EUCLIDEAN POLYNOMIALS FOR CERTAIN ARITHMETIC PROGRESSIONS AND THE MULTIPLICATIVE GROUP OF F-p2

Berktav K. I., ÖZBUDAK F.

QUAESTIONES MATHEMATICAE, vol.46, no.7, pp.1283-1292, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 46 Issue: 7
  • Publication Date: 2023
  • Doi Number: 10.2989/16073606.2022.2077261
  • Journal Name: QUAESTIONES MATHEMATICAE
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.1283-1292
  • Keywords: Primes in arithmetic progression, Euclidean polynomials, Euclidean proofs, finite fields, PRIME-NUMBERS
  • Middle East Technical University Affiliated: Yes

Abstract

Let f(x) be a polynomial with integer coefficients. We say that the prime p is a prime divisor of f(x) if p divides f(m) some integer m. For each positive integer n, we give an explicit construction of a polynomial all of whose prime divisors are +/- 1 modulo (8n + 4). Consequently, this specific polynomial serves as an "Euclidean" polynomial for the Euclidean proof of Dirichlet's theorem on primes in the arithmetic progression +/- 1 (mod 8n + 4). Let F-p2 be a finite field with p(2) elements. We use that the multiplicative group of F-p2 is cyclic in our proof.