# Urgent: Number Theory-Wilson's Theorem

1. Nov 2, 2008

### mathsss2

Urgent:

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

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

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

Thanks.

Last edited: Nov 3, 2008
2. Nov 2, 2008

### robert Ihnot

For p=3+4k, then (p-1)/2 is an odd number. Thus $$1\equiv (-a)^\frac{p-1}{2} =-a^\frac{p-1}{2}$$

Consequently the right side would be -1 under the circumstances that both powers of a equalled 1.

Last edited: Nov 2, 2008
3. Nov 3, 2008

### mathsss2

I got the rest, I still need help in #2.