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A couple conjectures for fun

  1. Apr 15, 2013 #1

    Entropee

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    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.

    [itex]\forall[/itex]n[itex]\epsilon[/itex]integers, n is not prime (n! is not prime)
    [itex]\forall[/itex]n[itex]\epsilon[/itex]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. jcsd
  3. Apr 16, 2013 #2
    [itex]\forall[/itex] n [itex]\epsilon[/itex] 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.
     
  4. Apr 16, 2013 #3

    Bacle2

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    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
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