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.

Clearly for two quadratic equations x^2+m*x+n=o and x^2+m'*x+n'=o to have identical couples of roots m must be equal to m' and n=n'.

In our case A=B.

So x1+x2=A  and x1*x2=A

The above set has only one solution in integers: A=B=4:  x1=x2=2.

'Prove that there are no others."-

 AFTER FINDING ALL SOLUTIONS this is an  unnecessary/redundant demand and should be avoided in the future.

n.b.   please see my next comment

Edited on October 18, 2016, 7:33 pm

Posted by Ady TZIDON on 2016-10-18 19:05:13