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.

x^x^x is called tetration I will use the ascii notation on the wikipedia entry x^^3
likewise y^y I will write as y^^2

I tried working by analogy using multiplication and then exponents

3x=2y

works for (2n,3n) for any n
for example if n=2 we have (4,6)
3*4=2*6
12=12

x^3 = y^2

works for (n^2,n^3)
for example if n=2 we have (4,8)
4^3=8^2
64=64

x^^3 = y^^2
I was hoping would work for (n^^2,n^^3)
but if n=2 we have (4,16)
4^^3 = 4^4^4 = 1.3*10^154
16^^2 = 16^16 = 1.8*10^19

So this idea did not work.
I may as well give the trivial solution (1,1)

Posted by Jer on 2009-03-18 13:06:25