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Cubic reciprocity and explicit primality tests for $$h\cdot 3^k\pm1$$. (English) Zbl 1103.11039
van der Poorten, Alf (ed.) et al., High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Selected papers from the international conference on number theory, Banff, AB, Canada, May 24–30, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3353-7/hbk). Fields Institute Communications 41, 77-89 (2004).
This is a thematical paper which succeeds to captivate with a simple classical problem. The use of quadratic reciprocity leads to deterministic tests for the Mersenne numbers $$M_n = 2^n - 1$$ and the so-called Proth numbers, $$P(n, k) = k 2^n \pm 1$$, which are generalizations thereof. By replacing the power of two by a power of three, one has a variant problem setting, which can be addressed by use of the cubic reciprocity. This is an obvious observation but has been seldom pursued concretely.
The author addresses the question in a general form, aiming to find covering sets of conditions for fixed $$n$$ or $$k$$ and touches thus en passant another interesting combinatorial problem going back to Sierpiński and which also interested Erdős. As a byproduct, he encounters values of $$k$$ such that $$k \cdot 3^n - 1$$ is composite for all $$n$$, a frequent question of the covering set problems.
For the entire collection see [Zbl 1051.11008].

##### MSC:
 11Y11 Primality 11A07 Congruences; primitive roots; residue systems 11A15 Power residues, reciprocity