Integral points on rational curves of the form y=(x^2+bx+c)/(x+a); a,b,c integers

TL;DR

This paper characterizes the integral points on rational curves of the form (x^2+bx+c)/(x+a), providing explicit parametrizations and counting formulas based on number theoretic properties of the coefficients.

Contribution

It offers a complete classification and parametrization of integral points on these rational curves, extending previous results to a broader class with explicit formulas.

Findings

Finitely many integral points when b^2-4c=0, with explicit parametrizations.

Exactly 4N integral points in the general case, with N being the number of divisors.

Special cases where the count differs by 2, depending on perfect square conditions.

Abstract

The subject matter of this work is the set of integral points(i.e. points with both coordinates integers) on the graphs of rational functions of the form f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we establish Proposition1 in Section2. This proposition plays a key role in the proof of Theorem1 in Section5. Proposition1 is proved with the aid of Euclid's lemma and another well known result in number theory; see reference [1].In Sections3 and4, we focus on the special case b^2-4c=0; which implies b=2d and c=d^2, for some integer d. If d and a are distinct; then there are finitely many integral points, parametrically described in Results1 and2. Theorem1 in Sec.5 deals with the general case.Accordingly, if a^2-ab+c is not zero; there are exactly 4N distinct integral points parametrically described.Except in the cases where a^2-ab+c is a perfect square, or minus a perfect square; in which cases thare are exactly 4N-2 distinct integral points. Here N stands for the number of positive divisors, not exceeding the square root of the absolute value of a^2-ab+c. (divisors of that absolute value).The paper concludes with Th.2, which is a direct application of Th.1 in the cases where the above absolute value is 1, p, or p^2; p a prime.