2018/07/05 by Füsun Akman, Akman, Fusun
Mathematics · #11A41 #11A63 #11N32 #11Y05 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1807.02195
openalex publication_date 2018/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable x. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique "base-x" representation of such polynomials that makes the ring ℤ[x] into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes p and certain infinite sets of irreducible polynomials f(x) that attain the value p at sufficiently large x, each generated in finitely many steps from the pth cyclotomic polynomial. The base-x representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.