I guess this would be an elementary number theory question, but it's in Advanced Algebra by Rotman, so I figured it would go here. I apologize if it's wrong.(adsbygoogle = window.adsbygoogle || []).push({});

If p is an odd prime and [itex]a_{1},...,a_{p-1}[/itex] is a permutation of [itex]1,2,...,p-1[/itex] then there exist [itex]i \neq j[/itex] with [itex]ia_{i} \equiv ja_{j} modp [/itex].

It says to use Wilson's Theorem, but I can't seem to be getting anywhere with it.

I thought that for any i we have

[itex]ia_{i} = -i(a_{1}...a_{i-1}a_{i+1}...a_{p-1})^{-1} modp[/itex] by Wilson.

I figured the j=-i modp, which is not equal to i since p is odd. But I'm not sure how to show

[itex] (a_{1}...a_{i-1}a_{i+1}...a_{p-1})^{-1} = a_{-imodp} [/itex]

if the assumption j=-i is even correct. Any help, please?

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# Wilson's Theorem

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