Define a sequence $a_n$ as follows: for each positive integer $n$, set $a_n$ equal to the remainder of $n^n$ when it is divided by 101. What is the smallest positive integer $d$ such that $a_n = a_{n+d}$ for all $n$?

I have a feeling perhaps this would have to deal with orders, as a number raised to the order of it modulo $101$ would be congruent to the number added to it's order, still modulo $101$. However, I don't know how to go about this, as the powers change with terms.

Can someone provide me with a solution? Thanks so much :)


$d$ must be a multiple of $101$ because $101$ is prime. Otherwise, $101-d$ is not a multiple of $101$, so $a_{101-d}\not=0$, while $a_{101}=0$.

Now, write $d=101k$. Consider $a_{n+d}\equiv(n+d)^{n+d}\equiv n^{n+101k}=n^nn^{101k}\pmod{101}$. Therefore, we must guarantee that $n^{101k}=(n^{101})^k\equiv 1\pmod{101}$.

Using Fermat's little theorem, we know that $n^{101}\equiv n\pmod{101}$. Therefore, we must conclude that $n^k\equiv 1\pmod{101}$.

Finally, since $101$ is prime, $\mathbb{Z}/101^\times\simeq\mathbb{Z}/100$, so there is an element of (multiplicative) order $100$. Let $g$ be that element of order $100$, then we know that $k\geq 100$, but by Fermat's little theorem, $k=100$ is enough.


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