Integral points on hyperbolas: A special case

TL;DR

This paper investigates integral solutions on hyperbolas defined by quadratic equations with integer coefficients, providing methods to determine and explicitly find all solutions in various special cases.

Contribution

It introduces a new algebraic approach to reduce the problem to linear factors and explicitly characterizes solutions for specific parameter cases.

Findings

When I≠0, only finitely many integer solutions exist.

Explicit formulas for solutions when I=2^n, n≥2.

Hyperbola solutions are finite or infinite depending on I=0 or not.

Abstract

The subject matter of this work is integral points on conics described by the general equation, ax^2+bxy+cy^2+dx+ey+f=0 (1) where the six coefficients are integers satisfying the conditions, b^2-4ac=k^2, with a and c being nonzero and k a positive integer. It is well known the when b^2-4ac>0, equation (1) describes either a hyperbola on the plane or a pair of two straight lines(the degenerate case). The key integer is the number, I=k^2(d^2-4af)-(2ae-bd)^2. In Section 2, we show via a straightforward algebraic method that equation (1) can be put in the form, g(x,y)h(x,y)=I, where g(x,y) and h(x,y) are linear polynomials in x and y with integer coefficients. Thus, when I is not zero, equation (1) has only finitely many integer solutions (x,y). The process of finding these solutions is outlined in Section 3. In Section 4,we give a detailed numerical example. In Section 5, we offer some observations and remarks. In Section 6, we discuss the case b=d=e=0. In Section 7 we discuss the case a=1=k, and we show that when I=2^n, n>or=2, the above hyperbola has exactly 2(n-1) integral points, all explicitly given. In Section 10, we prove that these points are distinct. In Section 8 we discuss the case I=0, the hyperbola in (1) has either infinitely many integral points: or no integral points. Finally in Section 9, we discuss the special case d=e=f=0.