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 p-cyclotomic integers. However, 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 p is irregular iff p divides the class number of the cyclotomic field generated by 
.
with 
(Broadhurst).
An infinite number of irregular primes exist, as proven in 1915 by Jensen (Vandiver and Wahlin 1928, p. 82; Carlitz 1954; Carlitz 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, ... (Sloane's 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 n = 0, 1, 2, ... are 0, 0, 3, 64, 497, ... (Sloane's A092901).
The largest known irregular prime is 
,
Bernoulli Number, Fermat's Last Theorem, Integer Sequence Primes, Irregular Pair, Regular Prime
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