find the number of pairs using modular arithmetics

idontknow

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
$$\displaystyle N=4,~ (\pm 1, \pm 3)$$

I don't see how the mod operator comes into it.

idontknow

$$\displaystyle N=4,~ (\pm 1, \pm 3)$$

I don't see how the mod operator comes into it.
How to find the pairs with method ?

romsek

Math Team
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.