What is the best method for finding integer solutions to a challenging function?

[drosser]

Is there a quick way to find integer values of x that give integer values for y?

(x^2-R)/(P-2x)=y

sqrt(R) rounded down<x<P/2

an equivalent equation is

x^2+Px+R=y y= a perfect square

sqrt(x^2+Px+R)= integer

P and R are integer values. They are very large.
P=1.720901664588208977632751606930114527882871349707453690712637328347852193783039275682367157744911327176901e+106

R=1.611966555644167779663503662738501807226651661942209780569274299995114404468640924608971613224013135298666e+105

Maybe a generalized equation or a program?

 

[fresh_42]

You could search for it in a brute force approach. Otherwise we probably need more information about $P,R$, e.g. if $P^2-4R$ is a square in which case we have Pythagorean triplets as solutions, or the prime factor decomposition of them.