Specific equations of the form below (for different given values of p, a prime number) will either have positive integer solutions for $x$ & $y$, or will not have any integer solutions.
$${2^{p-1} -1 \over p} = 2pxy + x + y$$
Is there a way to determine when an equation of this form has integer solutions and when it does not? (Other than just trying values in a brute force manner and eventually giving up?) A repeatable algorithmic method of determining whether there are integer solutions would be ideal, does this exist? If not, are there any tricks or techniques that work sometimes, or any suggested approaches?
For example, these have positive integer solutions:
when p=11, (1/11)*(2^(10)-1)=2*11*x*y+x+y has integer solution x=1 & y=4
when p=23, (1/23)*(2^(22)-1)=2*23*x*y+x+y has integer solution x=1 & y=3880
when p=29, (1/29)*(2^(28)-1)=2*29*x*y+x+y has integer solution x=4 & y=39727
when p=37, (1/37)*(2^(36)-1)=2*37*x*y+x+y has integer solution x=3 & y=8328624
And these do not have (positive) integer solutions:
when p=7, (1/7)*(2^(6)-1)=2*7*x*y+x+y has no integer solution
when p=13, (1/13)*(2^(12)-1)=2*13*x*y+x+y has no integer solution
when p=19, (1/19)*(2^(18)-1)=2*19*x*y+x+y has no integer solution
when p=31, (1/31)*(2^(30)-1)=2*31*x*y+x+y has no integer solution