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Korselt numbers and sets. (English) Zbl 1214.11013
This paper introduces a generalization of Carmichael numbers. For \(\alpha \in \mathbb{Z}\setminus \{0\}\) a positive integer \(N\) is said to be an \(\alpha \)-Korselt number if \(N\neq \alpha \) and \(p - \alpha \) divides \(N -\alpha \) for each prime divisor \(p\) of \(N\). Carmichael numbers are \(1\)-Korselt numbers. The authors are concerned with both a numerical and theoretical study of composite squarefree \(\alpha \)-Korselt numbers.
The paper contains two main results. The first one gives the following properties.
(i) If \(\alpha \leq 1\), then each composite squarefree \(\alpha \)-Korselt number has at least three prime factors.
(ii) Suppose that \(\alpha > 1\). Let \(p < q\) be two prime numbers and \(N:= pq\). If \(N\) is an \(\alpha \)-Korselt number, then \(p < q \leq 4\alpha - 3\). In particular, there are only finitely many \(\alpha \)-Korselt numbers with exactly two prime factors.
A positive integer which is both an \(\alpha \)-Korselt and a \((-\alpha)\)-Korselt number is referred to as an \(\alpha \)-Williams number. The second main result of this paper shows that if \(p, 3p - 2, 3p + 2\) are all prime, then their product is a (\(3p\))-Williams number.
Some conjectures are presented, e.g. that for each \(\alpha\neq 0\) there are infinitely many \(\alpha \)-Korselt numbers. Further, the open question of whether there exists a \(1\)-Williams number still remains open.

MSC:
11A51 Factorization; primality
11Y11 Primality
11Y16 Number-theoretic algorithms; complexity
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