# 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 loglogx.

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 37100 is divided by 29.

solution Find the remainder when 451000 is divided by 31.

solution Find the remainder when 137153 is divided by 18.

solution Prove that n3n is divisible by 6.

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