Show that the sequence 1^1, 2^2, 3^3 ,... considered mod p is periodic with the least period p(p-1). This was on my midterms, and I couldnt solve it.... Could you please help me.

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- Oct 4th 2017, 11:59 PM #1

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- Oct 5th 2017, 03:41 AM #2

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## Re: from private message

You are trying to show $n^n=(n+k)^{n+k} \pmod{p} $. Then show that $k=p(p-1) $ is a value that makes it true, then show it is the minimum value. I'll think about it more, but I have a feeling it is gonna be fairly straightforward.

- Oct 5th 2017, 05:46 AM #3

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## Re: from private message

Since $p$ is prime, we know that $\mathbb{Z}_p^*$ has elements with order $p-1$. So, the period cannot be smaller than $p$, and $p$ must divide the length of the smallest period.

Next, $(n+kp)^{n+kp} = (n+kp)^n(n+kp)^{kp} \equiv (n+kp)^n\left[(n+kp)^p\right]^k \pmod p \equiv (n+kp)^n(n+kp)^k \pmod p \equiv n^{n+k} \pmod p$

In order for $n^{n+k} \equiv n^n \pmod p$, it must be that $n^k \equiv 1 \pmod p$, and by Fermat's Little Theorem, $k = p-1$ is the smallest such $k$ that this is true for all $n$.

Therefore, the smallest period is $p(p-1)$.

- Oct 6th 2017, 07:13 AM #4

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- Oct 6th 2017, 07:35 AM #5

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## Re: from private message

- Oct 6th 2017, 05:18 PM #6

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- Oct 13th 2017, 08:49 PM #7

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- Oct 13th 2017, 08:53 PM #8

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- Oct 13th 2017, 10:14 PM #9

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- Oct 14th 2017, 05:32 AM #10

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