20050629, 17:22  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Prime free sequence.
Much more is known about how far apart primes are than about how close they are. By choosing the number n as large as you want how can we have a prime free sequence of consecutive whole numbers as long as you want? Mally 
20050629, 19:16  #2  
Feb 2005
2^{2}×3^{2}×7 Posts 
Quote:
They all are composite since n!+i has nontrivial divisor i. 

20050629, 21:38  #3 
Jun 2005
Near Beetlegeuse
2^{2}·97 Posts 
As this is so obviously true, and n + 1 + 2 + 3... is an infinite series, doesn't that imply that there is out there somewhere an infinite gap with no primes in it?
I find it quite difficult to reconcile that idea with the proof that the primes themselves are infinite. So where does this infinite gap fit in? 
20050629, 22:19  #4  
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Quote:
Last fiddled with by jinydu on 20050629 at 22:20 

20050629, 22:22  #5  
Nov 2003
2^{2}×5×373 Posts 
Quote:
(0) From where did you get the expression n + 1 + 2 + 3 +....??? It has ZERO connect with any prior discussion. (1) There is no such thing as an infinite prime. There are infinitely *many*, but all primes are *finite* (2) There is no such thing as an "infinite gap". The gap (equal to the difference) between any two integers is also an integer. All integers are finite. The gap between primes can be arbitrarily large. That is, for any integer M, you can find a gap between primes that is larger than M. Period. 

20050629, 22:31  #6 
Jan 2005
Caught in a sieve
5·79 Posts 
Welcome to the wonderful (and crazy) world of infinity and limits! The increase of n! is much faster than the increase of the gap size, so there's always plenty of room outside the gaps left for primes.
Here's a proof that there are infinitely many primes, which actually uses these gaps! 
20050629, 22:40  #7 
Jun 2003
Oxford, UK
3×661 Posts 
Arbitrary large
Take an integer = x#, where # is the symbol primorial, such that x#= 2*3*5*7*...*x
x can be any prime number, and there are an infinite number of those. Lets take a really big x (i.e. largest possible prime ie. infinitely large) The gap between x# and x#+x+2 is prime free, and this gap is arbitrarily and infinitely large. Regards Robert Smith "Play with fire, its safer than playing with infinity" 
20050629, 22:56  #8  
Apr 2004
Copenhagen, Denmark
2^{2}×29 Posts 
Quote:
 Cheers, Jes 

20050629, 23:36  #9 
Jun 2005
Near Beetlegeuse
604_{8} Posts 
Dr. Silverman,
“From where did you get the expression n + 1 + 2 + 3 +....??? It has ZERO connect with any prior discussion.” I beg to differ. In his post, Maxal quite clearly defined n as an integer > 1. He used this definition to explain the sequence n!+2, n!+3, n!+4, ..., n!+n. Well, is it really too much to expect that someone interested in maths would recognise that n! is itself a valid value of n ? In a post on June 9th in a thread entitled “rsa640 challenge”, you told Mr CedricVonck that “The way to learn is to start by asking questions,” I did exactly as you recommended Dr Silverman. I asked a question. I ended it with a question mark to indicate that it was a question, and I added a smilie character called “Unsure” to indicate that I really was unsure about this. Can we assume from your response to my question that you have perhaps changed your mind since June 9th ? In a post on June 14th in a thread entitled “stats question” you said, “N.B. We who have been on the Internet for a long time see this frequently. It seems that sometimes people deliberately look for excuses to be offended.” Well, now we know whom you were talking about, don’t we. Your last bullet point answers my question quite succinctly. Thank you for the interest you take in my continuing mathematical education. 
20050629, 23:40  #10  
∂^{2}ω=0
Sep 2002
República de California
2^{2}×3×7×139 Posts 
Quote:
Quote:


20050630, 11:02  #11  
Aug 2003
Upstate NY, USA
2×163 Posts 
Quote:
this is either a gap of x terms, or a subsequence of a larger gap Last fiddled with by tom11784 on 20050630 at 11:03 

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