(HP15C)(HP67)(HP41C) Bernoulli Polynomials

[Albert Chan]

(09-02-2023 05:44 PM)Albert Chan Wrote:  Denominator likely smaller, but required work. d = product(p, for (p-1) | even m)

Proof: B(even m) denominator d, divides D = 2*(2^m-1)

If p=2, (p-1)=1, always divides m, and D also have factor 2.
With even denominator, B(even m) numerator must be odd.

All odd p, such that (p-1) | m, then m = k*(p-1)

2^m - 1 ≡ (2^(p-1))^k - 1 ≡ 1^k - 1 ≡ 0 (mod p)      // Fermat's little theorem

D have all d prime factors --> d divides D