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

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The discussion focuses on finding integer solutions for the equation (x^2 - R) / (P - 2x) = y, where P and R are extremely large integer values. The participants suggest that the equivalent equation x^2 + Px + R = y must yield a perfect square for y to be an integer. They emphasize the importance of analyzing the discriminant P^2 - 4R to identify potential integer solutions, particularly through the lens of Pythagorean triplets and prime factor decomposition.

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drosser
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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?
 
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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.
 

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