Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
Let be a prime number. Let such that
Prove that .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
First show that the problem is equivalent to the problem of showing that
Pair up the terms by looking at
In the words of Tom Hanks,
... Try to use Wilson's Theorem.
Lastly, you will want to show that
Do this by showing that .
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
First, notice that
Now, if we are to show that then it suffices to show that since the term given by is corpime to p. Cross multiplying shows us that it suffices to show that
Let's now pair terms with their additive inverses:
and thus it now suffices to show that the sum
To do this, we use the trick common in evaluating Gauss sums and looking at . this gives
Now note that , and all form complete residue systems for . Thus, the above sum is
Here is where we use the fact that . Notice that as p is odd, we have that . Further, we know that either or . If then . If then and so . In either case, we see that and thus . Hence, we see that . Again as p is odd, we have that .
This completes the proof.
For those of you wondering, this is also called Wolstenholme's Theorem.