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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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References:
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