# 6.3: Fermat Primes, Mersenne Primes and Primes of the other forms

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In this section, we consider special kinds of prime numbers.

## Fermat Primes and Mersenne Primes

Definition:

1. The prime numbers of the form $$2^k+1$$, where $$k\in \mathbb{Z_+}$$, are called Fermat primes.

2. The prime numbers of the form $$2^k-1$$, where $$k\in \mathbb{Z_+}$$, are called Mersenne primes.

They are named after the French mathematicians Fermat and Mersenne.

Example $$\PageIndex{1}$$:

1. $$2^1+1=3, 2^2+1=5,2^4+1=17$$ are Fermat primes. Notice that $$2^3+1=9$$ is not prime.

2. $$2^2-1=3, 2^3-1=7, 2^5-1=31$$ are Mersenne primes. Notice that $$2^1-1=1, 2^4-1=15$$ are not prime.

Theorem $$\PageIndex{1}$$

If $$2^k+1$$ is a prime,$$k\in \mathbb{Z_+}$$, then $$k$$ is a power of $$2$$.

Proof

Left as an exercise.

Theorem $$\PageIndex{2}$$

If $$2^k-1$$ is a prime,$$k\in \mathbb{Z_+}$$, then $$k$$ is also a prime.

Proof

Left as an exercise.

## Primes of the form $$4k-1$$

Example $$\PageIndex{2}$$:

$$(4)(1)-1=3, (4)(2)-1=7, (4)(3)-1=11, 4(5)-1=19, (4)(6)-1=23$$ are primes of the form $$4k-1$$. Notice that $$(4)(4)-1=15$$ is not a prime.

How many are there?

Theorem $$\PageIndex{3}$$

There are infinitely many primes of the form $$4k-1$$, $$k\in \mathbb{Z_+}$$.

The proof of this theorem is beyond the scope of this class.

## Primes of the form $$6k-1$$

Example $$\PageIndex{3}$$:

$$(6)(1)-1=5, (6)(2)-1=11, (6)(3)-1=17, 6(5)-1=29$$ are primes of the form $$6k-1$$. Notice that $$(6)(6)-1=35$$ is not a prime.

6.3: Fermat Primes, Mersenne Primes and Primes of the other forms is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.