Fermat’s Last Theorem after 356 Years

The purpose of this expository lecture is to explain the basic ideas underlying the final resolution of “Fermat’s Last Theorem” after 356 years as a consequence of the reported establishment by Andrew Wiles of a sufficient portion of the “Shimura-Taniyama-Weil” conjecture. As these notes are being written, the work of Wiles is not available, and the sources of information available to the author are (1) reports by electronic mail, (2) the AMS Notices article [16] of K. Ribet, and (3) a preprint [18] by K. Rubin and A. Silverberg based on the June, 1993 lectures of Wiles at the Newton Institute in Cambridge, England. It should be noted that the fact that “Fermat’s Last Theorem” is a consequence of sufficient knowledge of the theory of “elliptic curves” has been fully documented in the publications ([14], [15]) of K. Ribet. “Fermat’s Last Theorem” is the statement, having origin with Pierre de Fermat in 1637, that there are no positive integers x, y, z such that x + y = z for any integer exponent n > 2. Obviously, if there are no positive integer solutions x, y, z for a particular n, then there are certainly none for exponents that are multiples of n. Since every integer n > 2 is divisible either by 4 or by some odd prime p, it follows that “Fermat’s Last Theorem” is true if there are no solutions in positive integers of the equation x + y = z when n = 4 and when n = p for each prime p > 2. The cases n = 3, 4 are standard fare for textbooks (e.g., see Hardy & Wright [6]) in