The author stated that in consequence of Legendre's work, the proof of Fermat's Theorem is reducible to showing the impossibility of

xm = ym+zm,

when m is an odd prime, x, y, z being integers.

Talbot has shown that in this case x, y, z are necessarily composite numbers.

The author shows, among other results of very elementary processes, that if numbers can be found to satisfy the above equation, x and y leave the remainder 1 when divided by m; and that z has m as a factor. Many farther limitations are given on possible values of x, y, z—the process being based on the consideration of their prime factors, and on Fermat's Elementary Theorem Nm − N = Nm.