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Curious congruences for cyclotomic polynomials.
- Source :
- Research in Number Theory; 11/152022, Vol. 8 Issue 4, p1-10, 10p
- Publication Year :
- 2022
-
Abstract
- Let Φ n (k) (x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Φ n (k) (1) for fixed positive integers n. D. H. Lehmer proved that Φ n (k) (1) / Φ n (1) is a polynomial of the Euler totient function ϕ (n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Φ n (k) (1) / Φ n (1) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: 2 Φ n (3) (1) is divisible by ϕ (n) - 2 . Moreover, if k is greater than 1, then Φ n (2 k + 1) (1) is divisible by ϕ (n) - 2 k . The proof depends on a new combinatorial identity for general self-reciprocal polynomials over Z , which gives rise to a formula that expresses the value Φ n (k) (1) as a Z -linear combination of the coefficients in the minimal polynomial of 2 cos (2 π / n) - 2 . As a supplement, we show the monotonic increasing property of Φ n (x) on [ 1 , ∞) in two ways. [ABSTRACT FROM AUTHOR]
- Subjects :
- EULER polynomials
POLYNOMIALS
CONGRUENCE lattices
Subjects
Details
- Language :
- English
- ISSN :
- 25220160
- Volume :
- 8
- Issue :
- 4
- Database :
- Complementary Index
- Journal :
- Research in Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 160254840
- Full Text :
- https://doi.org/10.1007/s40993-022-00410-0