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In another question I was asking if there are any different $x,y>2$ primes such that $xy+5=a(x+y)$.

Where $a=2^r-1$, and $r>2$.

I was thinking if it is able to find a Pell equation or a similar pattern of $xy+5=a(x+y)$ to say what are and how many integer solutions are there (in particular prime solutions).

Thanks.

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$xy-5=a(x+y)$ can be rewritten as $(x-a)(y-a)=a^2+5$ so for any fixed $a$ solving it just amounts to finding all the ways to factor $a^2+5$. So how many solutions depends on the prime factorization of $a^2+5$. I don't think there will be any formula for how many of those solutions have $x$ and $y$ prime.

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    These does not need to be common to be infinitely many..2011-05-08