The Diophantine Equation arctan(1/x)+arctan(m/y)= arctan(1/k)

TL;DR

This paper investigates the solutions to a specific Diophantine equation involving arctangent functions, establishing a precise count of solutions based on divisors of a related number, and extends geometric arctangent identities.

Contribution

It provides a complete characterization of positive integer solutions to the arctangent-based Diophantine equation, linking solutions to divisors of a quadratic-related number.

Findings

Number of solutions equals the number of divisors of k^2+1

Solutions are explicitly given by x=k+m(k^2+1)/d, y=km+d

Includes listing of nine arctangent identities

Abstract

In the fall 2011 issue of the Journal'Mathematics and Computer Education', author Unal Hasan, in the one page article "Proof without Words", gives a purely geometric proof of the equality, arctan(1/3)+ arctan(1/7) = arctan(1/2) (1) (See reference [1]) Now consider the two-variable diophantine equation(x and y being positive integer variables), arctan(1/x) + arctan(m/y) = arctan(1/k) (2), where m and k are given or fixed positive integers with gcd(m,k^2+1)=1;and also with gcd(m,y)=1. Equality (1) then says that the pair (3,7)is a positive integer solution to (2) in the case m=1=k. We prove, in Theorem1(a,) that equation (2) has exactly N(number of positive divisors of k^2+1) distinct positive integer solutions (x,y), given by x=k+m(k^2+1)/d, y=km+d; d a positive divisor of k^2+1. As a result of Th.1, we list nine arctangent equalities in Section5 of this article, including inequality (1) above.