Let S[x] be the digital root function (also known as the repeated digital sum function), where one adds the digits of positive integer x, then adds the digits of the sum until obtaining a single-digit number. (For example, S[975] = 3 because 9 + 7 + 5 = 21 and 2 + 1 = 3).

Given S[a^a] = 2, what is the smallest positive integer that a can be such that a is a perfect power?


Note: a is a perfect power if there exist natural numbers m > 1, and k > 1 such that m^k = a.

here is a list of the digit counts for the first 4 a^a

a=5^7 and a^a has 382,250 digits

a=5^13 and a^a has 11,092,053,292 digits

a=11^11 and a^a has 3,268,336,354,411 digits

a=5^19 and a^a has 253,304,101,607,978 digits

at this rate I don't suspect it would take very long for the number of digits in a^a to surpass a google or even a googleplex

Posted by Daniel on 2008-12-29 06:55:56