Totient of a prime

Builds on Euler's totient function Prime number

Statement

Let $p$ be a prime number. Then $\varphi(p) = p - 1$ , where $\varphi$ is Euler's totient function.

Remarks

The proof: the only divisors of $p$ are $1$ and $p$ . So each of $1, \dots, p - 1$ is coprime to $p$ , and $p$ is not. The converse holds: if $n > 1$ and $\varphi(n) = n - 1$ , then $n$ is prime. The result is the case $k = 1$ of the totient of a prime power.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Greatest common divisor definition
Prime number definition
Coprime integers definition
Euler's totient function definition
Totient of a prime statement

Tag	`0HY1MXMEN9`
URI	`m2://0HY1MXMEN9`
Kind	statement, proposition
Writing	W0 stub
MSC	11A25
Status	This copy of the site has no build data. Run `m2 status 0HY1MXMEN9`.
Uses	Euler's totient function, Prime number

Totient of a prime

Statement

Remarks

Reading path

Further reading

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