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Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$.

Is there any infinite class of primes for which FLT could be known to hold fully (not just the first case) before Wiles's proof?

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    $\begingroup$ You need to distinguish between (1) a set $P$ of primes for which FLT was proved and which is probably infinite and (2) a set $P$ of primes for which FLT was known and which was proved to be infinite. Kummer proved FLT for regular primes, and it is conjectured that there are infinitely many regular primes, but it hasn't been proved that there are infinitely many. $\endgroup$ Commented 19 hours ago
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    $\begingroup$ If there are Infinitely many Sophie Germain primes( Its Still a conjecture). Sophie Germain was first who proved FLT for Infinitely many primes. $\endgroup$ Commented 19 hours ago
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    $\begingroup$ Some time in the past I have tried to look for an answer to this question, and have turned out empty handed. It definitely appears to not have been known, but I have not found in print an explicit mention of that. $\endgroup$ Commented 19 hours ago
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    $\begingroup$ @AchimKrause When investigating FLT for exponent $p$, it is for a number of reasons helpful to separate the cases where $p\nmid xyz$, so called first case which is almost always easier, and $p\mid xyz$, which is harder. Fouvry et al. have shown that there are infinitely many primes for which FLT holds under the assumption we are in the first case. The last sentence is written confusingly, but I'm assuming OP just means that FLT holds for exponent $p$ without that assumption. $\endgroup$ Commented 18 hours ago
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    $\begingroup$ @Guruprasad: No, Germain only proved the first case of FLT for Sophie Germain primes (as well as for some other primes): see mathwomen.agnesscott.org/women/germain-FLT/SGandFLT.htm for a detailed description of her contributions $\endgroup$ Commented 18 hours ago

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According to the 2007 survey of Darmon, Diamond, and Taylor, in 1985 "Fermat’s Last Theorem was still not known to be true for an infinite set of prime exponents". [Note: I found this paper using ChatGPT. Normally I would have found it using a simple Google search, but with the advent of AI searching that way has become much worse!]

I suspect that the later FLT developments pre-Wiles also did not establish infinitely many prime exponent cases, but those more familiar with the Taniyama-Shimura connection can weigh in.

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    $\begingroup$ A google search has a carbon footprint of about 0.2g of CO2. A chatgpt request about 4g. Better or worse depends on your criteria. $\endgroup$ Commented 14 hours ago
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    $\begingroup$ If your numbers are right, it sounds like it is worse relative to both criteria, unless you want more pollution or you like ineffective search engines. $\endgroup$ Commented 12 hours ago
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    $\begingroup$ It may be worth pointing out (as this survey does) that the work of Faltings implied that the set of all exponents (not just prime exponents) for which FLT holds has density 1, and in particular is infinite. $\endgroup$ Commented 11 hours ago
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    $\begingroup$ Is ChatGPT a fan of NCIS? $\endgroup$ Commented 11 hours ago
  • $\begingroup$ @Aurel Just to put this in perspective, burning a gallon of gasoline produces almost 9000g of CO2. So, driving a mile in a car that gets say 25 mpg produces 350g of CO2. $\endgroup$ Commented 5 hours ago
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Fermat's last theorem was known for regular primes (Kummer, 1850). It was conjectured that there are infinitely many regular primes (Siegel 1964), moreover that a majority of primes (about 65%) are regular. For any specific prime, one can check whether it is regular or not, and there are tables of them up to 25000.

Thus it is likely, but not proved, that Kummer established FLT for infinitely many primes.

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    $\begingroup$ If X proved that $P(n)\implies Q(n)$, and if it is "likely but not proved" that $P(n)$ holds for infinitely many $n$, is it then "likely but not proved" that X established $Q(n)$ for infinitely many $n$? Consider the degenerate case $P(n)=Q(n)$. $\endgroup$ Commented 10 hours ago
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    $\begingroup$ I guess there is a difference between "(X proved $P(n)$) for infinitely many $n$" and "X proved ($P(n)$ for infinitely many $n$)". $\endgroup$ Commented 10 hours ago

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