2009-08-02T17:14:35.000+0200},
archiveprefix = {arXiv},
author = {Euler, Leonhard},
biburl = {https://www.bibsonomy.org/bibtex/2a71aa1fe15f39415c36c2b3c88e1f0fd/rwst},
citeulike-article-id = {3036301},
citeulike-linkout-0 = {http://arxiv.org/abs/math/0501118},
citeulike-linkout-1 = {http://arxiv.org/pdf/math/0501118},
description = {my bookmarks from citeulike},
eprint = {math/0501118},
interhash = {a7e105b8bd2b28be68f68e5f793bfdbb},
intrahash = {a71aa1fe15f39415c36c2b3c88e1f0fd},
keywords = {Vor1750 available-in-tex-format mathematics number-theory pre1750},
month = Apr,
posted-at = {2008-07-23 08:57:19},
priority = {2},
timestamp = {2009-08-06T10:28:48.000+0200},
title = {Observations on a certain theorem of Fermat and on others concerning prime numbers},
url = {http://arxiv.org/abs/math/0501118},
year = 2008
}