The following appendix accompanies the main article, "Fermat Meets
SWAC: Vandiver, the Lehmers, Computers, and Number Theory," by Leo
Corry, which appears in the January-March 2008 issue of the *IEEE Annals of the History of Computing*. The material here was omitted from the main article because of space constraints.

Ernst E. Kummer's results are interestingly related to a discussion on the problem held in 1847 at the Paris Academy among prominent mathematicians that included Gabriel Lamé as well as Augustin Louis Cauchy (1789–1857) and Joseph Liouville (1809–1882). A possible proof of the theorem was suggested, based on representing a sum of integers as a product of certain complex numbers, as follows:

*x ^{p}* +

Here *p* is an odd prime number, and *r* is a complex number called a primitive *p*^{th}root of unity, namely, a number that satisfies the condition: *r ^{p}*= 1 and

Several years prior to that, however, as part of his research on higher reciprocity, Kummer had investigated the behavior of cyclotomic fields, and he knew well that this assumption is not generally valid for such domains. On hearing about their intended proof, he wrote to Liouville, informing him that in 1844 he had already published a counterexample to that assumption. He also wrote that his new theory of "ideal complex numbers" restored a somewhat different kind of unique prime factorization into these fields. While working on his theory, Kummer came up with the idea of the regular primes.

The basic definition of a regular prime uses the concept of "class number" *h _{p}* of a cyclotomic field

The Bernoulli numbers appeared for the first time in 1713 in the
pioneering work of Jakob Bernoulli on probabilities, and thereafter in
several other contexts. Leonhard Euler, for instance, realized that they
appear as coefficients *B _{n}* of the following Taylor expansion:

He was also the first to calculate actual values of the coefficients.
There are also several, well-known recursion formulas to calculate them.
Given that for all odd indexes *n* greater than 1, *B _{n}*
= 0, I followed in this article a simplifying convention adopted by
Harry Schultz Vandiver and Dick and Emma Lehmer, namely, to consider
only even indexes. In these terns, the first few values of

*B*_{1}= 1/6

*B*_{2}= –1/30

*B*_{3}= 1/42

*B*_{4}= –1/30

*B*_{5}= 5/66

*B*_{6}= –691/273

Kummer showed that a prime *p* is regular *iff* it does not divide the numerators of any of the Bernoulli numbers *B*_{0}, *B*_{2}, … , *B*_{(p-3)/2}. Already in the lower cases one sees that *B*_{6}= –691/2,730, which shows directly that 691 is an irregular prime.

As a measure of Kummer's willingness to undertake arduous computations
with individual cases it is worth noticing that the lowest case for
which unique factorization fails in the cyclotomic fields *k*(*z _{p}*), is

Kummer's three criteria for the validity of the theorem for irregular primes involve the class number *h _{p}* as well as several divisibility relations vis-à-vis certain Bernoulli numbers. One case to which they do not apply is