Kummer's congruence

Template:Short description In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Template:Harvs.

Template:Harvtxt used Kummer's congruences to define the p-adic zeta function.

Statement

The simplest form of Kummer's congruence states that

$\frac{B_h}{h}\equiv \frac{B_k}{k}(\mathrm{mod}p)\text{ whenever }h\equiv k(\mathrm{mod}p-1)$

where p is a prime, h and k are positive even integers not divisible by p−1 and the numbers B_h are Bernoulli numbers.

More generally if h and k are positive even integers not divisible by p − 1, then

$(1-p^{h-1})\frac{B_h}{h}\equiv (1-p^{k-1})\frac{B_k}{k}(\mathrm{mod}{p}^{a+1})$

whenever

$h\equiv k(\mathrm{mod}\phi (p^{a+1}))$

where φ(p^a+1) is the Euler totient function, evaluated at p^a+1 and a is a non negative integer. At a = 0, the expression takes the simpler form, as seen above. The two sides of the Kummer congruence are essentially values of the p-adic zeta function, and the Kummer congruences imply that the p-adic zeta function for negative integers is continuous, so can be extended by continuity to all p-adic integers.

See also

• Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers
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References

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