6.3: Fermat Primes, Mersenne Primes and Primes of the other forms
- Page ID
- 26366
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.