Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod { (a-b)}$ for all $a>b$. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto\lfloor e^{ 1/a}\;a^x\;x!\rfloor$, with $a\in\Z\setminus\{ 0,1\}$, and a function equal to $\lfloor e\;x!\rfloor$ except on 0. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.
Categories: cs.DM math.NT