Lifting-the-exponent lemma

Statement

Let $p$ be an odd prime, let $x$ and $y$ be integers with $p \nmid x$ , $p \nmid y$ and $p \mid x - y$ , and let $n$ be a positive integer. Then $\nu_p(x^n - y^n) = \nu_p(x - y) + \nu_p(n)$ , where $\nu_p$ is the p-adic valuation.

Remarks

For $p = 2$ the formula needs a stronger hypothesis. If $x$ and $y$ are odd and $4 \mid x - y$ , then $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(n)$ . For $p = 2$ , odd $x$ and $y$ , and even $n$ , the general formula is $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(x + y) + \nu_2(n) - 1$ . For example, $\nu_3(10^9 - 1) = \nu_3(9) + \nu_3(9) = 4$ , and indeed $10^9 - 1 = 3^4 \cdot 37 \cdot 333667$ .

History

Édouard Lucas had some of the ideas by 1878. Mihai Manea published the modern form in 2006.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility definition
Prime number definition
P-adic valuation definition
Lifting-the-exponent lemma statement

Tag	`0JYF6PTKEW`
URI	`m2://0JYF6PTKEW`
Aliases	`lte-lemma`
Kind	statement, lemma
Writing	W1 stated
MSC	11A51
Status	This copy of the site has no build data. Run `m2 status 0JYF6PTKEW`.
Uses	P-adic valuation
Same as	Wikidata Q97704248 Wikipedia Lifting-the-exponent lemma

Lifting-the-exponent lemma

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

What links here