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Math 2.0

Number theory Corollary

Fermat's little theorem in the form a^p = a

Builds on Fermat's little theorem

Statement

Let $p$ be a prime number and let $a$ be an integer. Then $a^p \equiv a \pmod p$ .

Remarks

If $p \nmid a$ , multiply Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ by $a$ . If $p \mid a$ , both sides are $0$ modulo $p$ . Conversely, this form implies the form $a^{p-1} \equiv 1$ : cancel $a$ , which is allowed when $p \nmid a$ . In the ring $\mathbb{Z}/p\mathbb{Z}$ , the form says that $x^p = x$ for each element $x$ .

Reading path

To understand this page from the ground up, read these 5 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
Congruence modulo n definition
Prime number definition
Fermat's little theorem statement
Fermat's little theorem in the form a^p = a statement

Further reading

Web https://en.wikipedia.org/wiki/Fermat%27s_little_theorem, lead section

About this page

Tag	`0BMXWXDVJZ`
URI	`m2://0BMXWXDVJZ`
Kind	statement, corollary
Writing	W0 stub
MSC	11A07
Status	This copy of the site has no build data. Run `m2 status 0BMXWXDVJZ`.
Uses	Fermat's little theorem

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