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.