# modular multiplicative inverse

I have a homework problem that I've attempted for days in vain... It's asking me to find an n so that there is exactly one element of the complete residue system mod n that is its own inverse apart from 1 and n-1. It also asks me to construct an infinite sequence of n's so that the complete residue system mod n has elements that are their own inverses apart from 1 and n-1.

For the first part, I tried all n from 3 up to 40, but none worked... For the second part, I'm really confused...

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For the second: What about $3$ and $5 \pmod8$?

For the first: if $x^2\equiv 1$ then $(-x)^2\equiv 1$, too, so if there is only one (besides $\pm 1$), then $x\equiv -x \pmod{n}$, that is $2x=n$.

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If $n=2x$ then $x^2 \equiv 0 \text{ or } x \mod n$ $(=2x)$ – Mark Bennet Oct 28 '12 at 20:11
It shows that there is no such $n$. – Berci Oct 28 '12 at 20:20
This is really helpful, thanks very much! – hello.world Oct 28 '12 at 22:35

Hint for the second part. You can choose the values of $n$ as you please and want to satisfy $x^2=kn+1$ for some integer $k$. Try simple cases first.

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thanks! now i get it. – hello.world Oct 28 '12 at 22:47