A prime which does not divide the class number of the cyclotomic field
obtained by adjoining a primitive p th
root of unity to the field of rationals .
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 proven that there are an infinite
number of regular primes (Guy 1994, p. 145). Of the primes , (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, ... (OEIS A007703 ).

See also Bernoulli Number ,

Fermat's
Theorem ,

Irregular Prime
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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. Sequence A007703 /M2411
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Regular Prime
Cite this as:
Weisstein, Eric W. "Regular Prime." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RegularPrime.html

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