Does Lamé's proof of FLT hold for some n? Despite the fact that Lamé's proof of FLT fails in general, is it correct that it holds for every n where the corresponding ring of integers of cyclotomic field is a UFD (which is all n with class numbers equal 1, like 3-22, for example)?
Thank you for your help.
 A: Your question is a bit imprecise, because it seems that Lamé has presented, at intervals of a few years, at least two "proofs" of special cases of FLT, see e.g. Catherine Goldstein historical note Gabriel Lamé et la théorie des nombres : « une passion malheureuse » ?  pp. 131 - 139 https://doi.org/10.4000/sabix.690.
His first communication, around 1840, was about the exponent 7, of which H. M. Edwards says in his book "A genetic approach to ANT" that [Lamé's arguments] are difficult, unmotivated and, worst of all, seem to be hopelessly tied to the case n =7. In his second, around 1847, after a reduction to the case of a prime exponent p, he made the famous mistake that $\mathbf Z[\zeta_p]$ is always a UFD. Later in his career Lamé came back to the particular cases $p=3,5$, with great care concerning the factoriality property (see Goldstein, op. cit.), but it does not seem that he went beyond these cases. Recall that $p$ is called a regular prime if $p$ does not divide the class number of $\mathbf Z[\zeta_p]$, and the smallest irregular prime is 37. So we can reasonably attribute to Kummer (and not Lamé) the theorem saying that FLT holds for regular primes (see e.g. Washington's "Introduction to cyclotomic fields", chap.1).
