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On the Finiteness of Bernoulli Polynomials Whose Derivative Has Only Integral Coefficients
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Bernd C. Kellner

Göppert Weg 5

37077 Göttingen

Germany

**Abstract:**

It is well known that the Bernoulli polynomials **B**_{n}(*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.

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(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.

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