[fatdave]

Isn't "SOD(SOD(SOD(4444^4444)))" redundant? Since SOD(4444^4444) is a single digit number, then SOD(4444^4444) should be the same as SOD(SOD(4444^4444)).

Anyways, without going through all the work above.... just know that SOD(n) = n mod 9.

"n mod 9" means if you divide "n" by 9, the remainder is the modulus. So... the sum of digits of a number is equal to the remainder of dividing the number by 9.

If you want to cheat, here's a quick and easy formula to use in PHP for SOD(x^y):

$sod = bcmod(bcpow(x, y), 9);

For example: bcmod(bcpow(4444, 4444), 9);