Question 143
Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m} + 1$ is a prime number. Prove that
a. $2^{2^{m+1}} p^k \equiv 1 \pmod{p^{k+1}}$;
b. $2^{m+1} p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1 \pmod{p^{k+1}}$.
a. $2^{2^{m+1}} p^k \equiv 1 \pmod{p^{k+1}}$;
b. $2^{m+1} p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1 \pmod{p^{k+1}}$.