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Fermat announced many of his results, but released very few details of his proofs, so his methods could be considered for the most part to have been kept secret.

Often he challenged others in written correspondence to solve problems that he claimed to be able to do, such as finding nontrivial integral solutions to some Pell equations (he offered the example of $x^2 - 61y^2 = 1$, where he said he chose the coefficient 61 because it is small and shouldn't be too much work, whereas in fact the first solution in positive integers is unusually large, so it was not a random example) or finding all solutions in positive integers to $y^2 = x^3 - 2$ (the only one is $(x,y) = (3,5)$) or showing every prime that is 1 mod 4 is a sum of two squares.

He hoped these public challenges would inspire others to get interested in number theory, but this goal was largely unsuccessful in his lifetime. It took about 100 more years for anyone to really pick up where Fermat left off, and that person was Euler. All he had to start off with were the claims by Fermat of what he could do, and proofs of everything had to be reconstructed, assuming Fermat had proofs at all. To this day only one proof of Fermat in his own writing is known (that 1 and 2 are not "congruent numbers" based on his method of descent).

Of course the most famous challenge by Fermat was Fermat's last theorem, but that is not in the same category as his usual challenges because it was a result he wrote to himself in one of his books and was not intended for others. So in a sense the statement of FLT itself could be considered secret, but it doesn't really fit the parameters of the question as Fermat was almost certainly wrong about having a proof in this case.