next up previous contents
Next: Prime Numbers Up: Fermat's Last Theorem Previous: Related Conjectures

Did Fermat prove this theorem?

No he did not. Fermat claimed to have found a proof of the theorem at an early stage in his career. Much later he spent time and effort proving the cases n=4 and n=5. Had he had a proof to his theorem earlier, there would have been no need for him to study specific cases.

Fermat may have had one of the following ``proofs'' in mind when he wrote his famous comment.

Another argument for the belief that Fermat had no proof --and, furthermore, that he knew that he had no proof-- is that the only place he ever mentioned the result was in that marginal comment in Bachet's Diophantus. If he really thought he had a proof, he would have announced the result publicly, or challenged some English mathematician to prove it. It is likely that he found the flaw in his own proof before he had a chance to announce the result, and never bothered to erase the marginal comment because it never occurred to him that anyone would see it there.

Some other famous mathematicians have speculated on this question. Andre Weil, writes:

Only on one ill-fated occasion did Fermat ever mention a curve of higher genus , and then hardly remains any doubt that this was due to some misapprehension on his part [ ] for a brief moment perhaps [ ] he must have deluded himself into thinking he had the principle of a general proof.

Winfried Scharlau and Hans Opolka report:

Whether Fermat knew a proof or not has been the subject of many speculations. The truth seems obvious [Fermat's marginal note] was made at the time of his first letters concerning number theory [1637] as far as we know he never repeated his general remark, but repeatedly made the statement for the cases n=3 and 4 and posed these cases as problems to his correspondents [ ] he formulated the case n=3 in a letter to Carcavi in 1659 [ ] All these facts indicate that Fermat quickly became aware of the incompleteness of the [general] ``proof" of 1637. Of course, there was no reason for a public retraction of his privately made conjecture.

However it is important to keep in mind that Fermat's ``proof" predates the Publish or Perish period of scientific research in which we are still living.


From Fermat to Minkowski: lectures on the theory of numbers and its historical development. Winfried Scharlau, Hans Opolka. New York, Springer, 1985.

Basic Number Theory. Andre Weil. Berlin, Springer, 1967

next up previous contents
Next: Prime Numbers Up: Fermat's Last Theorem Previous: Related Conjectures

Alex Lopez-Ortiz
Mon Feb 23 16:26:48 EST 1998