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Given a prime number $p$, find the number of pairs of integers $(a, b)$ such that $p \lt a$, $p \lt b$ and $ab$ is divisible by $(a-p)(b-p)$.

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Welcome to MathSE. I see that this is your first question. So I wanted to let you know a few things about MathSE. We like to know the sources of questions. We also like to know what you've tried on a problem or what your thoughts are, so that the answer does not re-invent the wheel. Also, many users find questions posted in the imperative ("Show that", "Prove", "Do", "Find") unpleasant and somewhat rude. These sort of pleasantries usually result in more and better answers. Thank you! – Arturo Magidin Dec 1 '11 at 18:06
If $f(p)$ is the number of pairs you're counting, I computed $f(2)=14$, $f(3)=26$, $f(5)=38$, $f(7)=44$, $f(11)=56$, $f(13)=50$, $f(17)=62$, $f(19)=68$, $f(23)=80$. The dip at $13$ is surprising. I plugged that sequence into and got back nothing. How did this problem come up? – Dimitrije Kostic Dec 1 '11 at 20:27
The "dip" at $13$ is not surprising, since $14$ has few divisors. – André Nicolas Dec 1 '11 at 20:43
@AndréNicolas: I don't understand your comment. The number of divisors of $14$ is only relevant if one of $a$ or $b$ is $14$. – Dimitrije Kostic Dec 2 '11 at 20:07
From divisors of $p^2+p$ one gets the simplest solutions. – André Nicolas Dec 3 '11 at 3:38

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