Connecting primitive roots and permutations

V. P. Ramesh (), M. Makeshwari () and Saswati Sinha ()
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V. P. Ramesh: Central University of Tamil Nadu
M. Makeshwari: Central University of Tamil Nadu
Saswati Sinha: Central University of Tamil Nadu

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 513-516

Abstract: Abstract Let n be any natural number such that 2 is a primitive root of $$2n+1$$ 2 n + 1 . In this article, we prove that the permutation (n!) has two orbits if and only if $$2n+1=p^2$$ 2 n + 1 = p 2 for some odd prime p, where $$(n!)={\prod \limits _{k=0}^{n-1}(1,2,\dots ,(n-k))}$$ ( n ! ) = ∏ k = 0 n - 1 ( 1 , 2 , ⋯ , ( n - k ) ) .

Keywords: Primitive root; Permutation; Orbits of permutation; 11A07; 20B30 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00384-4

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