Find all pairs (A, B) of positive integers such that each of the equations
x² - A*x + B = 0 and x² - B*x + A=0 has positive integer roots.

Prove that there are no others.

I can find three pairs that work:

(A, B) = (4, 4), (5, 6) and (6, 5),

but I can't yet prove that no others exist.

They must lie in the 1st quadrant of the A-B plane 'outside' the parabolas a^2 = 4b and b^2 = 4a and there's plenty of room to explore there...

Posted by Harry on 2016-10-19 17:15:01