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On the equation \(a^x\equiv x\pmod b\). (English) Zbl 1205.11007

Summary: Recently, J. Jiménez-Urroz and J. L. A. Yebra [J. Integer Seq. 12, No. 8, Article ID 09.8.8, 8 p., electronic only (2009; Zbl 1202.11006)] 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 \(X/b + O_b(1)\) solutions \(x \leq X\). We also show how solutions may be lifted in more generality. Moreover we show that the construction of Jiménez-Urroz and Yebra [loc. cit.] (and obvious modifications) cannot always find all solutions to \(a^x \equiv x \pmod b\).

MSC:

11A07 Congruences; primitive roots; residue systems

Citations:

Zbl 1202.11006
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