You can speed up brute force search by starting somewhere before where log10(x^y) * 4.5 is roughly equal to x. For example 2222^140 is 2110 which is close to 2222. Then brute force search from there and you get:

2222 ^ 145

How much does this speed up brute force search, though? You have to actually compute the exponent and digit sums. If I iterate over the powers of N from N^I to N^J, I do something like O((J-I)^2*log N) work doing multiplications the naive way.

Rewrite

    log(x^y)*4.5 = x
    y*log(x)*4.5 = x
    y = x/(4.5*log x)

So picking a random number like 234, we get y=21.94819 and the digit sum of 234^22 == 234. In this case we actually just had to compute one exponent and one digit sum.

Hint: try checking other numbers.

I think one reason to limit to base 10 is because we don't have an arbitrary number of digits, we only have maybe 36 if we use the full alphabet. And it will be difficult for people to follow along with if it's something they aren't used to calculating in. But I'm also guessing the results will become a lot less "rare" which is part of the cool factor here.

Omg, you're not thinking he's done all this by hand, right? People here can code such simple things. And you don't need no specific symbols, just arrays of ints (or any suitable type).