von Staudt-Clausen Theorem

The von Staudt-Clausen theorem, sometimes also known as the Staudt-Clausen theorem (Carlitz 1968), states that

 B_(2n)=A_n-sum_(p_k; (p_k-1)|2n)1/(p_k),

where B_(2n) is a Bernoulli number, A_n is an integer, and the p_ks are the primes satisfying (p_k-1)|(2n), i.e., p_k-1 divides 2n.

For example, for n=1, the primes included in the sum are 2 and 3, since (2-1)|2 and (3-1)|2, giving


Similarly, for n=6, the included primes are (2, 3, 5, 7, 13), since (1, 2, 4, 6, 12) divide 12=2·6, giving


The first few values of A_n for n=1, 2, ... are 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, ... (OEIS A000146), and the lists of primes appearing in successive sums are 2, 3; 2, 3, 5; 2, 3, 7; 2, 3, 5; 2, 3, 11; ... (OEIS A080092).

The theorem was rediscovered by Ramanujan (Hardy 1999, p. 11) and can be proved using p-adic Numbers.

See also

Bernoulli Number, p-adic Number

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Carlitz, L. "Bernoulli Numbers." Fib. Quart. 6, 71-85, 1968.Clausen, T. "Theorem." Astron. Nach. 17, 351-352, 1840.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 109, 1996.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Wright, E. M. "The Theorem of von Staudt" and "Proof of von Staudt's Theorem." §7.9-7.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 90-93, 1979.Rado, R. "A New Proof of a Theorem of V. Staudt." J. London Math. Soc. 9, 85-88, 1934.Rado, R. "A Note on the Bernoullian Numbers." J. London Math. Soc. 9, 88-90, 1934.Sloane, N. J. A. Sequences A000146/M1717 and A080092 in "The On-Line Encyclopedia of Integer Sequences."Staudt, K. G. C. von. "Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend." J. reine angew. Math. 21, 372-374, 1840.

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von Staudt-Clausen Theorem

Cite this as:

Weisstein, Eric W. "von Staudt-Clausen Theorem." From MathWorld--A Wolfram Web Resource.

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