find the number of pairs using modular arithmetics

Dec 2015
1,068
164
Earth
find the number of pairs \(\displaystyle (x,y)\in\mathbb{N}\) such that \(\displaystyle 8+x^2 =y^2 \). using mod-operator. :)
N=?
 

romsek

Math Team
Sep 2015
2,954
1,666
USA
the pairs themselves are by inspection.

that there are no other solutions comes from noting that \(\displaystyle 25-16 > 8 \) and that's as small as the difference of squares is going to get for any $x\neq y,~x,y\geq 4,~x,y\in\mathbb{Z^+}$
you can check the others by eye and see there are none.