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.
This does not work if one of the fractions is less than or equal to -2.
If equal to -2, then one of the cross-products is 0.
If less than -2, then one (but not both) of the cross-products is negative.
This does appear to work with two negative fractions, both of which are greater than -2.
For instance, -4/3 and -3/2 lead to cross products of 3 and 4.