Fermat's Fallibility

Fermat discussed the proposition that "every number of the form  is a prime" in several letters to several different people (including Frenicle, Pascal, Huygens, Brouckner, Wallis, etc) over a period of many years. In most cases he carefully pointed out that he had no proof. For example, (from Ore's "Number Theory and Its History") in August 1640 he wrote

Je n'en ai pas la demonstration exacte, mais j'ai exclu si grande quantite' de diviseurs par demonstrations infaillibles, et j'ai de si grandes lumieres, qui etablissent ma pensee que j'aurois peine a' me dedire.  [I do not have the exact demonstration of it, but I have excluded so many divisors by infallible demonstrations, and it seems so intuitively clear to me, that I would be very reluctant to recant.]

In "A Source Book In Mathematics", Struik writes

...in August 1640, in a letter to Frenicle, Fermat had turned to numbers of the form 2ⁿ + 1, writing that he was "almost convinced" [quasi persuade'] that these numbers are prime when n is a power of 2.

Here is what A. Weil says on this subject in his book "Number Theory, an Approach Through History"

Writing to Frenicle in 1640, Fermat enumerates such numbers up to n = 6, and then conjectures that all are prime. It is hard to believe that he did not try to apply, at least to the sixth one 2³² + 1, the method he had used to factorize 2³⁷ - 1; it shows that any prime divisor of 2³² + 1 must be of the form 64k + 1, which leaves the candidates 193, 257, 449, 577, 641, etc.; 641 divides 2³² + 1; in fact this is how Euler proceeded... What is even more surprising is that Frenicle, who had also factorized 2³⁷ - 1, did not at once point out the error, as (judging from the general tone of their correspondence) he would have been only too pleased to do; on the contrary, Frenicle expressed agreement. Fermat persisted in his conjecture to the end of his days, usually adding that he had no full proof for it (cf Fe II, 309-310.)

In Dickson's "History of the Theory of Numbers" we find

Fermat expressed his belief that every F_n is a prime, but admitted that he had no proof. Elsewhere he said that he regarded the theorem as certain. Later he implied that it may be proved by descent. It appears that Frenicle de Bessy confirmed this conjectured theorem of Fermat's. On several occasions Fermat requested Frenicle to divulge his proof, promising important applications.

Of all his letters on number theory, the last was apparently the 1659 letter to Carcavi in which Fermat "implied" the primeness of all F_n could be proved by descent, although whether he was referring to his own proof or to Frenicle's claimed (but undivulged) proof is unclear. Here's the text of the letter, quoted from Mahoney's "The Mathematical Career of Pierre de Fermat"

J'ai ensuite considere certaines questions qui bien que negatives, ne restent pas de recovoir tres grande difficulte, la method pour y pratiquer la descente etant tout a fait diverse des precedentes, comme il sera aise d'eprouver. Telles sont les suivants: Il n'y a aucun cube divisible en deux cubes. Il n'y qu'un seul quarre en entiers, qui augmente du binaire, fasse un cube. Le dit quarre' est 25. Il n'y a que deux quarres en entiers, lesquels, augmentes de 4, fassent un cube. Les dits quarres sont 4 et 121. Toutes les puissances quarrees de 2, augmentees de l'unite, sont nombres premiers. Cette derniere question est d'une tres subtile et tres ingenieuse recherche et, bien qu'elle soit concue affirmativement, elle est negative, puisque dire qu'un nombre est premier, c'est dire qu'il ne peut etre divise par aucun nombre.

Very roughly translated, this says

I have then considered certain questions which, although negative, are nevertheless of great difficulty, the method of applying the descent in these cases being completely different from the preceding cases, as is easy to see. Among these are the following: There is no cube equal to a sum of two cubes. There is only one square which, increased by two, equals a cube, namely, the square 25. There are only two squares which, increased by 4, make a cube, namely, the squares 4 and 121. All the square(?) powers of 2, increased by one, are prime numbers. This last proposition results from a very subtle and ingenious research and, although it is expressed in the affirmative, it is negative, since it says asserts that certain numbers can be divided by no number.

So he included the proposition that "toutes les puissances quarres de 2, augmentees de l'unite, sont nombres premiers" as the last item on a list of things that can be proven by his method of descent, and says "this last problem results from very subtle and very ingenious research...", but frankly, I have trouble with this whole passage. Doesn't "toutes les puissances quarres de 2" mean "all square powers of 2"? If so, then Fermat would be claiming that all numbers of the form 2^(n^2) + 1 are prime...which makes no sense at all. And yet even in Mahoney's biography (page 350) we find this proposition translated as "all square powers of 2 increased by 1 are prime". Perhaps this is why Bell referred to this as "an obscure statement". Is it possible that "puissances quarres de 2" was intended to mean something like "iterated powers of two"?

In any case, it's worth noting that Fermat first worked on this problem in 1640, and as late as June 1658 he was admitting to Brouckner and Wallis that he had no proof (according to Dickson's references). Is it likely that, after working on the problem for 18 years, Fermat finally convinced himself some time between June 1658 and August 1659 that he had found a proof? It seems more probable that he just momentarily abandoned his usual caution in the letter to Carcavi. This would accord with Weil's statement that, to the end of his life, Fermat "usually" (but not always) acknowledged that he had no proof. On balance, I think it's fair to say that Fermat believed every F_n is prime, but he repeatedly stated over a period of 18 years that he was not able to rigorously prove it. He did, however, on at least one occasion state the proposition as a proven, or at least provable, result, but bear in mind he was trying to interest people in the power of his method of descent, so it's perhaps not surprising that he overstated his results in that particular letter.

Incidentally, another example of Fermat in error is described in Weil's "Number Theory". After proving every prime p has a unique minimal representation of the form 2b² – a² with 0 < a < b, Fermat adds off-handedly [in a letter to Frenicle] that the same can be proved for composite numbers, which is false.

One of the reasons some people have always been interested in Fermat’s fallibility is for the light it sheds on his most famous claim, i.e., the impossibility of integer solutions to xⁿ + yⁿ = zⁿ for n greater than 2. Here is Fermat's famous "marginal note" in his copy of Diophantus, describing what became known as his Last Theorem:

It is impossible to divide a cube into two cubes, a fourth power into two fourth powers, and in general any power except the square into two powers with the same exponents,...I have discovered a truly wonderful proof of this, but the margin is too narrow to hold it.

This note appeared in Fermat's edited works, published by Fermat's son Samuel in 1670. It's worth mentioning that this manner of announcing results (stating the result, and that he had a proof of it, but that he lacked the time or the paper or whatever to actually spell out the proof) was frequently used by the senior Fermat. As a result, Euler and others had to reconstruct the proofs for most of Fermat's claimed theorems more than a century later.

It is sometimes said that Fermat's marginal comments were merely intended as private notes to himself, and that he never imagined they would be published, but this is not really correct. It's true that the senior Fermat never got around to publishing his comments on Diophantus, but it's misleading to call it a private notebook. It was the style of his day to write (and publish) commentaries and observations on ancient works. Typically the original text was printed with wide margins in which the modern commentator would expound on the subject and, if possible, show that he was even smarter than the revered ancient author. The impulse to "outdo the ancients" was a major psychological factor driving the scientific revolution in the West during the 16th and 17th centuries.

With this in mind, it's clear that Fermat's notes were in the form of a commentary to be published. Obviously no one would write a note to himself to inform himself that he had discovered a truly wonderful proof, but that unfortunately he didn't have room to tell himself what the proof was. This is somewhat reminiscent of Galois' comments scribbled in the margins of one of his papers the night before his death:

There are a few things left to be completed in this proof. I have not the time. (Author's note.)

Clearly this note was not written by Galois to himself, especially since he appended the parenthetical “Author’s note”. The point is that it's usually fairly easy to tell the difference between something that is really a personal note someone has written to himself and a note intended to be read by others. Clearly Fermat's commentary on Diophantus was written with the idea of it being read by others.

Of course, this is not to suggest that Fermat ever made a decision to actually publish the commentary. In fact, for various reasons, Fermat almost never decided to publish anything. (This had both advantages and disadvantages for his reputation; it denied him some of the credit for differential calculus, but it also preserved "plausible deniability" for his various mis-statements.) Needless to say, if he had ever published his Observations on Diophantus he would have omitted the above-quoted comment, having long since realized that his wonderful proof didn't work. So, in a sense, it was fortunate for the development of number theory that he never edited his papers.

One might (irresponsibly) speculate that the Pierre Fermat never wrote that particular comment at all, and that it was in fact an original contribution of Samuel's. This may seem implausible, but it would account for the otherwise puzzling fact that the senior Fermat never once mentioned the fully general problem in any conversation or letter, even as a challenge question to other mathematicians - despite the fact that he loved to stump other mathematicians. Indeed his life's passion seems to have been number theory, and he was always striving (usually in vain) to cite examples of propositions, theorems, and conjectures from this field of research that would impress people with the depth and subtlety of the subject. It seems exceedingly strange that, with such a striking proposition, one that is supported by a lot of numerical evidence, and for which he had once believed he had a truly marvelous proof - that he would never have returned to the question again, even in passing. This is especially true considering that he returned many times in his correspondence to the question for cubes and 4th powers. It is really quite incongruous. Even in its technical substance it was out of character because, as Weil notes, the "Last Theorem" was the only time that Fermat ever mentioned a problem that involves a curve of genus greater than 1. And we know from Fermat’s correspondence regarding the Fermat numbers that he was not averse to advancing interesting hypotheses, even if he was unsure of their truth.

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