2014 UMO Problems/Problem 2
(a) Find all positive integers and that satisfy or prove that there are no solutions.
(b) Find all positive integers and that satisfy or prove that there are no solutions.
(a) We see that we can rewrite as . Since and are perfect squares, their modulo can only be . Since none of those two combinations make , there are no solutions to such that .
(b) Similarly, we can rewrite as and therefore it also does not have integer solutions.
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