Abstract We determine the solutions of Goła̧b-Schinzel type functional equation in class non-periodic functions. Applying this result we give a positive answer to problem raised by E. Jabłońska (Aequationes Math 87:125–133, 2014).

Abstract We point out to a connection between problem of invariance power series families probability distributions under binomial thinning and functional equations which generalize both the Cauchy an additive form Gołąb–Schinzel equation. solve these in several settings with no or mild regularity assumptions imposed on unknown functions.

An old conjecture of Sierpiński asserts that for every integer k > 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particul...