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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 14 hours ago
  • 7
    $\begingroup$ 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 14 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 14 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 13 hours ago
  • 8
    $\begingroup$ @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 13 hours ago