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Regular Prime

A Prime which does not Divide the Class Number $h(p)$ of the Cyclotomic Field obtained by adjoining a Primitive Root of unity to the rational Field. A Prime $p$ is regular Iff $p$ does not divide the Numerators of the Bernoulli Numbers $B_0$, $B_2$, ..., $B_{p-3}$. A Prime which is not regular is said to be an Irregular Prime.

In 1915, Jensen proved that there are infinitely many Irregular Primes. It has not yet been proven that there are an Infinite number of regular primes (Guy 1994, p. 145). Of the 283,145 Primes $<4\times 10^6$, 171,548 (or 60.59%) are regular (the conjectured Fraction is $e^{-1/2} \approx 60.65\%$). The first few are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (Sloane's A007703).

See also Bernoulli Number, Fermat's Theorem, Irregular Prime


Buhler, J.; Crandall, R. Ernvall, R.; and Metsankyla, T. ``Irregular Primes and Cyclotomic Invariants to Four Million.'' Math. Comput. 61, 151-153, 1993.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 145, 1994.

Ribenboim, P. ``Regular Primes.'' §5.1 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 323-329, 1996.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 153, 1993.

Sloane, N. J. A. Sequence A007703/M2411 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.

© 1996-9 Eric W. Weisstein