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n is perfect square, 1/a+1/b=1/n for 2017 distinct (a,b)

Source: Switzerland - 2017 Swiss MO Final Round p7

January 14, 2023
number theoryPerfect SquarediophantineDiophantine equation

Problem Statement

Let nn be a natural number such that there are exactly2017 2017 distinct pairs of natural numbers (a,b)(a, b), which the equation 1a+1b=1n\frac{1}{a}+\frac{1}{b}=\frac{1}{n} fulfilld. Show that nn is a perfect square .
Remark: (7,4)≠(4,7)(7, 4) \ne (4, 7)
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