Least Positive Residues

If you do (a) through (d), the answer to e is obvious. Note that all the values used for the modulus are prime. Non prime values yield zero except for when the modulus is 4.

Engineer

I agree on the conjecture on the prime modulus. But I don't think 4 is unique, special, maybe, as the 2nd even number, but not unique.

Rap

I think that four is unique as a non-prime that does not yield a zero modulus value. Here is my logic:

For non prime numbers that are not squares of a prime number, the factors making up that number are multiplied together in the factorial term, so (n-1)! mod n is zero.

When n is a perfect square of a prime number (k) then the factorial series doesn't contain k twice, but it does contain k and 2k. The product 2k^2 mod n=0, so those values also yield a zero modulus. The only case where a perfect square does not contain a multiple of k is where k is 2 and n=4.

OK. So what is the conjecture? I don't see it.

Does't this just lead to Wilson's Theorem:

(n - 1)! = -1 (mod n) ???

You know the answer, RK4.

Well, I wasn't sure. So it is Wilson's Thm? Right?

Right.

Thank you!

But only for prime values of n.