Finding prime numbers such that....

2, 3, 5 and 7 are prime numbers.

5 - 2 = 3, 5 - 3 = 2, 7 - 2 = 5 and 7 - 5 = 2 produce prime numbers.

Now, I'm interested to find all pairs of primes (x, y) for which

(5^x - 2^x)(5^y - 2^y) /(xy) is an integer
(5^x - 3^x)(5^y - 3^y) / (xy) is an integer

(7^x - 2^x)(7^y - 2^y) /(xy) is an integer

(7^x - 5^x)(7^y - 5^y) / (xy) is an integer

Then, after that extend the research to other prime numbers