Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.8

On the Finiteness of Bernoulli Polynomials Whose Derivative Has Only Integral Coefficients

Bernd C. Kellner
Göppert Weg 5
37077 Göttingen


It is well known that the Bernoulli polynomials Bn(x) have nonintegral coefficients for n ≥ 1. However, ten cases are known so far in which the derivative B'n(x) has only integral coefficients. One may assume that the number of those derivatives is finite. We can link this conjecture to a recent conjecture about the properties of a product of primes satisfying certain p-adic conditions. Using a related result of Bordellès, Luca, Moree, and Shparlinski, we then show that the number of those derivatives is indeed finite. Furthermore, we derive other characterizations of the primary conjecture. Subsequently, we extend the results to higher derivatives of the Bernoulli polynomials. This provides a product formula for these denominators, and we show similar finiteness results.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000040 A001221 A027642 A064538 A094960 A144845 A195441 A324369 A324370 A324371.)

Received October 3 2023; revised version received February 15 2024. Published in Journal of Integer Sequences, February 19 2024.

Return to Journal of Integer Sequences home page