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In a letter to Christiaan Huygens entitled "on problems in the theory of numbers: a letter to Christiaan Huygens", Fermat claism that he solved the diophantine $x^2+2=y^3$ using infinite descent. Here is the letter:
"  For a long time I was unable to apply my method (i.e. infinite descent) to affirmative questions, because the twists and turns to get there are much more difficult than those which served me for the negative questions. [...] But finally a line of thought gone over many times showed me a light which did not fail, and affirmative questions surrendered to my method, with the help of some new principles which had to be joined with it of necessity. The progress in my thinking on these affirmative questions is this: if a prime number taken at one's discretion, which exceeds by one a multiple of four, is not a sum of two squares, there will be a prime of the same kind, less than the given one, and then a third still less etc., descending infinitely this way until you arrive at the number 5 which is the smallest of all those of this kind, which it follows cannot be the sum of two squares, which it is nonetheless. From which one must infer from that deduction of an impossibility that all those of this kind are consequently a sum of two squares. [...]  Finally I considered certain questions which, although negative, did not shrink from receiving very great difficulties, the way of applying the descent being quite as diverse as the preceding, as it will be easy to check. These are the following: No cube is a sum of two cubes. There is only one square in integers which, added to two, gives a cube. The said square is 25. There are only two squares in integers which, added to 4, give a cube. The said squares are 4 and 121. All the square powers of two, added to one, are prime numbers. This last question is of a very subtle and ingenious study and, even though it is posed affirmatively, it is negative; for to say that a number is prime is to say that it cannot be divided by any number."
So the question is: Did Fermat have an infinite descent proof of $x^2+2=y^3$ (i.e. that the only integer solutions are (x,y)=(5,3), (-5,3) ) ? Has anyone ever found such a proof?