Signed sums of polynomial values (Q2781330)

Let \(f(x)\) be an integer-valued polynomial of degree \(k\) satisfying the necessary condition that there exists no integer \(d>1\) dividing the values \(f(x)\) for all integers \(x\). The author proves that for any given integer \(\ell\), every positive integer \(n\) can be represented in the form \(n=\sum_{i=\ell}^m \varepsilon_i f(i)\) with suitable \(\varepsilon_i=\pm 1\), where \(m\geq \ell\) is an integer depending on \(\ell\), \(n\), and \(f(x)\).NEWLINENEWLINENEWLINEThis generalizes a result of \textit{M. N. Bleicher} [J. Number Theory 56, 36-51 (1996; Zbl 0841.11002)].