Mathematical Citation Quotient (MCQ) 2017: 0.20
On the Equation ax ≡ x (mod b)
Recently Jiménez-Urroz and Yebra [On the Equation ax ≡ x (mod bn )] constructed, for any given a and b, solutions x to the title equation. Moreover they showed how these can be lifted to higher powers of b to obtain a b-adic solution for certain integers b. In this paper we find all positive integer solutions x to the title equation, proving that, for given a and b, there are X/b + Ob (1) solutions x ≤ X. We also show how solutions may be lifted in more generality. Moreover we show that the construction of [Jiménez-Urroz and Yebra, On the Equation ax ≡ x (mod bn )] (and obvious modifications) cannot always find all solutions to ax ≡ x (mod b).