A numerical counterexample, even of such a famous conjecture, that has been believed (apparently unjustifiably) to have been proved, is not appropriate for such a prestigious journal like the PNAS. It should be submitted to a specialized journal like Mathematics of Computation, and if the latter journal would reject it, the author may try the journal Experimental Mathematics.

My reason for not recommending publication in PNAS has to do with the result not adding new mathematical insight. Of course, it is interesting for historical reasons, and the description of the algoirthm may be of some interest to specialists in computational number theory, and hence it should be published in one of the journals mentioned above.

Reviewer Comments:
Reviewer #1:

Suitable Quality?:
No

Sufficient General Interest?:
Yes

Conclusions Justified?:
Yes

Clearly Written?:
Yes

Procedures Described?:
Yes

Comments (Required):
Fermat's Last Theorem is a very longstanding conjecture that states that it is impossible to find three strictly positive integers a,b,c and another integer, n, stictly larger than two, such that the n-th power of a plus the n-th power of b equals exactly the n-th power of c.

Pierre Fermat famously believed to have had a proof, that, unfortunately, his margin was not wide enough to contain. In 1994, also famously, Andrew Wiles published a "proof" that was later "fixed" in collaboration with Richard Taylor. The present submission indicates that this second published attempt is flawed as well, but this fact is only of anecdotal and historical interest, and does not add one iota of mathematical insight

On the other hand, the original papers by Wiles and Wiles/Taylor are full of wonderful new insights, and the fact that there is (I am sure a minor) gap somewhere, does not at all detract from their conceptual tour-de-force.

The merits of the present submission:

1) The actual counterexample seems to be correct.

2) The algorithm leading to the counterexample may be of some independent interest (but only to specialists). The science is about figuring out the truth. But in this case the "truth" is a conceptually trivial numerical identity.

3) Whether PNAS is a proper home for this paper is the question.

Here are this reviewer's thoughts.

a) Quite by accident, I have recently reviewed another numerical number theory paper, where I wrote

"if FLT were false, and someone found this to be the case because they found (with the help of a computer) some numbers a,b,c and a prime p for which

a^p+b^p=c^p

then would that be a PNAS paper? This referee does not."

To my surprise, this is exactly what happened. Since I have to be true to my word, I really must reject this submission.

Furthermore, because of the enormous size of the counterexample, one can only hope to have a probabilistic verification, that would validate it with probability larger than 0.99999999999999999999999. Nevertheless, this is still not a rigorous disproof, and pure mathematics insists on proofs that are believed to be purely rigorous (even though often they are not).

As a final word, I also think that it would be very impolite to Sir Andrew Wiles, (foreign) member of the NAS, to publicize the fact that his proof must be wrong. For this reason alone, this paper should be rejected.