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.

After 5^7 = 78,125, the next 4 values of a are:   

 11^5     =                   161,051    
 5^13     =          1,220,703,125
 11^11 ~=       285,312,000,000
 5^19   ~=  19,073,500,000,000

Posted by Steve Herman on 2008-07-10 11:35:28