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1. let [tex]p[/tex] be prime form [tex]4k+3[/tex] and let [tex]a[/tex] be an integer. Prove that a has order [tex]p-1[/tex] in the group [tex]U(\frac{\texbb{Z}}{p\texbb{Z}})[/tex] iff [tex]-a[/tex] has order [tex]\frac{(p-1)}{2}[/tex]

2. let [tex]p[/tex] be odd prime explain why: [tex]2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{(p-1)/2}*1*3*...*(p-2) [/tex]mod p.

3. Using number 2 and wilson's thereom [[tex](p-1)!\equiv-1[/tex] mod p] prove [tex]1^23^25^2*....*(p-2)^2\equiv(-1)^{(p-1)/2}[/tex] mod p

Thanks.

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# Homework Help: Urgent: Number Theory-Wilson's Theorem

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