Calculate $3^{64}\pmod{67}$.

Let $p$ be a prime. Show that there exist infinitely many positive integer $n$ such that $p\mid (2^n-n)$.

Show that there are infinite many composite numbers in the sequence $$1, 31, 331, 3331, 33331, \cdots$$

Let $N = 7\times 8\times 9\times 15\times 16\times 17\times 23\times 24\times 25\times 43$. Compute $N\pmod{11}$.

Find the remainder when $10^{10}+10^{100}+10^{1000}+\cdots+10^{\overbrace{\scriptsize{10\cdots 0}}^{2018}}$ is divided by $7$.

How many positive integers $N$, less than $2017$, satisfy $$N^{2016^{2016}}\equiv 1\pmod{2017}$$

Let $p$ is an odd prime, compute $1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1}\pmod{p}$.

Let $p$ is an odd prime, compute $1^{p}+2^{p}+3^{p}+\cdots+(p-1)^{p}\pmod{p}$.

(Fermat's little theorem) Show that $a^p\equiv a\pmod{p}$ holds if $p$ is a prime.

An integer in the form of $F_n=2^{2^n}+1$ where integer $n\ge 1$ is called a Fermat's number. Let $d_n$ be any divisor of $F_n$. Show that $d_n\equiv 1\pmod{2^{n+1}}$.

Assume positive integer $n > 1$ satisfies $n\mid (2^n+1)$, prove $n$ is a multiple of $3$.

Let $p$ be an odd prime, and $n=\frac{2^{2p}-1}{3}$ in an integer. Prove $2^{n-1}\equiv 1\pmod{n}$.

Show that for any integer $x$, the number $\left(\frac{x^5}{5}+\frac{x^3}{3}+\frac{7x}{15}\right)$ is an integer.

Let $m$ and $n$ be positive integers, $m$ be odd, and $(m, 2^{n} - 1)=1$. Show that $\displaystyle\sum_{k=1}^{m}k^n$ is a multiple of $m$.

Find the last $4$ digits of $2018^{2019^{2020}}$.

Determine whether a given integer can be expressed as the sum of two perfect squares.