Write down two fractions whose product is 2.
Add 2 to each. Keep them improper.
Cross multiply to get two whole numbers.
These numbers are the legs of a Pythagorean triangle!
Prove this always works.
The two fractions satisfying the given conditions must have the form p/q and 2q/p, where p and q are positive integers.
So adding 2 to each and simplifying, we have:
(2q+p)/q and 2(p+q)/p
Cross multiplying, we have:
p(2q+p) and 2q(p+q)
or, 2pq+p^2 and 2q(p+q)--(*)
we see that: x > q as p+q > q
and, (*) reduces to:
x^2 - q^2 and 2qx which is the well known form for the two legs of a pythagorean triangle.