Research Article

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

Published in: Quaestiones Mathematicae
Volume 47 , issue 12 : Entrepreneurship and Africa’s Cultural Context , 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 = pk 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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