Fermat's Fallibility

Fermat discussed the propositition that "every number of the form
2^2^n + 1 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."

which is roughly translated as

   "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 2^n + 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 
  2^32+1, the method he had used to factorize 2^37-1; it shows 
  that any prime divisor of 2^32+1 must be of the form 64k+1, which 
  leaves the candidates 193, 257, 449, 577, 641, etc.; 641 divides 
  2^32+1; in fact this is how Euler proceeded... What is even more 
  surprising is that Frenicle, who had also factorized 2^37-1, 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(!)...I must be misinterpreting.... 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".
Maybe this is why Bell refered to this as "an obscure statement".
Is it possible that "puissances quarres de 2" was intended to mean
something like "iterated powers of two"?

Anyway, 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 proveable,
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.

(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 2b^2-a^2 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.)