Haha fun Halloween? Not for me, either.

I've spent the last week being sick and in pain because of a perforated appendix. I'm in the hospital now and I'm so upset because I've never missed a Halloween before in my life. I was supposed to dress as Soren to go with my friend's Ike but here I am fighting multiple appendix-induced infections. I've decided my costume will consist of only a hospital gown and Soren's mark on my forehead.

That being said, happy Halloween, guys.

Ah, sorry. They are all integers. q has to be prime, since thats the result you want to get. In this concept you want to look up primes and write factorial representations of them. You'll then notice for each n up to but not q, both n! and p don't have common factors. The point is to write a prime in a different form like "5!-107 is a representation of 13". Yeah you can simplify it further using subtraction, but this is about writing primes in some factorial form. And those forms when q>n>=1 show unique representations.


When I say q(1), I mean that when you use n>=q, you can always pull out the prime itself from both n! and p, and the result inside the parenthesis results is 1 (With bigger and bigger factorials, of course the numbers tend to be pretty ridiculously high so you have to subtract to get back to the prime you want, but the subtraction cancels itself out back to 1 every time). Example:

7=7!-5033 -> 7=7(720-719)
7=8!-40313 -> 7=7(5760-5759)

When considering say, composite numbers, you tend to get a lot of results that have a similar way of writing it elsewhere. 10=2!+8, or 2(5). 3!+4 or 2(5) 4!-14 or 2(5) Then 5!-110 or 10(12-11) or 10(1). 6!-710 or 10(72-71) or 10(1). Follows some different rules. With composite numbers its more of the idea that once the factorial has numbers it can pull out the number you want (say 10 here) the parenthesis always results in 1. So for 10 its either 2(5) or 10(1), nothing very interesting. The focus was on primes because they have unique ways of writing them. I think it says a few things on the properties of primes possibly.

EDIT: Looking it over one more time, i'm wondering if I should omit forms of 2 and 3, because the only "unique" forms are forms in which the factorial could be placed elsewhere. Like 3=2!+1 or 1!+2. Thats not very interesting and isn't as unique as I thought.. I'll give it some more thought.

Well, the thing is that p being prime doesn't necessarily mean that q will be prime as well. Playing with the factorials shows that. Although its only a few examples, I did try a few numbers, thinking that "Well, I wonder...what if the factorial was so huge that the chances of any sort of prime being a part of the number is nullified?" Thats not true either, since there's infinitely many primes, and numbers all have sort of unique prime "sets" (Like, 900=9*2*5*2*5), so the factorial would have to be infinitely huge too, so that doesn't help..

I'm starting to reconsider keeping 1! and sets of 2 and 3. The 1! because its odd. So there's a ridiculously high chance you can have something like 1!+t! (as in, you hit an even number that can be represented by a factorial). So its safer to say n>=2. Anyways, tomorrow i'll be going over with it with my professor and see what he thinks about it.

I think I'm still missing something.

Never using a smaller factorial than I've used before:
3=1!+2
5=2!+3
7=2!+5
11=3!+5 (first one I can't use 2! with)
13=3!+7
17=3!+11
19=3!+13
23=3!+17
29=3!+23
31=4!+7 (first one I can't use 3! with)
37=4!+13
41=4!+17
43=4!+19
47=4!+23
53=4!+29
59 (first one I can't use 4! with, but 5! is too big)