mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2005-06-29, 17:22   #1
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22×33×19 Posts
Question 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
mfgoode is offline   Reply With Quote
Old 2005-06-29, 19:16   #2
maxal
 
maxal's Avatar
 
Feb 2005

22×32×7 Posts
Default

Quote:
Originally Posted by mfgoode

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
Take n!+2, n!+3, n!+4, ..., n!+n for any integer n > 1.
They all are composite since n!+i has non-trivial divisor i.
maxal is offline   Reply With Quote
Old 2005-06-29, 21:38   #3
Numbers
 
Numbers's Avatar
 
Jun 2005
Near Beetlegeuse

1100001002 Posts
Default

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?
Numbers is offline   Reply With Quote
Old 2005-06-29, 22:19   #4
jinydu
 
jinydu's Avatar
 
Dec 2003
Hopefully Near M48

2×3×293 Posts
Default

Quote:
Originally Posted by Numbers
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?
In order for that to be true, n (in fact, I think you mean n!) would have to be infinite, which would make it not an integer. No such infinite gap exists.

Last fiddled with by jinydu on 2005-06-29 at 22:20
jinydu is offline   Reply With Quote
Old 2005-06-29, 22:22   #5
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22×5×373 Posts
Talking

Quote:
Originally Posted by Numbers
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?
Gibberish. Illucid.

(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.
R.D. Silverman is offline   Reply With Quote
Old 2005-06-29, 22:31   #6
Ken_g6
 
Ken_g6's Avatar
 
Jan 2005
Caught in a sieve

18A16 Posts
Default

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!
Ken_g6 is offline   Reply With Quote
Old 2005-06-29, 22:40   #7
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

22·32·53 Posts
Default 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"
robert44444uk is offline   Reply With Quote
Old 2005-06-29, 22:56   #8
JHansen
 
JHansen's Avatar
 
Apr 2004
Copenhagen, Denmark

22×29 Posts
Default

Quote:
Originally Posted by robert44444uk
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.
Now you are just trying to tease Dr. Silverman. That is not nice of you.

--
Cheers,
Jes
JHansen is offline   Reply With Quote
Old 2005-06-29, 23:36   #9
Numbers
 
Numbers's Avatar
 
Jun 2005
Near Beetlegeuse

22×97 Posts
Default

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 “rsa-640 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.
Numbers is offline   Reply With Quote
Old 2005-06-29, 23:40   #10
ewmayer
2ω=0
 
ewmayer's Avatar
 
Sep 2002
República de California

2D3316 Posts
Default

Quote:
Originally Posted by robert44444uk
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)
No - it makes no sense to speak of primes (or composites, or even integers, for that matter) being "infinitely large". Primes, composites and integers are all *numbers* - infinity is not a number, although by convention it can be manipulated in some ways like finite numbers can.

Quote:
The gap between x# and x#+x+2 is prime free, and this gap is arbitrarily and infinitely large.
Again, you're blithely mixing concepts that sound superficially similar but are profoundly different. "Arbitrarily" in your sense means "you give me any finite natural N, I can find a prime-free gap whose length exceeds N." But any such gap will clearly not be infinite in length, since the very definition of a gap between primes implies the existence of a next-larger prime, i.e. one which bounds the gap from above. And even without an explicit gap we know that gaps cannot be arbitrarily large with respect to the primes bracketing them, since by Bertrand's Postulate (a.k.a. Chebyshev's theorem) if n > 1, then there is always at least one prime p such that n < p < 2*n. So one first needs to be precise about what one means by "arbitrarily". And irrespective of whether one is referring to the absolute or relative size of prime gaps, "arbitrarily" in this context does not mean "infinitely."
ewmayer is offline   Reply With Quote
Old 2005-06-30, 11:02   #11
tom11784
 
tom11784's Avatar
 
Aug 2003
Upstate NY, USA

2×163 Posts
Default

Quote:
Originally Posted by robert44444uk
The gap between x# and x#+x+2 is prime free...
not to be picky, but only x#+2 to x#+x+1 need be composite (take x=5)
this is either a gap of x terms, or a subsequence of a larger gap

Last fiddled with by tom11784 on 2005-06-30 at 11:03
tom11784 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Unhappy in choosing prime sequence :-) pepi37 Lounge 9 2017-07-15 19:53
Mersenne Prime Sequence Stan Miscellaneous Math 34 2013-08-25 17:35
A Prime Sequence davar55 Puzzles 16 2009-07-02 19:58
Prime creator through sequence roger Puzzles 25 2007-02-09 15:50
Catalan sequence (is C5 prime?) Orgasmic Troll Math 10 2003-10-03 15:45

All times are UTC. The time now is 23:18.

Fri Dec 4 23:18:35 UTC 2020 up 1 day, 19:29, 0 users, load averages: 1.38, 1.44, 1.45

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.