Determine all possible pair(s) (X, Y) of positive integers that satisfy this equation.

                                        X^XX = Y^Y

Note: The order of calculation in X^XX is as given in this article.

(In reply to computer exploration by Charlie)

Since the LHS increases so much more rapidly than the RHS, it doesn't make sense to treat x and y equivalently. So:

10   point 16
20   X=1:Y=1
30   loop
40   Lhs=(X^X)*log(X):Rhs=Y*log(Y)
50   Diff=abs(Lhs-Rhs)
60   if Diff<0.000000001 then print X;Y
70   if Lhs=Rhs then inc X:inc Y:else if Lhs<Rhs then inc X:else inc Y
80   endloop

had still found only (1,1) by the time it was stopped with x=10 and y=57,090,100

Posted by Charlie on 2009-03-18 15:19:52