MathNet 2006 — Part 03 Question 174

For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. (For example $f_1(123) = (1+2+3)^2 = 36$.) Let $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim.