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This section was written by Ralph Greenberg.

**Definition 4.4.1** (Irregular Prime)
A prime

is said to be

*irregular* if

divides the numerator of a
Bernoulli number

, where

and

is even.
(For odd

, one has

.)

The *index of irregularity* for a prime
is the number of such
.'s There is considerable numerical data concerning the statistics
of irregular primes - the proportion of
which are irregular or
which have a certain index of irregularity. (See *Irregular
primes and cyclotomic invariants to four million*, Buhler et al., in
*Math. of Comp.*, vol. **61**, (1993), 151-153.)
Let

for each
as above. According to the Kummer congruences,
is a
-integer, i.e., its denominator is not divisible by
. But its
numerator could be divisible by
. This happens for
and
.

**Problem 4.4.2**
Obtain numerical data for the divisibility of
the numerator of

by a prime

analogous to that for the

's.

Motivation: It would be interesting to find an example of a prime
and an index
(with
,
even) such that
divides the numerator of both
and
. Then the
-adic
-function for a certain even character of conductor
(namely,
the
-adic valued character
, where
is the
character characterized by
for
) would have at least two zeros. No such example exists for
. The
-adic
-functions for those primes have at
most one zero. If the statistics for the
's are similar to those
for the
's, then a probabilistic argument would suggest that
examples should exist.

**Problem 4.4.3**
Computation of

for a specific

is very efficient in PARI,
hence in

*SAGE* via the command

`bernoulli`. Methods for
computation of

for a large range of

are described
in

*Irregular primes and cyclotomic invariants to four million*,
Buhler et al. Implement the method of Buhler et al. in

*SAGE*.

** Next:** Half Integral Weight Modular
** Up:** Computing with Classical Modular
** Previous:** Weight
** Contents**
William Stein
2006-10-20