I have encountered a problem, that is:
Find all integers $a,b,c,d$, so that $$ \frac{\pi}{4}=a\arctan\left(\frac1b\right)+c\arctan\left(\frac1d\right). $$
Is it possible to find all solutions satisfying the upper equation?
the equation is equivalent to $(b+i)^a(d+i)^c=r(1+i)$, which $r$ some unknown integers. Using the theory of Gaussian Integers, and define $N(x+yi)=x^2+y^2$, so $N(\alpha\beta)=N(\alpha)N(\beta)$, for $N(1+i)$ is prime number, so $1+i$ is prime in Gaussian, all these result is fundamental. so there must be $(1+i)\mid(b+i)$ or $(1+i)\mid(d+i)$. but this conclusion is not sufficient to get $(b+i)^a(d+i)^c$ have the same real part and imaginary part. so... any help?