This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to different epistemological point of views. One of the hard-line stances we discussed in the course was the one which considers an assertion as being a mathematical theorem only if it comes with a completely rigorous proof.

In 1990 and 1995, the papers *On modular representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ arising from modular forms*, *Modular elliptic curves and Fermat's Last Theorem* and *Ring-theoretic properties of certain Hecke algebras* by K.Ribet, A.Wiles and R.Taylor and A.Wiles respectively were published. Presupposing previous mathematical knowledge, they form a complete proof of Fermat's Last Theorem.

My question is the following:

Are there any mistakes in these 3 papers of some mathematical significance?

Of course, it might be hard ot judge what is of significance, so let me mention a couple of things which I believe are not significant.

- Typographical errors are not significant, even if they alter the mathematical content (so if it is written $r\leq s$ while the proof shows $s\leq r$, that is not significant).
- Reference mistakes are not significant (so if there is a sentence like "by lemma 12.8 of [6], we conclude" and lemma 12.8 of [6] allows no such conclusion, but it is easy to see that lemma 4.2 of [11] would do equally well, then that is not significant).
- For the same reason, I will not count an easily filled incomplete argument as significant, even if strictly faulty (so for instance if there is a sentence like "X is a cosy zip, and every cosy zip is a zap, therefore X is a zap" and in fact it is not true that every cosy zip is a zap, but it is true and equally well-known that every strongly cosy zip is a zap and it easy to check that X is a strongly cosy zip, I won't consider this significant).

I would count as significant anything that requires more than a few sentences of essentially novel content (compared to the content provided). If for instance, it is claimed that a certain $x$ can be of 3 different types, each of which satisfy property P, so $x$ satisfies P, but it turns out that $x$ can be of a fourth type. Then, I would count this as a significant mistake even if $x$ satisfies P in this fourth case as well, but for reasons different from the first three cases. In particular, an error does not need to threaten the complete proof nor even the local part it is contained in to be considered significant.

Basically, I'm thinking of a standard much below what formal certification of the proof would require, but still quite stringent. I think that most authors and referees typically do not achieve that standard in most published papers, though some regularly do even in proofs of deep and hard theorems (I think it is believed that Serre achieved it throughout his career with very few exceptions).

With such a standard, are there known mistakes in these 3 papers?

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