# Number Theory

### From Exampleproblems

## Prime Numbers

solution Prove that there are infinitely many primes.

solution Prove that there are infinitely many primes of the form .

solution Prove that the number of primes less than *x* is bounded below by loglog*x*.

solution Prove that there are consecutive composite numbers, for any .

solution Prove that any number 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.