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 20090318 13:06:25 