# Cython wrapper for bernmm library¶

AUTHOR:

• David Harvey (2008-06): initial version
sage.rings.bernmm.bernmm_bern_modp(p, k)

Computes $$B_k \mod p$$, where $$B_k$$ is the k-th Bernoulli number.

If $$B_k$$ is not $$p$$-integral, returns -1.

INPUT:

p – a prime k – non-negative integer

COMPLEXITY:

Pretty much linear in $p$.

EXAMPLES:

sage: from sage.rings.bernmm import bernmm_bern_modp

sage: bernoulli(0) % 5, bernmm_bern_modp(5, 0)
(1, 1)
sage: bernoulli(1) % 5, bernmm_bern_modp(5, 1)
(2, 2)
sage: bernoulli(2) % 5, bernmm_bern_modp(5, 2)
(1, 1)
sage: bernoulli(3) % 5, bernmm_bern_modp(5, 3)
(0, 0)
sage: bernoulli(4), bernmm_bern_modp(5, 4)
(-1/30, -1)
sage: bernoulli(18) % 5, bernmm_bern_modp(5, 18)
(4, 4)
sage: bernoulli(19) % 5, bernmm_bern_modp(5, 19)
(0, 0)

sage: p = 10000019; k = 1000
sage: bernoulli(k) % p
1972762
sage: bernmm_bern_modp(p, k)
1972762

sage.rings.bernmm.bernmm_bern_rat(k, num_threads=1)

Computes k-th Bernoulli number using a multimodular algorithm. (Wrapper for bernmm library.)

INPUT:

• k – non-negative integer

COMPLEXITY:

Pretty much quadratic in $k$. See the paper “A multimodular algorithm for computing Bernoulli numbers”, David Harvey, 2008, for more details.

EXAMPLES:

sage: from sage.rings.bernmm import bernmm_bern_rat

sage: bernmm_bern_rat(0)
1
sage: bernmm_bern_rat(1)
-1/2
sage: bernmm_bern_rat(2)
1/6
sage: bernmm_bern_rat(3)
0
sage: bernmm_bern_rat(100)
-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330
sage: bernmm_bern_rat(100, 3)
-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330