Learn – Math circle – Prime numbers

Table of Contents

1 Prime factorization and primality

1.1.1 Problem set

1.1.2 Problem set

2.1.1 Problem set

2.1.2 Problem set

3 Mersenne primes

3.1.1 Problem set

3.1.2 Problem set

1 Prime factorization and primality

1.1 Exercises

1.1.1 Problem set

Find the prime factorization of the following numbers

$1350$

1.1.2 Problem set

For each of the following, answer if the number is prime or not.

$73$
$383$
$1349$
$2309$
$2581$

2 Some theory

2.1 Exercises

2.1.1 Problem set

For each of the following, answer if what is given is true or not.

$a \mbox{ is a prime } \implies \forall 0 < b < a, \,\,a \bmod b \,\,\ne \,\, 0$

2.1.2 Problem set

Prove the following.

If $a$ is a prime number and $a \nmid b$ , then $a$ and $b$ are relatively prime.
If $p$ is a prime number and $p\mid ab$ , then either $p\mid a$ or $p\mid b$ .
If $a\mid bc$ and $a, b$ are relatively prime, then $a\mid c$ .

3 Mersenne primes

3.1 Exercises

3.1.1 Problem set

What is the smallest composite Mersenne number?
What is the smallest composite number of the form $Mp = 2^p-1$ , $p$ being a prime?
How many Mersenne primes are known so far?
What is the biggest known prime?

3.1.2 Problem set

Prove the following.

If $x^n-1$ is prime, then $x = 2$ .
If $2^n-1$ is prime, then $n$ is prime.