$$\\\mathbf{How\ to\ calcultate\ the\ \textit{Euler phi function} \ \phi(n):}\\
$ We have the prime factorization of n = p_1\cdot p_2\cdot p_3 \cdots\\
\phi(n) = n \cdot (1-\frac{1}{p_1})\cdot (1-\frac{1}{p_2}) \cdot (1-\frac{1}{p_3}) \cdots$$
$$\\\mathbf{Example\ 1: n = 6 } $\\
The prime factorization of 6 = 2 * 3 = p_1*p_2 \\
\phi(6) = 6 \cdot (1-\frac{1}{2}) \cdot (1-\frac{1}{3}) \\
\phi(6) = 6 \cdot \frac{1}{2} \cdot \frac{2}{3} \\
\phi(6) = \frac{6}{3} \\
\phi(6) = 2$$
$$\\\mathbf{Example\ 2: n = 9 } $\\
The prime factorization of 9 = 3^2 = p_1^2 \\
\phi(9) = 9 \cdot (1-\frac{1}{3}) \\
\phi(9) = 9 \cdot \frac{2}{3} \\
\phi(6) = 3 \cdot 2 \\
\phi(6) = 6$$
$$\\\mathbf{Example\ 3: n = 7 } $\\
The prime factorization of 7 = 7 = p_1 \qquad 7 $ is a prime number!$ \\
\phi(7) = 7 \cdot (1-\frac{1}{7}) \\
\phi(7) = 7 \cdot \frac{6}{7} \\
\phi(7) = 6$$
$$\\\mathbf{Example\ 4: n = 11 } $\\
The prime factorization of 11 = 11 = p_1 \qquad 11 $ is a prime number!$ \\
\phi(11) = 11 \cdot (1-\frac{1}{11}) \\
\phi(11) = 11 \cdot \frac{10}{11} \\
\phi(11) = 10$$
$$\boxed{\text{ In general $ \phi(p) = p-1 $, if p is a prime number }}\\\\
\begin{array}{lr}
p = 2: &\phi(2) = 1 \qquad =(2-1)\\
p = 3: &\phi(3) = 2 \qquad =(3-1)\\
p = 5: &\phi(5) = 4 \qquad =(5-1)\\
p = 7: &\phi(7) = 6 \qquad =(7-1)\\
p = 11: &\phi(11) = 10 \qquad =(11-1)\\
p = 13: &\phi(13) = 12 \qquad =(13-1)\\
\cdots & \phi(p) = p-1
\end{array}$$
The first 99 values of the Phi function are:
+0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | |
---|---|---|---|---|---|---|---|---|---|---|
0+ | 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | |
10+ | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 |
20+ | 8 | 12 | 10 | 22 | 8 | 20 | 12 | 18 | 12 | 28 |
30+ | 8 | 30 | 16 | 20 | 16 | 24 | 12 | 36 | 18 | 24 |
40+ | 16 | 40 | 12 | 42 | 20 | 24 | 22 | 46 | 16 | 42 |
50+ | 20 | 32 | 24 | 52 | 18 | 40 | 24 | 36 | 28 | 58 |
60+ | 16 | 60 | 30 | 36 | 32 | 48 | 20 | 66 | 32 | 44 |
70+ | 24 | 70 | 24 | 72 | 36 | 40 | 36 | 60 | 24 | 78 |
80+ | 32 | 54 | 40 | 82 | 24 | 64 | 42 | 56 | 40 | 88 |
90+ | 24 | 72 | 44 | 60 | 46 | 72 | 32 | 96 | 42 | 60 |
what is Φ6?
$$\!\ \varphi(6) = 2. \qquad 1 \text{ and } 5 \\\\
6 = 2 \cdot 3\\
\!\ \varphi(6)}=6\cdot (1-\frac{1}{2})( 1-\frac{1}{3} )}=2$$
Heureka, could you (or someone else) explain the phi function to me please?
I thought phi was an irrational number like pi.
Hello Melody,
$$\\\mathbf{De{finition:}}\\
\begin{text}
L{et} $ n \ge 1$ be an integer. Then we de{fine} the \\
\textit{Euler phi function} $\phi$ by\\
$\phi(n)=$ the number of positive integers \\
less than $n$ that are relatively prime to $n$
\end{text}$$
$$\\\mathbf{Relatively Prime:}\\
$
\begin{text}
Describes two numbers for which\\
the only common factor is 1. \\
In other words, relatively prime numbers have\\
a greatest common factor $(gcf)$ of $1$. \\
For example, $6$ and $35$ are relatively prime $(gcf = 1)$.\\
The numers $6$ and $8$ are not relatively prime $(gcf = 2)$.
\end{text}$$
$$\\\mathbf{Example\ 1: n = 6 }
$ and n is not a prime number $\\
\samll{\text{
$
\begin{array}{rr}
\textcolor[rgb]{1,0,0}{1}& gcf(6,1) = \textcolor[rgb]{1,0,0}{1}\\
2& gcf(6,2) = 2\\
3& gcf(6,3) = 3\\
4& gcf(6,4) = 2\\
\textcolor[rgb]{1,0,0}{5}& gcf(6,5) = \textcolor[rgb]{1,0,0}{1}\\
6& gcf(6,6) = 6\\
\end{array}
$
}}\\
6 \text{ has } \textcolor[rgb]{0,0,1}{2} \text{ relative primes } 1 \text{ and } 5 \text { see the red color. So } \phi(6) = \textcolor[rgb]{0,0,1}{2}$$
$$\\\mathbf{Example\ 2: n = 9 }
$ and n is not a prime number $\\
\samll{\text{
$
\begin{array}{rr}
\textcolor[rgb]{1,0,0}{1}& gcf(9,1) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{2}& gcf(9,2) = \textcolor[rgb]{1,0,0}{1}\\
3& gcf(9,3) = 3\\
\textcolor[rgb]{1,0,0}{4}& gcf(9,4) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{5}& gcf(9,5) = \textcolor[rgb]{1,0,0}{1}\\
6& gcf(9,6) = 3\\
\textcolor[rgb]{1,0,0}{7}& gcf(9,7) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{8}& gcf(9,8) = \textcolor[rgb]{1,0,0}{1}\\
9& gcf(9,9) = 9\\
\end{array}
$
}}\\
9 \text{ has } \textcolor[rgb]{0,0,1}{6} \text{ relative primes } 1,2,4,5,7,8 \text { see the red color. So } \phi(9) = \textcolor[rgb]{0,0,1}{6}$$
$$\\\mathbf{Example\ 3: n = 7 }
$ and n is a prime number $\\
\samll{\text{
$
\begin{array}{rr}
\textcolor[rgb]{1,0,0}{1}& gcf(7,1) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{2}& gcf(7,2) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{3}& gcf(7,3) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{4}& gcf(7,4) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{5}& gcf(7,5) = \textcolor[rgb]{1,0,0}{1}\\
\textcolor[rgb]{1,0,0}{6}& gcf(7,6) = \textcolor[rgb]{1,0,0}{1}\\
7& gcf(7,7) = 7\\
\end{array}
$
}}\\
\small{
7 \text{ has } 7-1=\textcolor[rgb]{0,0,1}{6} \text{ relative primes } 1 \text{ until } 6 \text { see the red color. So } \phi(7) = \textcolor[rgb]{0,0,1}{6}.}\\
\text{ So } \phi \text{ of a prime number } \phi(p) = p-1$$
I have a further question Heureka.
$$\!\ \varphi(6) = 2. \qquad 1 \text{ and } 5 \\\\
6 = 2 \cdot 3\\
\!\ \varphi(6)}=6\cdot (1-\frac{1}{2})( 1-\frac{1}{3} )}=2$$
Now I do get why phi(6)=2
But what is it about from 6=2.3
What is the relvance of the brackets. How can this be used on other examples?
$$\\\mathbf{How\ to\ calcultate\ the\ \textit{Euler phi function} \ \phi(n):}\\
$ We have the prime factorization of n = p_1\cdot p_2\cdot p_3 \cdots\\
\phi(n) = n \cdot (1-\frac{1}{p_1})\cdot (1-\frac{1}{p_2}) \cdot (1-\frac{1}{p_3}) \cdots$$
$$\\\mathbf{Example\ 1: n = 6 } $\\
The prime factorization of 6 = 2 * 3 = p_1*p_2 \\
\phi(6) = 6 \cdot (1-\frac{1}{2}) \cdot (1-\frac{1}{3}) \\
\phi(6) = 6 \cdot \frac{1}{2} \cdot \frac{2}{3} \\
\phi(6) = \frac{6}{3} \\
\phi(6) = 2$$
$$\\\mathbf{Example\ 2: n = 9 } $\\
The prime factorization of 9 = 3^2 = p_1^2 \\
\phi(9) = 9 \cdot (1-\frac{1}{3}) \\
\phi(9) = 9 \cdot \frac{2}{3} \\
\phi(6) = 3 \cdot 2 \\
\phi(6) = 6$$
$$\\\mathbf{Example\ 3: n = 7 } $\\
The prime factorization of 7 = 7 = p_1 \qquad 7 $ is a prime number!$ \\
\phi(7) = 7 \cdot (1-\frac{1}{7}) \\
\phi(7) = 7 \cdot \frac{6}{7} \\
\phi(7) = 6$$
$$\\\mathbf{Example\ 4: n = 11 } $\\
The prime factorization of 11 = 11 = p_1 \qquad 11 $ is a prime number!$ \\
\phi(11) = 11 \cdot (1-\frac{1}{11}) \\
\phi(11) = 11 \cdot \frac{10}{11} \\
\phi(11) = 10$$
$$\boxed{\text{ In general $ \phi(p) = p-1 $, if p is a prime number }}\\\\
\begin{array}{lr}
p = 2: &\phi(2) = 1 \qquad =(2-1)\\
p = 3: &\phi(3) = 2 \qquad =(3-1)\\
p = 5: &\phi(5) = 4 \qquad =(5-1)\\
p = 7: &\phi(7) = 6 \qquad =(7-1)\\
p = 11: &\phi(11) = 10 \qquad =(11-1)\\
p = 13: &\phi(13) = 12 \qquad =(13-1)\\
\cdots & \phi(p) = p-1
\end{array}$$
The first 99 values of the Phi function are:
+0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | |
---|---|---|---|---|---|---|---|---|---|---|
0+ | 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | |
10+ | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 |
20+ | 8 | 12 | 10 | 22 | 8 | 20 | 12 | 18 | 12 | 28 |
30+ | 8 | 30 | 16 | 20 | 16 | 24 | 12 | 36 | 18 | 24 |
40+ | 16 | 40 | 12 | 42 | 20 | 24 | 22 | 46 | 16 | 42 |
50+ | 20 | 32 | 24 | 52 | 18 | 40 | 24 | 36 | 28 | 58 |
60+ | 16 | 60 | 30 | 36 | 32 | 48 | 20 | 66 | 32 | 44 |
70+ | 24 | 70 | 24 | 72 | 36 | 40 | 36 | 60 | 24 | 78 |
80+ | 32 | 54 | 40 | 82 | 24 | 64 | 42 | 56 | 40 | 88 |
90+ | 24 | 72 | 44 | 60 | 46 | 72 | 32 | 96 | 42 | 60 |