Regular Prime

A prime which does not divide the class number h(p) of the cyclotomic field obtained by adjoining a primitive pth root of unity to the field of rationals. 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 283145 primes <4×10^6, 171548 (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, ... (OEIS A007703).

See also

Bernoulli Number, Fermat's Theorem, Irregular Prime

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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 "The On-Line Encyclopedia of Integer Sequences."

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Regular Prime

Cite this as:

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

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