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
*m*th 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