Smooth values of some quadratic polynomials

TL;DR

This paper identifies all integer values for which specific quadratic polynomials are smooth with respect to certain bounds, employing a method developed by Luca and the author.

Contribution

It determines all solutions where these quadratic polynomials are 200-smooth or 100-smooth, expanding understanding of smooth values of quadratic polynomials.

Findings

All x such that x^2+1, x^2+4, x^2+2, x^2-2 are 200-smooth are found.

All x such that x^2-4 is 100-smooth are identified.

The method of Luca and the author is applied to solve these smoothness problems.

Abstract

In this paper, using a method of Luca and the author, we find all values $x$ such that the quadratic polynomials $x^2+1,$ $x^2+4,$ $x^2+2$ and $x^2-2$ are 200-smooth and all values $x$ such that the quadratic polynomial $x^2-4$ is 100-smooth.