Fermat's Last Theorem was open for more than 350 years until Andrew Wiles proved it in 1995. Are there possible (historical or other) reasons why David Hilbert did not include this famous open problem in his problem list in 1900?
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10$\begingroup$ Perhaps Hilbert did not consider the problem conceptually important. Gauss had no interest in it. Until the 1980s it was not at all clear the possibility of a counterexample would have any relation to important themes in mathematics even though Hilbert considered FLT important as motivation for the creation of some very useful ideas by Kummer (whether or not that is historically accurate): are you aware that Hilbert does mention FLT in the introduction of his speech, preceding the list of problems? $\endgroup$– KCdCommented May 12, 2022 at 7:14
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Hilbert included in his list a much more general Problem 10:
Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
Probably he did not think that Fermat's equation by itself is sufficiently important to deserve a separate entry in his list.
answered May 14, 2022 at 3:07
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1$\begingroup$ Problem 21 does not generalise FLT. If Problem 21 had a solution, the FLT would have a mechanical solution. My hunch is that Hilbert hoped Problem 21 had a solution, but we now know that it provably does not. $\endgroup$ Commented May 29, 2022 at 23:06
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$\begingroup$ @Rob Arthan: Problem 21 has been solved, and the answer is negative. Probably this was not what Hilbert hoped for. $\endgroup$ Commented Nov 9, 2022 at 14:42
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$\begingroup$ Actually, I think what you have quoted is Hilbert's 10th problem, which as you say was answered in the negative (by Matiyasevitch). But even if there was such a process all it would tell us about FLT is that one can decide for each $n$ whether $a^n + b^n = c^n$ has solutions over the integers. $\endgroup$ Commented Nov 9, 2022 at 23:50
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$\begingroup$ @Rob Arthan: Thanks, I corrected the problem number. Its solution, positive or negative does not imply Fermat's theorem. But probably Hilbert considered his problem more important than the answer for one special family of equations. $\endgroup$ Commented Nov 9, 2022 at 23:55
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$\begingroup$ Thanks for making the correction. Given that FLT has been proved, what we now know is that for that family of equations there is a decision process: the answer is yes iff $n \le 2$. You may well be right about Hilbert's priorities. $\endgroup$ Commented Nov 10, 2022 at 0:13