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 $(a-p)(b-p)$.
Number of integral solutions
2
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elementary-number-theory
prime-numbers
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7Welcome 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! – 2011-12-01
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0If $f(p)$ 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 oeis.org and got back nothing. How did this problem come up? – 2011-12-01
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0The "dip" at $13$ is not surprising, since $14$ has few divisors. – 2011-12-01
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0@AndréNicolas: I don't understand your comment. The number of divisors of $14$ is only relevant if one of $a$ or $b$ is $14$. – 2011-12-02
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0From divisors of $p^2+p$ one gets the simplest solutions. – 2011-12-03