Math Is Fun Forum

ratbagp

Member

Registered: 2013-08-15

Posts: 6

another view of twin primes

As a new member, I have a favour to ask. Everybody should have at least one prime number theory and I have developed one that I can't find flaws in. After years of writing buggy computer programs, I am enough of a pessimist and realist to recognize that there has to be some error in my logic. Could you show me where I went wrong? The theory is for twin primes.

ray

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All prime numbers except for 2 and 3 belong to one of two sequences, 6n + 1 and 6n - 1.

Let's look at Euclid's famous proof about the number of primes being infinite. It becomes obvious that Euclid's proof applies only to the 6n + 1 series.

In more detail, the Euclid proof of multiplying primes together, including the 2 and 3, and then adding 1 always creates a result belonging to the 6n+1 series, prime or otherwise.

Therefore Euclid’s proof applies only to the primes belonging to the 6n+1 series. Are there an infinite number of 6n-1 primes? Could we change Euclid’s method to subtract 1 instead of adding 1 to include those 6n-1 primes? As far as I can determine, the -1 proof works just as well as the +1 proof.

Essentially for every possible twin prime, there are four possibilities with only option 1 producing a twin prime.
1.      6n-1 prime                   6n+1 prime
2.      6n-1 prime                   6n+1 composite
3.      6n-1 composite            6n+1 prime
4.      6n-1 composite            6n+1 composite

Now we can look at the problem in a different light, essentially like looking at a pair of scissors with two infinitely long blades instead of a sword with one infinitely long blade. The traditional sword approach presumes that something special happens for a twin prime to exist at higher values. This alternate scissors approach requires nothing special for twin primes to exist at higher values and would actually require some pattern to perpetually disallow twin primes at very high values.

If we can say that Euclid’s proof applies to both the 6n-1 as well as the 6n+1 series, then unless there is a recurring pattern to the distribution of primes, there is nothing to prevent a twin prime from existing at any stage.

Paradoxically, it is the lack of the so far unobserved recurring pattern for prime numbers that allows the twin primes to exist.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

Hi;

It is well known that all the primes are of the form 6n+1 and 6n-1 if I remember because we used to use that fact in programs.

In more detail, the Euclid proof of multiplying primes together, including the 2 and 3, and then adding 1 always creates a result belonging to the 6n+1 series, prime or otherwise.

This is not true. My primes could be

2*3*5*7*11+1 the 11 is of the form 6n-1. Euclid's proof works for all primes of any type.

---

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.

ratbagp

Member

Registered: 2013-08-15

Posts: 6

Re: another view of twin primes

Maybe I was not clear.

2*3*5*7*11 = 2310. Adding 1 gives 2311. Dividing by 6 gives 385 and +1 remainder, hence 6n+1.

Think of it as 2*3*( any sequence of primes ) + 1

ray

Last edited by ratbagp (2013-08-15 04:48:13)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

Hi;

I think you are stating it incorrectly. The product of of the primes although it is of the form 6n+1, may not even be a prime. Euclid's proof has nothing to say about that number. Except that it is larger then Pn and does not have a factor in the sequence P1P2P3...Pn.

2*3*5*7*11*13 + 1 is not a prime.

Euclid's proof is for all primes up to Pn.

Let's look at Euclid's famous proof about the number of primes being infinite. It becomes obvious that Euclid's proof applies only to the 6n + 1 series.

It is easy to see that the product of P1*P2*P3...Pn + 1 will always be of the form 6n+1

This is known: All twin primes are of the form 6n + 1 and 6n -1 except for 2 and 3. For that matter, except for 2 and 3 all primes are of that form.

---

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.

ratbagp

Member

Registered: 2013-08-15

Posts: 6

Re: another view of twin primes

I see how I am stating Euclid incorrectly now. What has been bothering me is how to state that the prime numbers can be divided into two sequences 6n+1 and 6n-1 and more importantly that are both sequences have no limit.

If you have a few minutes, could you take a quick look at some other ideas that I have put together in a blog on how the primes can be thought of as two sequences. I see that I cannot include links so you would have to do a Google search with the following:

unprime6 A different way of looking at prime numbers

I have found the concept of logically dividing the primes into two sequences intriguing and useful. The 6n-1 and 6n+1 composites are perhaps even more interesting.

In any case, thanks for your help.

ray

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

These add up to 1665. In addition there is one unprime with only four combinations.

What is that one number?

---

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.

ratbagp

Member

Registered: 2013-08-15

Posts: 6

Re: another view of twin primes

I will have to fire up my SQ Server to find it for you. It will take a little while.

ray

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

Hi;

Okay, 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.

ratbagp

Member

Registered: 2013-08-15

Posts: 6

Re: another view of twin primes

This is going to take a bit longer than I thought. I haven't used the database in a couple of years and there seems to be a problem where I will have to reinstall. I have the code to recreate the numbers but it will be a few days.

You have my humble apologies.

ray

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

Okay, I am sorry. I know the feeling. Take your time and do it at your own convenience.

---

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.

ratbagp

Member

Registered: 2013-08-15

Posts: 6

Re: another view of twin primes

It came to me in the middle of the night. It would have to be the starting point of minus group 4 which is 6875.

A simple spreadsheet comes up with the factors
5 * 1375
11 * 625

and

25 * 275
55 * 125

I still need to rebuild my database. Thanks for taking the time to look at the blog.

ray

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another view of twin primes

Hi;

Okay, thanks for providing that.

---

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.