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EUCLIDEAN POLYNOMIALS FOR CERTAIN ARITHMETIC PROGRESSIONS AND THE MULTIPLICATIVE GROUP OF F-p2
Date
2022-06-01
Author
Berktav, Kadri İlker
Özbudak, Ferruh
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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.
Subject Keywords
Primes in arithmetic progression
,
Euclidean polynomials
,
Euclidean proofs
,
finite fields
,
PRIME-NUMBERS
URI
https://hdl.handle.net/11511/100510
Journal
QUAESTIONES MATHEMATICAE
DOI
https://doi.org/10.2989/16073606.2022.2077261
Collections
Department of Mathematics, Article
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K. İ. Berktav and F. Özbudak, “EUCLIDEAN POLYNOMIALS FOR CERTAIN ARITHMETIC PROGRESSIONS AND THE MULTIPLICATIVE GROUP OF F-p2,”
QUAESTIONES MATHEMATICAE
, pp. 0–0, 2022, Accessed: 00, 2022. [Online]. Available: https://hdl.handle.net/11511/100510.