So I have seen a lot of replies to this here but I want it to use 8th grade math like Fermat's theory, modules or even basic theories. Like I don't want Lagrange's theory and Quadratic residue because its not needed in my opinion. So if someone can send me proof and like really explain it well it would be really helpful. Thank you.
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should say: the usual argument is something like "if it were possible, then $a$ would have order $4\pmod p$ which would tell us that $4,|,(p-1)$ which, while elementary, isn't what I'd call $8^{th}$ grade level. – lulu Nov 30 '24 at 20:31
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1Is Fermat's little theorem allowed to used? In my country, I think secondary high school student taking part in Olympiad contest can use this, but not sure about your context. – Tri Nov 30 '24 at 20:37
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One of the most insightful elementary proofs is by pairing up negations and inverses into quads, e.g. see here in the dupe, and see my elaboration in a Remark here. Though the underlying ideas are innately group theoretic, they can be presented in this simple instance without any knowledge of group theory - as I explain in the prior linked post (in the analogous context of Wilson's theorem for groups). $\ \ $ – Bill Dubuque Nov 30 '24 at 21:18