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# A Problem About Bernoulli Numbers

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