# Tagged: prime number

## A One-Line Proof that there are Infinitely Many Prime Numbers

## Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.

## Problem 344

Let $a, b$ be relatively prime integers and let $p$ be a prime number.

Suppose that we have

\[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\]
for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

Add to solve later## Mathematics About the Number 2017

Happy New Year 2017!!

Here is the list of mathematical facts about **the number 2017** that you can brag about to your friends or family as a math geek.

## A Group with a Prime Power Order Elements Has Order a Power of the Prime.

## Problem 17

Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$.

Add to solve later