In [1] Almkvist and Meurman proved a result on the
values of the Bernoulli polynomials (Theorem 5 below). Subsequently,
Sury [5] and Bartz and Rutkowski [2]
have given simpler proofs.
In this paper we show how this theorem can be obtained from classical results on
the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a
polynomial with rational coefficients which is integer-valued on the integers is
ℤ(p)-valued on ℤ(p). Here
ℤ(p) denotes the ring of rational numbers whose
denominator is not divisible by the prime p. An application is given in
Section 3 to the arithmetic of generalised Bernoulli numbers.