# modular arithmetic – More elegant solution to \$5^{5^5}\$ mod 23

I needed to calculate $$5^{5^5}$$. This took about 5 minutes or so using the standard trick of building the answer out of smaller answers. My method was to:

1. calculate $$5^{25}=5^{23} 5^2$$, which gives 10, using Fermat’s Little Theorem
2. Calculate $$10^{25}$$, using the same trick, finding that it equals 11
3. Calculate $$11^5$$, by calculating $$11^2$$ mod 23, using this to get $$11^4$$, and finally $$11^5$$ mod 23

This gave 5, which is correct.

However, this method, while fairly quick, was quite unsatisfying. Is there a more elegant solution?