I'm trying to prove that there exists two distinct positive integers whose sum and difference are both perfect squares. I cannot find any pattern or characteristic between the pairs of numbers that work i.e.
- 4, 5
- 6, 10
- 8, 17
- 10, 6
- 10, 26
Any help will be greatly appreciated!
Thanks!
Best Answer
Let those positive integers be $a$ and $b$.Then,
$a+b=x^2$
$a-b=y^2$
Adding the two we get $a=\frac{x^2+y^2}{2}$ and $b=\frac{x^2-y^2}{2}$
Since, a and b are integers we must have $x^2+y^2$ and $x^2-y^2$ must be even, for that we must have x and y both even or both odd.
Now for finding such pairs take any even $x,y$ for example let x=8 and y=4
which gives $a=40$ and $b=24$, we have $a+b=64=8^2$ and $a-b=16=4^2$