Something to do with 2

Number Theory Level 2

How many ordered pair of integers ( x , y ) (x,y) satisfy

x 2 − 5 y 2 = 2 \large x^2-5y^2=2

This question belongs to the set Number theory best problems

1 0 5 2

Shreyansh Mukhopadhyay by Shreyansh Mukhopadhyay , 18, India

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2 solutions

x 2 − 5 y 2 = 2 x^2-5y^2=2

⇒ x 2 = 2 + 5 y 2 \Rightarrow x^2=2+5y^2

⇒ x 2 ≡ 2 + 5 y 2 ( m o d 5 ) \Rightarrow x^2\equiv2+5y^2 (\mod 5)

⇒ x 2 ≡ 2 ( m o d 5 ) \Rightarrow x^2\equiv2(\mod 5)

Therefore x 2 ≡ 2 o r 7 ( m o d 10 ) x^2\equiv2\space or\space 7(\mod10) But this is not possible for any integer x x .

So the answer is 0 . \boxed0.

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Mihir Mallick
Feb 7, 2018

Taking the equation m o d u l o 4 modulo 4 , The LHS can only be 0 , 1 , 3 0,1,3 whereas the RHS is 2 2 . Hence, there doesn't exist any solutions to the given equation in integers.

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