
TL;DR
This paper investigates the distribution of squarefree parts of polynomial values modulo primes, establishing conditions for divisibility, proposing a conjecture on residue classes, and proving special cases using elementary and advanced number theory.
Contribution
It introduces a conjecture about the distribution of squarefree parts of polynomial values modulo primes and proves it for degree 1 and 2 polynomials, linking higher degrees to elliptic curve conjectures.
Findings
Necessary and sufficient conditions for set S to contain multiples of p
Conjecture that S contains infinitely many representatives in each nonzero residue class for large p
Partial proofs for degrees 1, 2, and 3, and a local analogue for arbitrary degree
Abstract
Given a separable nonconstant polynomial with integer coefficients, we consider the set consisting of the squarefree parts of all the rational values of , and study its behavior modulo primes. Fixing a prime , we determine necessary and sufficient conditions for to contain an element divisible by . Furthermore, we conjecture that if is large enough, then contains infinitely many representatives from every nonzero residue class modulo . The conjecture is proved by elementary means assuming has degree 1 or 2. If has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the Parity Conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Coding theory and cryptography
