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The real answer is in fact 1. I will prove this through infinite descent. Clearly, ( 0 , 0 ) is a solution. Now, assume there exists a solution with ∣ x ∣ + ∣ y ∣ > 0 which is minimal. Consider x 2 = 5 y 2 Implying 5 ∣ x . Let's replace x with 5 x 1 . Substituting, we get 2 5 x 1 2 = 5 y 2 ⇒ 5 x 1 2 = y 2 Implying 5 ∣ y . Substituting y = 5 y 1 , we get 5 x 1 2 = 2 5 y 1 2 x 1 2 = 5 y 1 2 Notice that this implies ( x 1 , y 1 ) is also a solution, with ∣ x 1 ∣ + ∣ y 1 ∣ > 5 ∣ x ∣ + ∣ y ∣ . But we already assumed x , y was the minimal solution! Thus, by infinite descent, ( 0 , 0 ) is the only solution.