Irregular Prime

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. 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 e^(2pii/p). Equivalently, but more conveniently, an odd prime p is irregular iff p divides the numerator of a Bernoulli number B_(2n) with 2n+1<p.


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 283145 primes less than 4×10^6, 111597 (or 39.41%) are irregular. The conjectured fraction is 1-e^(-1/2) approx 39.35% (Ribenboim 1996, p. 415).

The numbers of irregular primes less than 10^n for n=0, 1, 2, ... are 0, 0, 3, 64, 497, ... (OEIS A092901).

The largest known proven irregular prime as of Apr. 2009 is 6B_(4306)/2153, which has 10342 decimal digits and was found by M. Oakes et al. on Apr. 4, 2009 ( The largest known irregular probable prime is the numerator of -B_(22808)/22808, which has 71290 digits and was found by T. D. Noe on Sep. 28, 2005. The values of n such that |numer(B_n/n)| is prime are n=12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808, ... (OEIS A112548), with the corresponding values necessarily being irregular.

See also

Bernoulli Number, Fermat's Last Theorem, Integer Sequence Primes, Irregular Pair, Regular Prime

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Irregular Prime

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Weisstein, Eric W. "Irregular Prime." From MathWorld--A Wolfram Web Resource.

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