Math Is Fun Forum

alexGo

Guest

Primes

I've been thinking a bit about this..

We can create an arbitrarily long string of length k of consecutive non-primes by considering (k+1)! + 2, (k+1)! + 3 .... (k+1)! + (k+1). We can make k as big as we want. Why doesn't this suggest there's a finite number of primes?

Thanks

alexGo

Guest

Re: Primes

Is it maybe because the actual size of the numbers increases more rapidly than the length of the string?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Primes

Because the list of numbers

for

does not exhaust the list of natural numbers. It just gives you finite sequences of

consecutive non-primes. There will always be gaps between the finite sequences where prime numbers can fit in.

For example,

gives the sequence

and

gives the sequence

. You can see that there is a big gap between 9 and 26 where prime numbers can be found.

Last edited by JaneFairfax (2010-01-08 00:45:52)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Primes

Hi Jane;

If I prove that some subset B of a set A is infinite as his set is, haven't I proved that the set A itself is also infinite?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Primes

Why doesn't this suggest there's a finite number of primes?

For the same reason that the sequence

Does not suggest there's a finite number of primes.

bobby: Yes.  This would suffice to prove that the integers are infinite, assuming that you've been able to define them without implicitly having this property.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Primes

Hi Ricky;

Thanks!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.