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New primality criteria and factorizations of 2^{𝑚}±1

1975; American Mathematical Society; Volume: 29; Issue: 130 Linguagem: Inglês

10.1090/s0025-5718-1975-0384673-1

1088-6842

John Brillhart, D. H. Lehmer, J. L. Selfridge,

Advanced Mathematical Theories and Applications

A collection of theorems is developed for testing a given integer N for primality. The first type of theorem considered is based on the converse of Fermat’s theorem and uses factors of N − 1 N - 1 . The second type is based on divisibility properties of Lucas sequences and uses factors of N + 1 N + 1 . The third type uses factors of both N − 1 N - 1 and N + 1 N + 1 and provides a more effective, yet more complicated, primality test. The search bound for factors of N ± 1 N \pm 1 and properties of the hyperbola N = x 2 − y 2 N = {x^2} - {y^2} are utilized in the theory for the first time. A collection of 133 new complete factorizations of 2 m ± 1 {2^m} \pm 1 and associated numbers is included, along with two status lists: one for the complete factorizations of 2 m ± 1 {2^m} \pm 1 ; the other for the original Mersenne numbers.