Research Article

Sophie Germain prime p and the permutation of product of first p cycles

Published in: Quaestiones Mathematicae
Volume 47 , issue 12 , 2024 , pages: 2479–2484

DOI: 10.2989/16073606.2024.2374787

Author(s): M. Makeshwari Central University of Tamil Nadu, India , V.P. Ramesh Central University of Tamil Nadu, India , R. Thangadurai Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, India

Keywords: Primary: Primary: 20B30 , Secondary: Secondary: 11A41 , Primitive root , Sophie Germain prime , permutation , orbits of permutation

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Abstract

For a natural number n, the permutation (n!) is defined as the left-to-right product of the first n cycles, namely, . In this article, we prove that for any natural number n, 2 is a primitive root of 2n + 1 if and only if 2n + 1 = p^k for some odd prime number p and for some natural number k such that the permutation (n!) has exactly k orbits. We also prove that a prime number p is a Sophie Germain prime if and only if the permutation (p!) has at most two orbits.

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