On primitive roots modulo \(p^{\ell}\)
In Adv. Math., Beijing 1987, No.2, 214-215 (1987) the first author announced that for arbitrary nonzero elements a,b and c in the finite field \(GF(p^ n)\), where \(p\neq 2\) and \(p^ n\geq 2^{60}\), there exist two primitive roots \(\alpha\) and \(\beta\) such that \(a\alpha +b\beta =c.\) In this note a similar result is proved for the ring \({\mathbb{Z}}/p^ n{\mathbb{Z}}\). If p is an odd prime, \(p^ n>2^{60}\) and a,b,c are given integers with \(p\nmid abc\), then the authors prove that there exist at least \((p-2)p^{n-2}\) pairs of primitive roots \(\alpha\) and \(\beta\) modulo \(p^ n\) such that \(a\alpha +b\beta \equiv c (\mod p^ n).\)
scientific article; zbMATH DE number 4101281