Observations on a certain theorem of Fermat and on others concerning prime numbers
| dc.creator | Euler, Leonhard | |
| dc.date | 2005-01-09 | |
| dc.date | 2008-04-15 | |
| dc.date.accessioned | 2026-07-07T09:32:31Z | |
| dc.date.available | 2026-07-07T09:32:31Z | |
| dc.description | 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. | |
| dc.description | 4 pages | |
| dc.identifier | https://arxiv.org/abs/math/0501118 | |
| dc.identifier | http://arxiv.org/abs/math/0501118 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158807 | |
| dc.subject | History and Overview | |
| dc.subject | Number Theory | |
| dc.subject | 01A50; 11A41 | |
| dc.title | Observations on a certain theorem of Fermat and on others concerning prime numbers | |
| dc.type | text |