Cubefree Trinomial Discriminants

2019/01/11 by William Craig, Craig, William
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1901.03653

openalex publication_date 2019/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The discriminant of a polynomial of the form ± xⁿ ± x^m ± 1 has the form nⁿ ± m^m(n-m)^n-m when n,m are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in the classification of these values of n,m. We prove that there are infinitely many pairs of integers n,m for which this discriminant has no prime cube divisors. This result is extended to show that for infinitely many fixed m, there are infinitely many n for which the discriminant has no prime cube divisor.

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