Bernoulli numbers modulo p¶
AUTHOR:
David Harvey (2006-07-26): initial version
William Stein (2006-07-28): some touch up.
David Harvey (2006-08-06): new, faster algorithm, also using faster NTL interface
David Harvey (2007-08-31): algorithm for a single Bernoulli number mod p
David Harvey (2008-06): added interface to bernmm, removed old code
- sage.rings.bernoulli_mod_p.bernoulli_mod_p(p)[source]¶
Return the Bernoulli numbers \(B_0, B_2, ... B_{p-3}\) modulo \(p\).
INPUT:
p– integer; a prime
OUTPUT:
list – Bernoulli numbers modulo \(p\) as a list of integers [B(0), B(2), … B(p-3)].
ALGORITHM:
Described in accompanying latex file.
PERFORMANCE:
Should be complexity \(O(p \log p)\).
EXAMPLES:
Check the results against PARI’s C-library implementation (that computes exact rationals) for \(p = 37\):
sage: bernoulli_mod_p(37) [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] sage: [bernoulli(n) % 37 for n in range(0, 36, 2)] [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2]
>>> from sage.all import * >>> bernoulli_mod_p(Integer(37)) [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] >>> [bernoulli(n) % Integer(37) for n in range(Integer(0), Integer(36), Integer(2))] [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2]
Boundary case:
sage: bernoulli_mod_p(3) [1]
>>> from sage.all import * >>> bernoulli_mod_p(Integer(3)) [1]
AUTHOR:
David Harvey (2006-08-06)
- sage.rings.bernoulli_mod_p.bernoulli_mod_p_single(p, k)[source]¶
Return the Bernoulli number \(B_k\) mod \(p\).
If \(B_k\) is not \(p\)-integral, an
ArithmeticErroris raised.INPUT:
p– integer; a primek– nonnegative integer
OUTPUT: the \(k\)-th Bernoulli number mod \(p\)
EXAMPLES:
sage: bernoulli_mod_p_single(1009, 48) 628 sage: bernoulli(48) % 1009 628 sage: bernoulli_mod_p_single(1, 5) Traceback (most recent call last): ... ValueError: p (=1) must be a prime >= 3 sage: bernoulli_mod_p_single(100, 4) Traceback (most recent call last): ... ValueError: p (=100) must be a prime sage: bernoulli_mod_p_single(19, 5) 0 sage: bernoulli_mod_p_single(19, 18) Traceback (most recent call last): ... ArithmeticError: B_k is not integral at p sage: bernoulli_mod_p_single(19, -4) Traceback (most recent call last): ... ValueError: k must be nonnegative
>>> from sage.all import * >>> bernoulli_mod_p_single(Integer(1009), Integer(48)) 628 >>> bernoulli(Integer(48)) % Integer(1009) 628 >>> bernoulli_mod_p_single(Integer(1), Integer(5)) Traceback (most recent call last): ... ValueError: p (=1) must be a prime >= 3 >>> bernoulli_mod_p_single(Integer(100), Integer(4)) Traceback (most recent call last): ... ValueError: p (=100) must be a prime >>> bernoulli_mod_p_single(Integer(19), Integer(5)) 0 >>> bernoulli_mod_p_single(Integer(19), Integer(18)) Traceback (most recent call last): ... ArithmeticError: B_k is not integral at p >>> bernoulli_mod_p_single(Integer(19), -Integer(4)) Traceback (most recent call last): ... ValueError: k must be nonnegative
Check results against
bernoulli_mod_p:sage: bernoulli_mod_p(37) [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] sage: [bernoulli_mod_p_single(37, n) % 37 for n in range(0, 36, 2)] [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] sage: bernoulli_mod_p(31) [1, 26, 1, 17, 1, 9, 11, 27, 14, 23, 13, 22, 14, 8, 14] sage: [bernoulli_mod_p_single(31, n) % 31 for n in range(0, 30, 2)] [1, 26, 1, 17, 1, 9, 11, 27, 14, 23, 13, 22, 14, 8, 14] sage: bernoulli_mod_p(3) [1] sage: [bernoulli_mod_p_single(3, n) % 3 for n in range(0, 2, 2)] [1] sage: bernoulli_mod_p(5) [1, 1] sage: [bernoulli_mod_p_single(5, n) % 5 for n in range(0, 4, 2)] [1, 1] sage: bernoulli_mod_p(7) [1, 6, 3] sage: [bernoulli_mod_p_single(7, n) % 7 for n in range(0, 6, 2)] [1, 6, 3]
>>> from sage.all import * >>> bernoulli_mod_p(Integer(37)) [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] >>> [bernoulli_mod_p_single(Integer(37), n) % Integer(37) for n in range(Integer(0), Integer(36), Integer(2))] [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] >>> bernoulli_mod_p(Integer(31)) [1, 26, 1, 17, 1, 9, 11, 27, 14, 23, 13, 22, 14, 8, 14] >>> [bernoulli_mod_p_single(Integer(31), n) % Integer(31) for n in range(Integer(0), Integer(30), Integer(2))] [1, 26, 1, 17, 1, 9, 11, 27, 14, 23, 13, 22, 14, 8, 14] >>> bernoulli_mod_p(Integer(3)) [1] >>> [bernoulli_mod_p_single(Integer(3), n) % Integer(3) for n in range(Integer(0), Integer(2), Integer(2))] [1] >>> bernoulli_mod_p(Integer(5)) [1, 1] >>> [bernoulli_mod_p_single(Integer(5), n) % Integer(5) for n in range(Integer(0), Integer(4), Integer(2))] [1, 1] >>> bernoulli_mod_p(Integer(7)) [1, 6, 3] >>> [bernoulli_mod_p_single(Integer(7), n) % Integer(7) for n in range(Integer(0), Integer(6), Integer(2))] [1, 6, 3]
AUTHOR:
David Harvey (2007-08-31)
David Harvey (2008-06): rewrote to use bernmm library
- sage.rings.bernoulli_mod_p.verify_bernoulli_mod_p(data)[source]¶
Compute checksum for Bernoulli numbers.
It checks the identity
\[\sum_{n=0}^{(p-3)/2} 2^{2n} (2n+1) B_{2n} \equiv -2 \pmod p\](see “Irregular Primes to One Million”, Buhler et al)
INPUT:
data– list; same format as output ofbernoulli_mod_p()function
OUTPUT: boolean;
Trueif checksum passedEXAMPLES:
sage: from sage.rings.bernoulli_mod_p import verify_bernoulli_mod_p sage: verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(3))) True sage: verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(1000))) True sage: verify_bernoulli_mod_p([1, 2, 4, 5, 4]) True sage: verify_bernoulli_mod_p([1, 2, 3, 4, 5]) False
>>> from sage.all import * >>> from sage.rings.bernoulli_mod_p import verify_bernoulli_mod_p >>> verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(Integer(3)))) True >>> verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(Integer(1000)))) True >>> verify_bernoulli_mod_p([Integer(1), Integer(2), Integer(4), Integer(5), Integer(4)]) True >>> verify_bernoulli_mod_p([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5)]) False
This one should test that long longs are working:
sage: verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(20000))) True
>>> from sage.all import * >>> verify_bernoulli_mod_p(bernoulli_mod_p(next_prime(Integer(20000)))) True
AUTHOR: David Harvey