A regular prime is a prime which does not divide the class number cyclotomic field
obtained by adjoining a primitive p th
root of unity to the field of rationals .
This definition includes 2, whose cyclotomic field
has class number 1 (Conrad). An odd
prime iff divide the numerator
of any Bernoulli number positive integer prime which is
not regular is said to be an irregular prime .
Jensen (1915) proved that there are infinitely many irregular primes . It has not yet been proven that there is an infinite
number of regular primes (Guy 1994, p. 145). Of the odd primes et al. 1993).
The conjectured fraction is odd
primes are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (OEIS A007703 ).
See also Bernoulli Number ,
Fermat's
Last Theorem ,
Irregular Prime
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References Buhler, J.; Crandall, R.; Ernvall, R.; and Metsänkylä, T. "Irregular Primes and Cyclotomic Invariants to Four Million." Math.
Comput. 61 , 151-153, 1993. https://doi.org/10.1090/S0025-5718-1993-1197511-5 . Conrad,
K. "Fermat's Last Theorem for Regular Primes." https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf . Guy,
R. K. Unsolved
Problems in Number Theory, 2nd ed. Jensen, K. L. "Om talteoretiske Egenskaber ved de Bernoulliske
Tal." Nyt Tidsskr. Math. Afd. B 26 , 73-83, 1915. https://mathematics.dk/fileadmin/Files/DMF/Matilde/Matilde_43.pdf . Ribenboim,
P. "Regular Primes." §5.1 in The
New Book of Prime Number Records. Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. 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 https://mathworld.wolfram.com/RegularPrime.html
Subject classifications
of the cyclotomic field
obtained by adjoining a primitive pth
root of unity to the field of rationals.
This definition includes 2, whose cyclotomic field
has class number 1 (Conrad). An odd
prime 
is regular iff 
does not divide the numerator
of any Bernoulli number 
with positive integer 
satisfying 
. A prime which is
not regular is said to be an irregular prime.
odd primes 
, 
(or 60.59%) are regular (Buhler et al. 1993).
The conjectured fraction is 
. The first few regular odd
primes are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, ... (OEIS A007703).
