Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $x$, $y$, $n$ be positive integers. Where $n$ and $y$ shall be constants and $x$ a variable. Then it is trivial that the period of

$x \bmod y$

in $x$ is $y$, since the function simply drops to zero and starts over when $x$ reaches $y$. Now, for me at least, it is much less obvious, what the period of

$x^n \bmod y$

in $x$ should be. Visually (looking at plots of the function at different $n$ and $y$) the results suggest that the period still stays $y$. How can I see that analytically?

share|cite|improve this question
What does period mean? – William Aug 20 '12 at 23:09
If by period you mean order in a group then you're mistaken if you mean multiplication or addition. For instance, $2+2 \equiv 0 \pmod 4$ and $2 \times 2 \equiv 1 \pmod 3$, so $2$ has order $2$ in modulo-$4$ arithmetic under addition, and order $2$ in modulo-$3$ arithmetic under multiplication. – Clive Newstead Aug 20 '12 at 23:11
@William: the period $p$ is the smallest positive integer such that $(x+p)^n \equiv x^n\pmod y$. – MJD Aug 20 '12 at 23:11
The OP wants the period as a function in $x$, not as a function in $n$. – Qiaochu Yuan Aug 20 '12 at 23:12
I am sorry, I am not a native speaker. I mean period just as $2\pi$ for $\sin(x)$. – Fejwin Aug 20 '12 at 23:12

2 Answers 2

up vote 5 down vote accepted

There are many cases where the period is $y$ (and, as Andre points out, many cases where it is not). If $y$ is squarefree (that is, if there is no integer $d\gt1$ such that $y$ is a multiple of $d^2$), then $x^n$ is zero if and only if $x$ is 0 (modulo $y$), so the period has to be $y$.

share|cite|improve this answer

The period need not be $y$. For example, let $y=9$, and consider the function $x^3$. This has period $3$. One can build many similar examples.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.