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Question Number 29035 by abdo imad last updated on 03/Feb/18

let give a prime number p>2  and a /D(a,p)=1 and   suppose that the equation x^2 ≡ a[p]have a solution1)  1) prove that   a^((p−1)/2)    ≡ 1 [p]  2)prove that  x^2 ≡ −1[p] ⇔   p≡ 1 [4]

$${let}\:{give}\:{a}\:{prime}\:{number}\:{p}>\mathrm{2}\:\:{and}\:{a}\:/{D}\left({a},{p}\right)=\mathrm{1}\:{and}\: \\ $$ $$\left.{suppose}\:{that}\:{the}\:{equation}\:{x}^{\mathrm{2}} \equiv\:{a}\left[{p}\right]{have}\:{a}\:{solution}\mathrm{1}\right) \\ $$ $$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:{a}^{\frac{{p}−\mathrm{1}}{\mathrm{2}}} \:\:\:\equiv\:\mathrm{1}\:\left[{p}\right] \\ $$ $$\left.\mathrm{2}\right){prove}\:{that}\:\:{x}^{\mathrm{2}} \equiv\:−\mathrm{1}\left[{p}\right]\:\Leftrightarrow\:\:\:{p}\equiv\:\mathrm{1}\:\left[\mathrm{4}\right] \\ $$

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