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 prime
- k– 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 of- bernoulli_mod_p()function
 - OUTPUT: boolean; - Trueif checksum passed- EXAMPLES: - 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