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24 Bernoulli and Euler PolynomialsProperties

§24.10 Arithmetic Properties

  1. §24.10(i) Von Staudt–Clausen Theorem
  2. §24.10(ii) Kummer Congruences
  3. §24.10(iii) Voronoi’s Congruence
  4. §24.10(iv) Factors

§24.10(i) Von Staudt–Clausen Theorem

Here and elsewhere in §24.10 the symbol p denotes a prime number.

24.10.1 B2n+(p1)|2n1p=integer,

where the summation is over all p such that p1 divides 2n. The denominator of B2n is the product of all these primes p.

24.10.2 pB2np1(modp+1),

where n2, and (1) is an arbitrary integer such that (p1)p|2n. Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference.

§24.10(ii) Kummer Congruences

24.10.3 BmmBnn(modp),

where mn0(modp1).

24.10.4 (1pm1)Bmm(1pn1)Bnn(modp+1),

valid when mn(mod(p1)p) and n0(modp1), where (0) is a fixed integer.

24.10.5 EnEn+p1(modp),

where p(>2) is a prime and n2.

24.10.6 E2nE2n+w(mod2),

valid for fixed integers (0), and for all n(0) and w(0) such that 2|w.

§24.10(iii) Voronoi’s Congruence

Let B2n=N2n/D2n, with N2n and D2n relatively prime and D2n>0. Then

24.10.7 (b2n1)N2n2nb2n1D2nk=1M1k2n1kbM(modM),

where M(2) and b are integers, with b relatively prime to M.

For historical notes, generalizations, and applications, see Porubský (1998).

§24.10(iv) Factors

With N2n as in §24.10(iii)

24.10.8 N2n0(modp),

valid for fixed integers (1), and for all n(1) such that 2n0 (modp1) and p|2n.

24.10.9 E2n{0(modp)if p1(mod4),2(modp)if p3(mod4),

valid for fixed integers (1) and for all n(1) such that (p1)p1|2n.