# Number Theory

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## Prime Numbers

solution Prove that there are infinitely many primes.

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

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

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

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

There are many problems available under Project PEN.

## Divisibility

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

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

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

solution Prove that $\displaystyle n^3 - n$ is divisible by 6.