Values of Bernoulli polynomials

Andrew Granville and Zhi-Wei Sun

Vol. 172 (1996), No. 1, 117–137

DOI: 10.2140/pjm.1996.172.117

Abstract

Let B_n(t) be the n-th Bernoulli polynomial. We show that B_p−1(a∕q) − B_p−1 ≡ q(U_p − 1)∕2p (mod p), where U_n is a certain linear recurrence of order [q∕2] which depends only on a,q and the least positive residue of p (mod q). This can be re-written as a sum of linear recurrence sequences of order ≤ ϕ(q)∕2, and so we can recover the classical results where ϕ(q) ≤ 2 (for instance, B_p−1(1∕6) − B_p−1 ≡ (3^p − 3)∕2p + (2^p − 2)∕p (mod p)). Our results provide the first advance on the question of evaluating these polynomials when ϕ(q) > 2, a problem posed by Emma Lehmer in 1938.

Mathematical Subject Classification 2000

Primary: 11B68

Milestones

Received: 21 July 1993

Published: 1 January 1996