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I also have to prove this for $$2^2 \times 4^2 \times 6^2 \cdots \times (p-1)^2 \equiv (-1)^{(p+1)/2} \pmod{p}$$ I made some progress so far and got stuck. I said that since p is odd, $(p+1)/2$ is even. Then we can say that $(-1)^{\mathrm{even}}= 1$, so $(-1)^{(p+1)/2}\pmod{p}$ can be written as $1\pmod{p}$. I know that the product of odds is odd and the product of evens is even, but I cant prove that the left side of this equation in either case is congruent to $1 \pmod{p}$. Any help here would be greatly appreciated.

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Homework I presume? –  BBischof Feb 16 '11 at 19:39
You would probably use Wilson's Theorem. –  PEV Feb 16 '11 at 19:42
Hint: if $z$ is even then $p-z$ is odd and vice versa. –  Yuval Filmus Feb 16 '11 at 19:43
Also, $(p+1)/2$ is not always even: try $p=5$. –  Yuval Filmus Feb 16 '11 at 19:44
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1 Answer

You can start with Wilson's theorem, $(p-1)! \equiv -1 \pmod{p}.$ Rearrange the factorial thus, with $p=2m+1,$ $$-1 \equiv 1(p-1)2(p-2) \ldots m(p-m) \equiv 1(-1)2(-2) \ldots m(-m) \equiv 1^22^2 \ldots m^2(-1)^m \pmod{p}.$$

So we have shown

$$1^22^2 \ldots m^2 \equiv (-1)^{m+1} $$

or, noting that $m=(p-1)/2,$ in terms of $p$

$$1^12^2 \ldots \left( \frac{p-1}{2} \right)^2 \equiv (-1)^{(p+1)/2}.$$

Now we construct the product on the LHS, so consider first $2^24^2 \ldots (p-1)^2$ and factor out $2^{2p-2}$ then use Fermat, $2^{p-1} \equiv 1 \pmod{p}.$

You should now have the result for the product with the even numbers, use that along with Wilson's theorem again to prove the desired result for the product with the odd numbers.

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I don't understand how you ignored the odd terms in the end of the second line and moved to the 3rd line that says "Now consider..." –  Katie Barosie Feb 17 '11 at 3:41
I think we may apply quadratic reciprocity law to do so, to classify some cases, can we do that? –  awllower Feb 17 '11 at 4:29
@Katie: Please see the additional information the edit. I tried to reach a balance between giving enough information to tackle it and ruining the problem for you. And that, unfortunately, involves a bit of guesswork on a site like this as we don't get to meet the OP. (The proof that intelligent life exists elsewhere in the universe is that they have not tried to contact us yet.) –  Derek Jennings Feb 17 '11 at 8:26
Hi Derek, I was just wondering how to factor out $2^{2p-2}$ as this factor might not exist - take $p=7$ for example. The even numbers are $2^2 \cdot 4^2 \cdot 6^2$ but this has the common factor $2^8$. Please advise. –  Tyler Hilton Feb 9 '13 at 6:29
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