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On the divisors of \(a^ k+b^ k\). (English) Zbl 0884.11019

Let \(a\), \(b\) be two fixed non-zero coprime integers with \(a\neq\pm b\). A positive integer \(n\) is called good if \(n|(a^k+ b^k)\) for some positive integer \(k\) and bad otherwise. The special case of characterizing and counting the good integers when \(a=2\) and \(b=1\) provides a motivation for this paper and is discussed in the last section.
Several authors have carried out investigations related to the study of good odd primes. In this paper, the author characterizes the good odd prime powers and goes on to obtain a necessary and sufficient condition in terms of its prime divisors for an odd integer to be good. He is then able to establish an asymptotic formula for the number of good integers \(n\leq x\), and this shows that almost all integers are bad. His proof uses results of K. Wiertelak on the density of some sets of primes [Acta Arith. 34, 183-196 (1978; Zbl 0373.10029), ibid. 197-210 (1978; Zbl 0373.10030); ibid. 43, 177-190 (1984; Zbl 0531.10049)] and a triple iterative process.

MSC:

11B83 Special sequences and polynomials
11N25 Distribution of integers with specified multiplicative constraints
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