Munk said that this must have been Fermat's proof, and I can only verify it up to the very last page of the alleged proof, which may be poorly stated due to "English as a second language," although it is mostly very elementary. Munk was well-qaulified as a student of Prandtl in aerodynamics, as far as mathematical physics goes, at Goettingen U.; he was one of the pioneers at NACA, the predecessor to NASA, albei mainly with planar slices of airfoils.
The book is actually quite amuzing, and he states in the chapters after the main result, that most mathematicians would think of his "congruence surds" as "p-adic numbers, which they are not." Yes, the p-adics are a big key to this, as they are in quantum mechanics, but it is really just a matter of "adding-machine mechanics," which a few of the classical math-folks had actually developed, before anyone else: Pascal, Liebniz and Fermat.
Although the congruence surds have the same problem as p-adics, namely that they have no simple, "archimedean valuation" or absolute scalar value, that is not important to this theorem, which merely proves the impossiblity of certain congruences, teh ones that are akin to the Pythagorean theorem, using exponents other than two.
Here, I will just supply a note as to the hearsay about Fermat, not having been able to prove this "miraculous" result, as he wrote in the margin of his _Bachet's Diophantus_ book of old number problems. Well, he made no other known errors, and this includes the sole remaining unsloved problem of Fermat, the characterization of the Fermat numbers (of the form, 2^(2^h) + 1, h = 0, 1 etc. Well, he had merely congectured that they were all prime, and he later retracted this ideal in a letter to Bernard de Frenicle; so, theresville.
Furthermore, this may have been one of his earliest proofs in the theory of numbers -- which science he created, mostly using right trigona -- and this is highlighted by the fact taht he *later* produced the proof for teh very special case of the exponent, n = 4. All of the other cases can be made for only prime exponents, since it is easy to prove that composite exponents -- other than 2x2 -- reduce to the case for prime ones.