Well

How many pairs of integers x x and y y that satisfies x 2 5 y 2 = 0 x^{2} - 5y^{2} =0 .


The answer is 1.

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1 solution

Sharky Kesa
Jan 22, 2016

The real answer is in fact 1. I will prove this through infinite descent. Clearly, ( 0 , 0 ) (0, 0) is a solution. Now, assume there exists a solution with x + y > 0 |x|+|y|>0 which is minimal. Consider x 2 = 5 y 2 x^2 = 5y^2 Implying 5 x 5|x . Let's replace x x with 5 x 1 5x_1 . Substituting, we get 25 x 1 2 = 5 y 2 25x_1^2 = 5y^2 5 x 1 2 = y 2 \Rightarrow 5x_1^2=y^2 Implying 5 y 5|y . Substituting y = 5 y 1 y=5y_1 , we get 5 x 1 2 = 25 y 1 2 5x_1^2=25y_1^2 x 1 2 = 5 y 1 2 x_1^2=5y_1^2 Notice that this implies ( x 1 , y 1 ) (x_1, y_1) is also a solution, with x 1 + y 1 > x + y 5 |x_1|+|y_1|>\dfrac{|x|+|y|}{5} . But we already assumed x , y x, y was the minimal solution! Thus, by infinite descent, ( 0 , 0 ) (0, 0) is the only solution.

Lovely solution! =D =D

Pi Han Goh - 5 years, 4 months ago

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