modular_bernoulli_numbers_numberphile.sf

#!/usr/bin/ruby

# A nice recursive formula for computing the n-th Bernoulli number (modulo another integer).

# Formula presented in the following Numberphile video:
#   Faulhaber's Fabulous Formula (and Bernoulli Numbers) - Numberphile
#   https://youtube.com/watch?v=83NFR7JDlww

func modular_bernoulli_numberphile(n, m) is cached {

    return 1          if (n == 0)
    return ((1/2)%m)  if (n == 1)
    return 0          if (n.is_odd)

    var x = PolyMod(1, m)
    var t = ((x - 1)**(n+1) - x**(n+1))

    var cf = t.coeffs
    var lt = cf.pop

    var term = cf.sum_2d {|a,b|
        mulmod(__FUNC__(a, m), b, m)
    }

    divmod(term, -lt[1], m)
}

var m = next_prime(1e9)

for n in (0..20) {
    var B = modular_bernoulli_numberphile(n, m)
    assert_eq(B, n.bernoulli % m)
    say "B_#{n} = #{B.as_rat}"
}