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Algebra Level pending

How many pairs of positive integers ( x , y ) , y ≤ 2018 (x,y) , \ \ y \leq 2018 satisfy x 2 − y = 2018 x^2 - y = 2018


The answer is 19.

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Stephen Brown
Dec 20, 2017

0 < y ≤ 2018 ⇒ 2018 < y + 2018 = x 2 ≤ 4036 ⇒ 2018 < x ≤ 4036 ⇒ 45 ≤ x ≤ 63 0 < y \leq 2018 \Rightarrow 2018 < y+2018 = x^2 \leq 4036 \Rightarrow \sqrt{2018} < x \leq \sqrt{4036} \Rightarrow 45 \leq x \leq 63

So there are 19 \boxed{19} possible choices for x x , and it is easy to see that we can pick y y for any x x in this range.

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