How do you solve for the number of integer solutions for $$\frac{1}{x}+\frac{a}{y}=\frac{1}{b} $$ From what I've tested it seems to be connected to the prime factorization of "a" and "b", as seen when setting $a=b$
$$\frac{1}{x}+\frac{P_n^k}{y}=\frac{1}{P_n^k} $$ the number of integer solutions seems to be $6k+1$
I've found that $$\frac{1}{x}+\frac{P_n^k\cdot P_m^j}{y}=\frac{1}{P_n^k\cdot P_m^j}$$ to have a pattern similar based on the exponents, but I was curious to what the theory is behind this and how to derive a general formula for $\frac{1}{x}+\frac{a}{y}=\frac{1}{b}$
See Marty Cohen's solution in this similar post https://math.stackexchange.com/a/1927080/399263
Transform your equation to $(x-b)(y-ab)=ab^2\iff XY=ab^2$ and you need to find how many ways to factorize $ab^2$.