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 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 proved Fermat's
last theorem for all prime number exponents in
the class of regular primes (Caldwell).
Jensen (1915) proved that there are infinitely many irregular primes congruent to 3 modulo 4, and hence infinitely many irregular primes (Vandiver and Wahlin 1928,
p. 82; Carlitz 1954). Montgomery (1965) proved a generalization that, in particular,
implies that infinitely many irregular primes are congruent to 5 modulo 6. The first
few irregular primes are 37, 59, 67, 101, 103, 131, 149, 157, ... (OEIS A000928).
Of the odd primes less than , (or 39.41%) are irregular (Buhler et al. 1993).
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 Oct. 2026 is , which has 35,160
decimal digits (Caldwell). The largest known irregular
probable prime reported in Sep. 2005 was the
numerator of , which has 71290 digits
and was found by T. D. Noe on Sep. 28, 2005. Reported values of such that is prime or a
probable prime include , 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870,
4306, 22808, ... (OEIS A112548), with the value
for
remaining a probable prime. Any prime
number in this list is necessarily irregular.
Buhler, J.; Crandall, R.; Ernvall, R.; and Metsänkylä, T. "Irregular Primes and Cyclotomic Invariants to Four Million." Math.
Comput.61, 151-153, 1993. https://doi.org/10.1090/S0025-5718-1993-1197511-5.Buhler,
J.; Crandall, R.; Ernvall, R.; Metsänkylä, T.; and Shokrollahi, M. A.
"Irregular Primes and Cyclotomic Invariants to 12 Million." J. Symb.
Comput.31, 89-96, 2001. https://doi.org/10.1006/jsco.1999.1011.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." https://t5k.org/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.Jensen, K. L.
"Om talteoretiske Egenskaber ved de Bernoulliske Tal." Nyt Tidsskr.
Math. Afd. B26, 73-83, 1915. https://mathematics.dk/fileadmin/Files/DMF/Matilde/Matilde_43.pdf.Johnson,
W. "Irregular Prime Divisors of the Bernoulli Numbers." Math. Comput.28,
653-657, 1974.Johnson, W. "Irregular Primes and Cyclotomic Invariants."
Math. Comput.29, 113-120, 1975.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.
that divides the class number
of the cyclotomic field generated by 
. Equivalently, an odd prime 
is irregular iff 
divides the numerator
of a Bernoulli number 
for some positive integer 
with 
.
-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 proved Fermat's
last theorem for all prime number exponents in
the class of regular primes (Caldwell).
odd primes less than 
, 
(or 39.41%) are irregular (Buhler et al. 1993).
The conjectured fraction is 
(Ribenboim 1996, p. 415).
for 
, 1, 2, ... are 0, 0, 3, 64, 497, ... (OEIS A092901).
, which has 35,160
decimal digits (Caldwell). The largest known irregular
probable prime reported in Sep. 2005 was the
numerator of 
, which has 71290 digits
and was found by T. D. Noe on Sep. 28, 2005. Reported values of 
such that 
is prime or a
probable prime include 
, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870,
4306, 22808, ... (OEIS A112548), with the value
for 
remaining a probable prime. Any prime
number in this list is necessarily irregular.
