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 $loglog x$ .

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.

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 $n 3 - n$ is divisible by 6.