# Theorems of Morley and Emma Lehmer and their generalizations.

Taking n = p-1 in (31) gives

which implies (15) after summing over each , applying (28) and then using Fermat's Theorem and the Von Staudt--Clausen Theorem.

In 1909, Wieferich showed that if the first case of Fermat's Last Theorem is false for prime exponent p then divides ; thus by (15). In 1914, Frobenius gave an induction hypothesis which should allow one to extend this to divides , for each successive integer m; however, because of the enormous amount of computation required for each step of the hypothesis, this is currently known to hold only up to m=89.

In 1938, Emma Lehmer used identities like (28) to show that if the first case of Fermat's Last Theorem is false for prime exponent p then for . Recently Skula has modified Frobenius's induction hypothesis so that the mth step might also show that for : (32) would then give (16) for . (Skula has done this for ; Zhong Cui--Xiang has obtained the same result for , independantly.)

The left side of (15) is

(33)

where .

Taking and n=p-2 in (31), and then using a number of the well--known congruences quoted in section 8 as well as (28), we obtain

and . Substituting these equations into (33), and using the fact that for , we see that the left side of (15) is