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congruence of Clausen and von Staudt
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(Theorem)
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Let $B_k$ denote the $k$ th Bernoulli number: $$B_0=1,\quad B_1=-\frac{1}{2},\quad B_2=\frac{1}{6},\quad B_3=0,\quad B_4=-\frac{1}{30},\ldots$$ In fact, $B_k=0$ for all odd $k\geq 3$ , so we will only consider $B_k$ for even $k$ . The following is a well-known congruence, due to Thomas Clausen and Karl von Staudt.
Theorem 1 (Congruence of Clausen and von Staudt)
For an even integer $k\geq 2$ ,
$$B_k \equiv -\sum_{p \text{ prime},\ (p-1)|k} \frac{1}{p} \mod \Ints$$ where the sum is over all primes $p$ such that $(p-1)$ divides $k$ . In other words, there exists an integer $n_k$ such that $$B_k=n_k -\sum_{p \text{ prime},\ (p-1)|k} \frac{1}{p}.$$
For example: $$B_2=\frac{1}{6}=1-\frac{1}{2}-\frac{1}{3}, \quad B_4=-\frac{1}{30}=1-\frac{1}{2}-\frac{1}{3}-\frac{1}{5}.$$ Sometimes the theorem is stated in this alternative form:
Corollary 1 For an even integer $k\geq 2$ and any prime $p$ the product $pB_k$ is $p$ -integral, that is, $pB_k$ is a rational number $t/s$ (in lowest terms) such that $p$ does not divide $s$ . Moreover:
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"congruence of Clausen and von Staudt" is owned by alozano.
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Cross-references: lowest terms, rational number, product, theorem, integer, divides, primes, sum, even integer, congruence, even, odd, Bernoulli number
There is 1 reference to this entry.
This is version 1 of congruence of Clausen and von Staudt, born on 2005-04-19.
Object id is 6957, canonical name is CongruenceOfClausenAndVonStaudt.
Accessed 2888 times total.
Classification:
| AMS MSC: | 11B68 (Number theory :: Sequences and sets :: Bernoulli and Euler numbers and polynomials) |
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Pending Errata and Addenda
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