Discriminators of quadratic polynomials

2013/08/17 by Soohyun Park, Park, Soohyun
Mathematics · #Advanced Mathematical Identities #Mathematics and Applications #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.1308.3754

Abstract

Given f ∈ ℤ[x] and n ∈ \mathbbZ⁺, the discriminator D_f(n) is the smallest positive integer m such that f(1), …, f(n) are distinct mod m. In a recent paper, Z.-W. Sun proved that D_f(n) = d^{\lceil log_d n \rceil} if f(x) = x(dx - 1) for d ∈ \2, 3\. We extend this result to d = 2^r for any r ∈ ℤ⁺ and find that D_f(n) = 2^{\lceil log₂ n \rceil} in this case. We also provide more general statements for d = p^r, where p is a prime. In addition, we present a potential method for generating prime numbers with discriminators of polynomials which do not always take prime values. Finally, we describe some general statements and possible topics for study about the discriminator of an arbitrary polynomial with integer coefficients.

Related

• Quadratic sequences with prime power discriminators
• On a discriminator for the polynomial f(x)=x³+x
• Squarefree discriminants of polynomials with prime coefficients
• Cubefree Trinomial Discriminants
• p-adic root separation and the discriminant of integer polynomials