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The study establishes divisibility conditions and derives coefficient formulas using Ramanujan sums, enhancing understanding of cyclotomic polynomials.
We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomial and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the coefficients of cyclotomic polynomials and propose a recursive relation between the coefficients of two cyclotomic polynomials whose indexes differ by a prime factor.
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Carli et al. (2026) studied this question.
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