Regular Prime

A Prime which does not Divide the Class Number of the Cyclotomic Field obtained by adjoining aPrimitive Root of unity to the rational Field. A Prime is regular Iff does not divide the Numerators of the Bernoulli Numbers ,, ..., . 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 proventhat there are an Infinite number of regular primes (Guy 1994, p. 145). Of the 283,145 Primes ,171,548 (or 60.59%) are regular (the conjectured Fraction is ). The first few are 3, 5, 7,11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (Sloane's A007703).

References

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. SequenceA007703/M2411in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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