# A couple conjectures for fun

1. Apr 15, 2013

### Entropee

I'm not sure if these have been looked into extensively (they probably have, I just have a hard time finding them), but I was just wondering about a couple conjectures I thought up and tested by hand for a bit. Matlab can't run it for very long. Let me know if anyone finds a contradiction or if there are any links to more information about these.

$\forall$n$\epsilon$integers, n is not prime (n! is not prime)
$\forall$n$\epsilon$integers, n is not prime ((n!+1) is not prime)

Also I'm new to proofs and this method of formatting, I'm just playing around with these for fun so let me know if I have my notation wrong. I couldn't find the correct capital Z that stands for integers. Also should I say n is composite or should I say n is not prime? Does it matter?

2. Apr 16, 2013

### ImaLooser

$\forall$ n $\epsilon$ natural numbers, n is not prime → n! is not prime

By the way, in graduate school in mathematics I was instructed not to use logic notation on homework. They wanted English.

I would avoid integers for this sort of thing. It complicates the issue and does you no good.

> Does it matter?

No. They are exactly the same. A natural number is composite if and only if it is not prime. So they are logically completely equivalent.

3. Apr 16, 2013

### Bacle2

Notice that n! is never a prime for n>2 : n!:= n(n-1)...2.1

If n is of the type p-1 , (with p a prime) , then you're right, e.g., using Wilson's Theorem,

which says that (p-1)!== 1(modp) , so that (p-1)!+1 == 0(modp);

notice 4!+1 is divisible by 5 ; 6!+1 is div. by 7 . But I don't know

otherwise.

I think you may find this one interesting: show n! is never a perfect square for n>1

(Hint: Bertrand's Postulate --actually a .theorem now )

Maybe you can experiment with Wolfram: http://www.wolframalpha.com/

EDIT: look at the page: http://primes.utm.edu/glossary/xpage/FactorialPrime.html

Nice conjecture, tho, the smallest counter is 116.

Last edited: Apr 16, 2013