## Rocky Mountain Journal of Mathematics

### On the middle coefficient of $\Phi _{3p_2p_3}$

Alaa Al-Kateeb

#### Abstract

The middle coefficient of a polynomial of an even degree $d$ is the coefficient of $x^{{d}/{2}}$. In this note we compute the middle coefficient of the cyclotomic polynomial $\Phi _{3p_2p_3}(x)$ when $p_3\equiv \pm 2,\pm 4, \pm 5\bmod 3p_2$.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 6 (2019), 1755-1767.

Dates
First available in Project Euclid: 3 November 2019

https://projecteuclid.org/euclid.rmjm/1572746416

Digital Object Identifier
doi:10.1216/RMJ-2019-49-6-1755

Mathematical Reviews number (MathSciNet)
MR4027231

#### Citation

Al-Kateeb, Alaa. On the middle coefficient of $\Phi _{3p_2p_3}$. Rocky Mountain J. Math. 49 (2019), no. 6, 1755--1767. doi:10.1216/RMJ-2019-49-6-1755. https://projecteuclid.org/euclid.rmjm/1572746416

#### References

• A. Al-Kateeb, Structures and properties of cyclotomic polynomials, Ph.D. thesis, North Carolina State University (2016).
• A.S. Bang, Om Ligningen $\phi_n(x) = 0$, Nyt Tidsskr. Math. 6 (1895), 6–12.
• M. Beiter, Coefficients of the cyclotomic polynomial $F_{3qr} (x)$, Fibonacci Quart. 16 (1978), 302–306.
• G. Dresden, On the middle coefficient of a cyclotomic polynomial, Amer. Math. Monthly 111 (2004), no. 6, 531–533.
• T.Y. Lam and K.H. Leung, On the cyclotomic polynomial $\Phi_{pq} (X)$, Amer. Math. Monthly 103 (1996), no. 7, 562–564.
• M. Beiter, The midterm coefficient of the cyclotomic polynomial $F_{pq}(x)$, Amer. Math. Monthly 71 (1964), no. 7, 769–770.
• R. Thangadurai, On the coefficients of cyclotomic polynomials, pp. 311–322 in Cyclotomic fields and related topics (Pune, 1999), Bhaskaracharya Pratishthana, Pune (2000).