Prime free sequence.

mfgoode · 2005-06-29, 17:22

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

maxal · 2005-06-29, 19:16

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.

Numbers · 2005-06-29, 21:38

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?

jinydu · 2005-06-29, 22:19

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.

R.D. Silverman · 2005-06-29, 22:22

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.

Ken_g6 · 2005-06-29, 22:31

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!

robert44444uk · 2005-06-29, 22:40

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"

JHansen · 2005-06-29, 22:56

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

Numbers · 2005-06-29, 23:36

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.

ewmayer · 2005-06-29, 23:40

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."

tom11784 · 2005-06-30, 11:02

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

2005-06-29, 17:22	#1
mfgoode Bronze Medalist Jan 2004 Mumbai,India 804₁₆ 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

2005-06-29, 22:40	#7
robert44444uk Jun 2003 Suva, Fiji 2³×3×5×17 Posts	Arbitrary large Take an integer = x#, where # is the symbol primorial, such that x#= 2357...*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"

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2005-06-29, 21:38	#3
Numbers Jun 2005 Near Beetlegeuse 2²·97 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?

2005-06-29, 22:31	#6
Ken_g6 Jan 2005 Caught in a sieve 18B₁₆ 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!

2005-06-29, 23:36	#9
Numbers Jun 2005 Near Beetlegeuse 388₁₀ 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 “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.