The two legs of a right triangle are consecutive natural numbers. What are the side lengths if...?

Question

What are the side lengths if it is the smallest triangle with hypotenuse > 10^15 ?
Please prove your answer mathematically.

I should have mentioned that the hypotenuse is a natural number.
Sorry.

To be fair, I am going to award best answer to Puzzling and re-ask the question. Others need not answer this one if you want to be considered for the 10 points.

Answer 1

On that scale, the legs are almost equal. Let's assume the legs were n and n. Then you would have:
n² + n² = (10^15)²
2n² = (10^15)²

n² = (10^15)²/2
n = sqrt(10^15²/2)
n = 10^15 / sqrt(2)
n = (sqrt(2) / 2) * 10^15
n = (1.4142135623730950488 / 2) * 10^15
n = (0.7071067811865475244) * 10^15
n = 707106781186547.5244

It looks like the numbers are:
707106781186548 and 707106781186549

Answer 2

x^2 + x^2+2x+1 >10^30
2x^2 +2x > 10^30 -1
x^2 +x > (10^30 -1)/2
(x+1)^2 > (10^30-1)/2 + 1
x+1 > +/- sqrt((10^30-1)/2 + 1)
x > -1 + sqrt((10^30-1)/2 + 1)
x>-1 +sqrt(2)/2 *(10^15)