Question 55
Let $p$ be a prime number and $x_1, x_2, \dots, x_p$ be integers. Show that if
$$
x_1^n + x_2^n + \dots + x_p^n \equiv 0 \pmod{p}
$$
for all positive integers $n$ then $x_1 \equiv x_2 \equiv \dots \equiv x_p \pmod{p}$.
$$
x_1^n + x_2^n + \dots + x_p^n \equiv 0 \pmod{p}
$$
for all positive integers $n$ then $x_1 \equiv x_2 \equiv \dots \equiv x_p \pmod{p}$.