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1 solution
Multiplying through by x − y and simplifying, we get x 5 − y 5 + x 3 − y 3 = 0 . But if x > y , x 3 > y 3 and x 5 > y 5 , so the left side is > 0 . Similarly x < y is impossible.
This leaves x = y , which leads in the original equation to 5 x 4 + 3 x 2 = 0 , which is only possible if x = 0 . So ( 0 , 0 ) is the only solution.