Explicit primality criteria for ℎ⋅2^{𝑘}±1

Abstract

Algorithms are described to obtain explicit primality criteria for integers of the form h ⋅ 2 k ± 1 h \cdot {2^k} \pm 1 (in particular with h divisible by 3) that generalize classical tests for 2 k ± 1 {2^k} \pm 1 in a well-defined finite sense. Numerical evidence (including all cases with h > 10 5 h > {10^5} ) seems to indicate that these finite generalizations exist for every h, unless h = 4 m − 1 h = {4^m} - 1 for some m, in which case it is proved they cannot exist.