Irregular Pairs
If p is an irregular prime and p divides the numerator of the Bernoulli number B2k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. In other words, an irregular pair is a book-keeping device to record, for an irregular prime p, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs are:
- (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), ... (sequence A189683 in OEIS).
For a given prime p, the number of such pairs is called the index of irregularity of p. Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.
It was discovered that (p, p − 3) is in fact an irregular pair for {{{1}}}. This is the first and only time this occurs for p < 30000.
Read more about this topic: Regular Prime
Famous quotes containing the word irregular:
“My father and I were always on the most distant terms when I was a boy—a sort of armed neutrality, so to speak. At irregular intervals this neutrality was broken, and suffering ensued; but I will be candid enough to say that the breaking and the suffering were always divided up with strict impartiality between us—which is to say, my father did the breaking, and I did the suffering.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)