If x and y are positive integers, L: LCM of x and y, G: GCD of x and y, then solve the following equations for x and y values:
1) x^y=L^G, L>x.
2) x²+y² = L²+G², L>x>G.

Let a = x/G, b = y/G

Then L = Gab.

Substituting into equation 2 yields

(Ga)² + (Gb)² = (Gab)² + G²

solving yields a = 1, b = 1 (if a,b are positive)

But then L = G, which violates L > x > G,

so there is no solution




        

Posted by Steve Herman on 2007-12-22 14:51:16