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.
Buhler, J.; Crandall, R.; Ernvall, R.; and Metsänkylä, T. "Irregular Primes and Cyclotomic Invariants to Four Million." Math.
Comput.61, 151-153, 1993.Buhler, J.; Crandall, R.; Ernvall,
R.; Metsankyla, T.; and Shokrollahi, M. "Irregular Primes and Cyclotomic Invariants
to 12 Million." J. Symb. Comput.11, 1-8, 2000.Buhler,
J. P.; Crandall, R. E.; and Sompolski, R. W. "Irregular Primes
to One Million." Math. Comput.59, 717-722, 1992.Caldwell,
C. K. "The Prime Pages. The Top 20: irregular Primes." http://primes.utm.edu/top20/page.php?id=26.Carlitz,
L. "A Note on Irregular Primes." Proc. Amer. Math. Soc.5,
329-331, 1954.Carlitz, L. "Bernoulli Numbers." Fib. Quart.6,
71-85, 1968.Hardy, G. H. and Wright, E. M. An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 202, 1979.Havil, J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 85,
2003.Hoffman, P. The
Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical
Truth. New York: Hyperion, p. 192, 1998.Johnson, W. "Irregular
Primes and Cyclotomic Invariants." Math. Comput.29, 113-120,
1975.Johnson, W. "Irregular Prime Divisors of the Bernoulli Numbers."
Math. Comput.28, 653-657, 1974.Montgomery, H. L.
"Distribution of Irregular Primes." Ill. J. Math.9, 553-558,
1965.Ribenboim, P. The
New Book of Prime Number Records. New York: Springer-Verlag, pp. 325-329
and 414-425, 1996.Siegel, C. L. "Zu zwei Bemerkungen Kummers."
Nachr. Akad. d. Wiss. Göttingen, Math. Phys. Kl.2, 51-62, 1964.Sloane,
N. J. A. Sequences A000928/M5260,
A092901, and A112548
in "The On-Line Encyclopedia of Integer Sequences."Stewart,
C. L. "A Note on the Fermat Equation." Mathematika24,
130-132, 1977.Vandiver, H. S. "On Developments in an Arithmetic
Theory of the Bernoulli and Allied Numbers." Scripta Math.25,
273-303, 1960.Vandiver, H. S. and Wahlin, G. E. "Algebraic
Numbers." Bull. Nat. Res. Council, No. 62, 1928.Wagstaff,
S. S. Jr. "The Irregular Primes to 125000." Math. Comput.32,
583-591, 1978.
-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.
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 
.
primes less than 
, 
(or 39.41%) are irregular. The conjectured fraction
is 
(Ribenboim 1996, p. 415).
for 
, 1, 2, ... are 0, 0, 3, 64, 497, ... (OEIS A092901).
, 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.
