6.3: Fermat Primes, Mersenne Primes and Primes of the other forms
- Page ID
- 26366
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
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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.