A solution to a problem and the Diophantine equation X^2+bX+c=Y^2

TL;DR

This paper analyzes the Diophantine equation x^2+bx+c=y^2, proving it has finitely many integer solutions, explicitly describing the solution set based on the discriminant, and identifying conditions for exactly two solutions.

Contribution

It provides a complete characterization of solutions for the equation, including explicit formulas and conditions for the number of solutions, which was not previously detailed.

Findings

Finitely many solutions for given b and c.

Explicit description of the solution set based on the discriminant.

Exactly two solutions occur when b^2-4c equals 1, 4, 16, -4, or -16.

Abstract

We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an explicit description of the solution set. Such a description depends on the form of the integer b^2-4c. Some Corollaries do follow. Furthermore, we show that the said equation has exactly two integer solutions, precisely when b^2-4c= 1,4,16,-4,or-16. In each case we list the two solutions in terms of the coefficients b and c.