# Number Theory

## Prime Numbers

solution Prove that there are infinitely many primes.

solution Prove that there are infinitely many primes of the form $p=6k-1\,$.

solution Prove that the number of primes less than $x$ is bounded below by $\log \log x$.

solution Prove that there are $n\,$ consecutive composite numbers, for any $n>0\,$.

solution Prove that any number $x\ {\boldsymbol {\epsilon }}\ {\mathbb {Z}}$ can be represented by the sum of Fibonacci numbers.

There are many problems available under Project PEN.

## Divisibility

solution Find the remainder when $37^{{100}}$ is divided by 29.

solution Find the remainder when $45^{{1000}}$ is divided by 31.

solution Find the remainder when $137^{{153}}$ is divided by 18.

solution Prove that $n^{3}-n$ is divisible by 6.