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Irregular Prime

Primes for which Kummer's theorem on the unsolvability of Fermat's Last Theorem does not apply. An irregular prime $p$ divides the Numerator of one of the Bernoulli Numbers $B_0$, $B_2$, ..., $B_{p-3}$, as shown by Kummer in 1850. The Fermat Equation has no solutions for Regular Primes.


An Infinite number of irregular primes exist, as proven in 1915 by Jensen. The first few irregular primes are 37, 59, 67, 101, 103, 131, 149, 157, ... (Sloane's A000928). Of the 283,145 Primes less than $4\times 10^6$, 111,597 (or 39.41%) are irregular. The conjectured Fraction is $1-e^{-1/2} \approx 39.35\%$ (Ribenboim 1996, p. 415).

See also Bernoulli Number, Fermat's Last Theorem, Irregular Pair, Regular Prime


Buhler, J.; Crandall, R.; Ernvall, R.; and Metsänkylä, T. ``Irregular Primes and Cyclotomic Invariants to Four Million.'' Math. Comput. 60, 151-153, 1993.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 202, 1979.

Johnson, W. ``Irregular Primes and Cyclotomic Invariants.'' Math. Comput. 29, 113-120, 1975.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 325-329 and 414-425, 1996.

Sloane, N. J. A. Sequence A000928/M5260 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Stewart, C. L. ``A Note on the Fermat Equation.'' Mathematika 24, 130-132, 1977.

© 1996-9 Eric W. Weisstein