Wilson's theorem

Hi,

Would the following constitute a proof of "if n is compositie then (n-1!)=0 mod n"

We can use the chinese remainder theorem to find

Suppose that

Then because in each case is less than (n-1) and so it divides (n-1!). We find that for all i. Then the CRT implies that n-1 =0 mod n


Does the above make sense??

Thanks for any help!

Quote:

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Originally Posted by Ant

in each case is less than (n-1)

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This is only true if it is not true if (i.e. if is a single-prime power).

By the way, The converse of Wilson’s theorem only states that if is composite, then

ah yes, I didn't deal with that case. Thank you

You were almost correct though. If is composite, we in fact have

If and where then (i) if we have straightaway (ii) if then:

I see, so you really needn't split it into coprime divisors of n like I did.

I had the above proof as an exam questions - hopefully I'll still get some marks for getting close!