Finding Squares in a Product of Squares

TL;DR

This paper investigates positive integer solutions to the equation where the product of terms of the form (k^2+1) equals a perfect square, using analytic number theory techniques.

Contribution

It introduces novel analytic number theory methods to analyze solutions to the product-of-squares Diophantine equation.

Findings

Identifies conditions under which solutions exist.

Provides bounds on solutions for certain n.

Establishes new connections between product equations and number theory.

Abstract

We wish to discuss positive integer solutions to the Diophantine equation $$\prod_{k=1}^n(k^2+1)=b^2.$$ Some methods in analytic number theory will be used to tackle this problem.