Irregular primes are prime numbers that divide the numerators of Bernoulli numbers, which are the coefficients of some Taylor series, and all of this has something to do with some work that was done on Fermat's Last Theorem in the 1840s. Rather than try to paraphrase -- and thereby mangle further -- the explanation, why don't I just display the entire message I got from alert contributor Dale Shoults:
Date: Fri, 16 May 1997 12:29:37 -0600 (MDT) Subject: Re: 37 Hello: I'm not an expert on the subject of irregular and regular primes, but here's a brief description as I understand it: Firstly, the Bernoulli numbers. The Bernoulli number are the coefficients in the Taylor series (about x = 0) for x/(exp[x] - 1). B0 = 1, B1 = -1/2, B2 = 1/6, B3 = 0, B4 = -1/30, B5 = 0, ... . Every 2nd one is zero starting with B3. Some definitions of Bernoulli numbers leave out the ones that are zero. In about the 1840's, Kummer made major advances in the study of Fermat's Last Theorem when he proved that the theorem is true for the exponents p (a prime) whenever p does not divide the numerator of any of B2, B4, B6, ... B(p-3). Primes like this are called regular primes. If a prime does divide one of the numerators, then it is an irregular prime. The first irregular prime is 37. 37 divides the numerator of B32. (The numerator of B32 = 7709321041217.) The only other irregular primes less than 100 are 59 and 67. Later the cases p = 37,59,67 were proven to hold, using extensions of Kummer's methods. Almost all of the work on FLT since the 1840's was based on Kummer's methods, although I think the successful complete proof by Wiles in 1995 used very different methods. Wiles proof has been verified now. There was one hole in his original "proof", but that was filled by Wiles and a former student of his, so it is now agreed that the problem has been completely solved, after months of careful scrutiny by experts. Anyways, it has been proven that there are infinitely irregular primes, of which 37 is the smallest. You could put that 37 is the smallest irregular prime, or maybe the smallest of Kummer's irregular primes, I guess, although I don't think there's any other context in which primes are called regular or irregular. A good reference is "Fermat's Last Theorem - A Genetic Introduction to Algebraic Number Theory" by Harold M. Edwards. By the way, I'm using a friends e-mail account at the University of Calgary. Have a good day, Dale Shoults.