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.

Using the Polynomial roots formula,

x^2 coefficient  = 1 
Roots (p+q) = A 
Roots (pq) = B 
 
x^2 coefficient  = 1 
Roots (r+s) = B 
Roots (rs) = A

it is easy to derive p=q and r=s; then 2p=r^2, 2r=p^2, p=q=r=s=2, {A,B} ={4,4} as the sole solution in the positive integers.

(Other solutions not compliant with the terms of the puzzle are also possible.)

Posted by broll on 2016-10-19 06:19:44