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In the fall 2011 issue of the Journal'Mathematics and Computer Education',author Unal Hasan, in the one page article "Proof without Words", gives apurely 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(xand 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 withgcd(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, inTheorem1(a,) that equation (2) has exactly N(number of positive divisors ofk^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 ninearctangent equalities in Section5 of this article, including inequality (1)above.

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