A note on cubic residues modulo $$\varvec{n}$$ n

V. P. Ramesh, R. Gowtham and Saswati Sinha
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V. P. Ramesh: Central University of Tamil Nadu
R. Gowtham: Indian Institute of Technology Madras
Saswati Sinha: Central University of Tamil Nadu

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 62-65

Abstract: Abstract The set of all cubic residues of any natural number n is denoted by $$S_n^{(3)}$$ S n ( 3 ) . In this article, we show that, $$\begin{aligned} \vert S_n^{(3)} \vert = \left\{ \begin{array}{ll} 3^{-\omega (n_1)}\phi (n_1)\phi (n_2), &{}\text {if} ~~ n=n_1n_2 \\ 3^{-\omega (n_1)}\phi (n_1)\phi (3n_2), &{}\text {if} ~~ n=3n_1n_2 \\ 3^{-\omega (n_1)}\phi (n_1)\phi (3^{a-1}n_2), &{}\text {if} ~~ n=3^an_1n_2, ~a>1, \end{array} \right. \end{aligned}$$ | S n ( 3 ) | = 3 - ω ( n 1 ) ϕ ( n 1 ) ϕ ( n 2 ) , if n = n 1 n 2 3 - ω ( n 1 ) ϕ ( n 1 ) ϕ ( 3 n 2 ) , if n = 3 n 1 n 2 3 - ω ( n 1 ) ϕ ( n 1 ) ϕ ( 3 a - 1 n 2 ) , if n = 3 a n 1 n 2 , a > 1 , where $$(n_1n_2,3)=1$$ ( n 1 n 2 , 3 ) = 1 , $$n_1$$ n 1 comprises of primes $$p\equiv 1 \pmod 3$$ p ≡ 1 ( mod 3 ) , $$n_2$$ n 2 comprises of primes $$p\equiv 2 \pmod 3$$ p ≡ 2 ( mod 3 ) and $$\omega (n_1)$$ ω ( n 1 ) denotes the number of distinct prime factors of $$n_1$$ n 1 . Consequently, every element of $$ {({\mathbb {Z}}/n{\mathbb {Z}})}^*$$ ( Z / n Z ) ∗ is a cubic residue modulo n if and only if, $$n=3,\prod \limits _{i=1}^{k} p_i^{\ell _i}$$ n = 3 , ∏ i = 1 k p i ℓ i or $$3\prod \limits _{i=1}^{k} p_i^{\ell _i}$$ 3 ∏ i = 1 k p i ℓ i where $$p_i$$ p i ’s are primes such that $$p_i\equiv 2 \pmod 3$$ p i ≡ 2 ( mod 3 ) and $$k,\ell _i$$ k , ℓ i ’s are natural numbers.

Keywords: Cubic residues; Congruences; Primes; 11A07; 11A41 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00230-z

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