nyuuzyou/wikis huggingface dataset:
Euler Phi Function of 168

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {168} = 48$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$168 = 2^3 \times 3 \times 7$
Thus:
Category:Euler Phi Function
Category:168
Euler Phi Function of 1798

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {1798} = 840$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$1798 = 2 \times 29 \times 31$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:1798
Euler Phi Function of 182

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {182} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$182 = 2 \times 7 \times 13$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:182
Euler Phi Function of 190

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {190} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$190 = 2 \times 5 \times 19$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:190
Euler Phi Function of 195

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {195} = 96$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$195 = 3 \times 5 \times 13$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:195
Euler Phi Function of 2

== Theorem ==
:$\map \phi 2 = 1$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
As $2$ is a prime number it follows that:
:$\map \phi 2 = 2 - 1 = 1$
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $27$
* : Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.1$
Category:Examples of Euler Phi Function
Category:2
Euler Phi Function of 2025

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {2025} = 1080$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$2025 = 3^4 \times 5^2$
Thus:
Category:Examples of Euler Phi Function
Category:2025
Euler Phi Function of 210

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {210} = 48$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$210 = 2 \times 3 \times 5 \times 7$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:210
Euler Phi Function of 216

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {216} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$216 = 2^3 \times 3^3$
Thus:
Category:Euler Phi Function
Category:216
Euler Phi Function of 222

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {222} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$222 = 2 \times 3 \times 37$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:222
Euler Phi Function of 228

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {228} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$228 = 2^2 \times 3 \times 19$
Thus:
Category:Euler Phi Function
Category:228
Euler Phi Function of 234

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {234} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$234 = 2 \times 3^2 \times 13$
Thus:
Category:Euler Phi Function
Category:234
Euler Phi Function of 248

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {248} = 120$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$248 = 2^3 \times 31$
Thus:
Category:Euler Phi Function
Category:248
Euler Phi Function of 25,930

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {25 \, 930} = 10 \, 368$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$25 \, 930 = 2 \times 5 \times 2593$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:25,930
Euler Phi Function of 25,935

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {25 \, 935} = 10 \, 368$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$25 \, 935 = 3 \times 5 \times 7 \times 13 \times 19$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:25,935
Euler Phi Function of 25,940

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {25 \, 940} = 10 \, 368$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$25 \, 940 = 2^2 \times 5 \times 1297$
Thus:
Category:Euler Phi Function
Category:25,940
Euler Phi Function of 25,942

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {25 \, 942} = 10 \, 368$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$25 \, 942 = 2 \times 7 \times 17 \times 109$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:25,942
Euler Phi Function of 252

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {252} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$252 = 2^2 \times 3^2 \times 7$
Thus:
Category:Euler Phi Function
Category:252
Euler Phi Function of 255

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {255} = 128$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$255 = 3 \times 5 \times 17$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:255
Euler Phi Function of 256

== Example of Use of Euler $\phi$ Function ==
The value of the Euler $\phi$ function for the integer $256$ is $128$.
== Proof ==
From the corollary to Euler Phi Function of Prime Power:
:$\phi \left({2^k}\right) = 2^{k-1}$
Thus:
Category:Euler Phi Function
Category:256
Euler Phi Function of 26

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {26} = 12$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$26 = 2 \times 13$
Thus:
Category:Examples of Euler Phi Function
Category:26
Euler Phi Function of 264

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {264} = 80$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$264 = 2^3 \times 3 \times 11$
Thus:
Category:Euler Phi Function
Category:264
Euler Phi Function of 270

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {270} = 72$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$270 = 3 \times 3^3 \times 5$
Thus:
Category:Euler Phi Function
Category:270
Euler Phi Function of 2 times Odd Prime

== Theorem ==
Let $n \in \Z_{>0}$ be a semiprime of the form $2 p$, where $p$ is an odd prime.
Let $\map \phi n$ denote the Euler $\phi$ function.
Then:
:$\map \phi n = p - 1$
== Proof ==
By definition $n$ is a semiprime.
As $p$ is an odd prime, $n$ is not square.
Thus from Euler Phi Function of Non-Square Semiprime:
:$\map \phi n = \paren {2 - 1} \paren {p - 1}$
Hence the result.
== Examples ==
Category:Euler Phi Function
Category:Semiprimes
Category:Euler Phi Function of 2 times Odd Prime
Euler Phi Function of 3

== Theorem ==
:$\map \phi 3 = 2$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
As $3$ is a prime number it follows that:
:$\map \phi 3 = 3 - 1 = 2$
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $27$
* : Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.1$
Category:Examples of Euler Phi Function
Category:3
Euler Phi Function of 30

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {30} = 8$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$30 = 2 \times 3 \times 5$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:30
Euler Phi Function of 315

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {315} = 144$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$315 = 3^2 \times 5 \times 7$
Thus:
Category:Euler Phi Function
Category:315
Euler Phi Function of 316

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {316} = 156$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$316 = 2^2 \times 79$
Thus:
Category:Euler Phi Function
Category:316
Euler Phi Function of 33,817,088

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {33 \, 817 \, 088} = 16 \, 842 \, 752$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$33 \, 817 \, 088 = 2^9 \times 257^2$
Thus:
Category:Euler Phi Function
Category:33,817,088
Euler Phi Function of 35

== Example of Euler $\phi$ Function of Non-Square Semiprime ==
:$\map \phi {35} = 24$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
We have that:
:$35 = 5 \times 7$
Thus:
Category:Euler Phi Function of Non-Square Semiprime
Category:35
Euler Phi Function of 357

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {357} = 192$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$357 = 3 \times 7 \times 17$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:357
Euler Phi Function of 362

== Example of Euler Phi Function of 2 times Odd Prime ==
:$\map \phi {362} = 180$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
We have that:
:$362 = 2 \times 181$
Thus:
Category:Euler Phi Function of 2 times Odd Prime
Category:362
Euler Phi Function of 4

== Example of Use of Euler $\phi$ Function ==
:$\map \phi 4 = 2$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
From Euler Phi Function of Prime Power: Corollary:
:$\map \phi {2^k} = 2^{k - 1}$
Thus:
:$\map \phi 4 = \map \phi {2^2} = 2^1 = 2$
They can be enumerated as:
:$1, 3$
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $27$
Category:Examples of Euler Phi Function
Category:4
Euler Phi Function of 418

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {418} = 180$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$418 = 2 \times 11 \times 19$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:418
Euler Phi Function of 42

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {42} = 12$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$42 = 2 \times 3 \times 7$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:42
Euler Phi Function of 420

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {420} = 96$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$420 = 2^2 \times 3 \times 5 \times 7$
Thus:
Category:Euler Phi Function
Category:420
Euler Phi Function of 5

== Theorem ==
:$\map \phi 5 = 4$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
From Euler Phi Function of Prime:
:$\map \phi p = p - 1$
As $5$ is a prime number it follows that:
:$\map \phi 5 = 5 - 1 = 4$
Category:Examples of Euler Phi Function
Category:5
Euler Phi Function of 5002

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {5002} = 2400$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$5002 = 2 \times 41 \times 61$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:5002
Euler Phi Function of 5186

== Example of Euler Phi Function of 2 times Odd Prime ==
:$\phi \left({5186}\right) = 2592$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
We have that:
:$5186 = 2 \times 2593$
Thus:
Category:Euler Phi Function of 2 times Odd Prime
Category:5186
Euler Phi Function of 5187

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {5187} = 2592$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$5187 = 3 \times 7 \times 13 \times 19$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:5187
Euler Phi Function of 5188

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {5188} = 2592$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$5188 = 2^2 \times 1297$
Thus:
Category:Euler Phi Function
Category:5188
Euler Phi Function of 52

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {52} = 24$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$52 = 2^2 \times 13$
Thus:
Category:Euler Phi Function
Category:52
Euler Phi Function of 524

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {524} = 260$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$524 = 2^2 \times 131$
Thus:
Category:Euler Phi Function
Category:524
Euler Phi Function of 525

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {525} = 240$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$525 = 3 \times 5^2 \times 7$
Thus:
Category:Euler Phi Function
Category:525
Euler Phi Function of 527

== Example of Euler $\phi$ Function of Non-Square Semiprime ==
:$\map \phi {527} = 480$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
We have that:
:$527 = 17 \times 31$
Thus:
Category:Euler Phi Function of Non-Square Semiprime
Category:527
Euler Phi Function of 56

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {56} = 24$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$56 = 2^3 \times 7$
Thus:
Category:Euler Phi Function
Category:56
Euler Phi Function of 570

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {570} = 144$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$570 = 2 \times 3 \times 5 \times 19$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:570
Euler Phi Function of 6

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi 6 = 2$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$6 = 2 \times 3$
and so is square-free.
Thus:
Category:Examples of Euler Phi Function
Category:6
Euler Phi Function of 60

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {60} = 16$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$60 = 2^2 \times 3 \times 5$
Thus:
Category:Euler Phi Function
Category:60
Euler Phi Function of 616

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {616} = 240$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$616 = 2^3 \times 7 \times 11$
Thus:
Category:Euler Phi Function
Category:616
Euler Phi Function of 630

== Example of Use of Euler $\phi$ Function ==
:$\map \phi {630} = 144$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$630 = 2 \times 3^2 \times 5 \times 7$
Thus:
Category:Euler Phi Function
Category:630
Euler Phi Function of 666 equals Product of Digits

== Theorem ==
The number $666$ has the following interesting property:
:$\map \phi {666} = 6 \times 6 \times 6$
where $\phi$ denotes the Euler $\phi$ function.
== Proof ==
From Euler Phi Function of Integer:
:$\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$666 = 2 \times 3^2 \times 37$
Thus:
== Also see ==
* Numbers for which Euler Phi Function equals Product of Digits
== Sources ==
* : $666$
Category:Euler Phi Function
Category:666
Euler Phi Function of 70

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {70} = 24$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$70 = 2 \times 5 \times 7$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:70
Euler Phi Function of 714

== Example of Euler $\phi$ Function of Square-Free Integer ==
:$\map \phi {714} = 192$
where $\phi$ denotes the Euler $\phi$ Function.
== Proof ==
From Euler Phi Function of Square-Free Integer:
:$\ds \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$
where $p \divides n$ denotes the primes which divide $n$.
We have that:
:$714 = 2 \times 3 \times 7 \times 17$
and so is square-free.
Thus:
Category:Euler Phi Function of Square-Free Integer
Category:714