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 April 22nd, 2011, 01:35 AM #1 Member   Joined: Jun 2010 Posts: 64 Thanks: 0 Are there infinitely many primes of the form (n!)/2+1 ? Are there infinitely many primes numbers of the form (n!)/2+1 , Thank you ..
 April 22nd, 2011, 05:32 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Are there infinitely many primes of the form (n!)/2+1 ? It's not known. See A082672 for a list of the known primes of this form (or rather, their n values). Generally no 'natural' sequence with a growth rate that high is known to contain infinitely many primes.
 April 22nd, 2011, 10:05 AM #3 Member   Joined: Jun 2010 Posts: 64 Thanks: 0 Re: Are there infinitely many primes of the form (n!)/2+1 ? Are there infinit primes numbers of the form (n!)/2+1 ?
 April 22nd, 2011, 10:13 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Are there infinitely many primes of the form (n!)/2+1 ? You must have somehow missed the reply from [color=#008000]CRGreathouse[/color]. He stated this is unknown, but generally thought not to be the case.
April 22nd, 2011, 10:57 AM   #5
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Re: Are there infinitely many primes of the form (n!)/2+1 ?

Quote:
 Originally Posted by MarkFL He stated this is unknown, but generally thought not to be the case.
It's unknown and generally considered to be a hard problem. I suspect they're infinite -- they are dense enough that if you took a random number of the same size as each member of that sequence, you'd get infinitely many primes (density log log x), and these are more likely than most to be prime since they're good mod 2, 3, 5, 7, ..., n.

 April 28th, 2011, 03:58 PM #6 Member   Joined: Jun 2010 Posts: 64 Thanks: 0 Re: Are there infinitely many primes of the form (n!)/2+1 ? n=5 we have (5!)/2+1=61 is prime number,, (7!)/2+1=2521 is prime number, (19!)/2+1=60822550204416001 IS PRIME NUMBER.Are there infinitely primes number of this form???.ca you help me please..
 April 28th, 2011, 08:47 PM #7 Senior Member   Joined: Nov 2010 Posts: 502 Thanks: 0 Re: Are there infinitely many primes of the form (n!)/2+1 ? I don't know. No one does. But I have a question for you - n = 2 we have$\Gamma (3)/2 + 1= 2$, a prime number. Do you think that there might be infinitely many primes of the form $\Gamma (n) / 2 + 1$ as well?
 April 28th, 2011, 09:16 PM #8 Newbie   Joined: Oct 2009 Posts: 26 Thanks: 0 Re: Are there infinitely many primes of the form (n!)/2+1 ? Friendlander and Iwaniec were able to prove there are infinitely many primes of the form a^2 + b^4, which is a fairly sparse set within the natural numbers. The set you're asking about is orders of magnitude more sparse than what can be proven at this time, unless there's some kind of proof trick that is unique to that set.
April 29th, 2011, 08:45 AM   #9
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Re: Are there infinitely many primes of the form (n!)/2+1 ?

Quote:
 Originally Posted by UnreasonableSin Friendlander and Iwaniec were able to prove there are infinitely many primes of the form a^2 + b^4, which is a fairly sparse set within the natural numbers. The set you're asking about is orders of magnitude more sparse than what can be proven at this time, unless there's some kind of proof trick that is unique to that set.
Heath-Brown (2001) gives an even sparser set: $a^3+2b^3,$ which has density on the order of $x^{2/3}.$ But the OP's set is of density roughly $\log x/\log\log x,$ much sparser (as you pointed out!).

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