In a 1847 talk to the Académie des Sciences in Paris, Gabriel Lamé (1795-1870) claimed to have proven Fermat's last
theorem. However, Joseph Liouville immediately pointed out an error in Lamé's
result by pointing out that Lamé had incorrectly assumed unique factorization
in the ring of -cyclotomic
integers. Kummer had already studied the failure of unique factorization in cyclotomic
fields and subsequently formulated a theory of ideals which was later further developed
by Dedekind.

Kummer was able to prove Fermat's last theorem for all prime exponents falling into a class he called "regular." "Irregular"
primes are thus primes that are not a member of this class, and a prime is irregular iff divides the class number of the cyclotomic
field generated by .
Equivalently, but more conveniently, an odd prime is irregular iff divides the numerator of a Bernoulli
number
with .

An infinite number of irregular primes exist, as proven in 1915 by Jensen (Vandiver and Wahlin 1928, p. 82; Carlitz 1954, 1968). In
fact, Jensen also proved the slightly stronger result that there are an infinite
number of irregular primes congruent to 5 (mod 6) (Carlitz 1968), a result subsequently
improved by Montgomery (1965). The first few irregular primes are 37, 59, 67, 101,
103, 131, 149, 157, ... (OEIS A000928). Of
the primes less than , (or 39.41%) are irregular. The conjectured fraction
is
(Ribenboim 1996, p. 415).

The numbers of irregular primes less than for , 1, 2, ... are 0, 0, 3, 64, 497, ... (OEIS A092901).

The largest known proven irregular prime as of Apr. 2009 is , which has 10342 decimal digits and was found
by M. Oakes et al. on Apr. 4, 2009 (http://primes.utm.edu/primes/page.php?id=87451).
The largest known irregular probable prime is the
numerator of , which has 71290 digits and was found by T. D. Noe
on Sep. 28, 2005. The values of such that is prime are , 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870,
4306, 22808, ... (OEIS A112548), with the corresponding
values necessarily being irregular.

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