If divides the numerator of the Bernoulli number for , then is called an irregular pair. For , the irregular pairs of various forms are for , for , none for , and for .

Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113-120, 1975.

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Weisstein, Eric W. "Irregular Pair." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IrregularPair.html