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 Titration or Tetration ? by ed bottemiller)

The link does lead you to the Identities and properties section of the article, and without scrolling you find

Similarly, while addition and multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either: 2³ to the 4th power is 8⁴ or 4096, but 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be top-down, not bottom-up:

<DL> <DD><IMG class=tex alt="a^{b^c}=a^{(b^c)}ne (a^b)^c=a^{(bcdot c)}=a^{bcdot c}." src="http://upload.wikimedia.org/math/d/8/b/d8b9a0216869860dcada1176ed40f5d7.png"> </DD></DL>

Posted by Charlie on 2009-03-18 15:54:16