Research Article

Euclidean polynomials for certain arithmetic progressions and the multiplicative group of ???? p 2

Published in: Quaestiones Mathematicae
Volume 46 , issue 7 , 2023 , pages: 1283–1292

DOI: 10.2989/16073606.2022.2077261

Author(s): Kadri İlker Berktav Institute of Applied Mathematics, Middle East Technical University, Turkey , Ferruh Özbudak Middle East Technical University, Turkey

Keywords: 11A41 , 11N13 , 11T06 , 12E20 , Primes in arithmetic progression , Euclidean polynomials , Euclidean proofs , finite fields

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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 be a finite field with p ² elements. We use that the multiplicative group of is cyclic in our proof.

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