Twin primes

Before you read this blog, you need to read this blog about prime numbers belonging 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. The Wikipedia article is a little obtuse so take at look at this simpler version. 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.

Multiplying any number of primes above 3 by 2 and then by 3 and then adding 1 invariably produces a 6n+1 result, 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.

We we go back to the alternate explanation of Euclid's method and in step 2 substitute -1 for +1.

If this is so, we then have another way of looking at the twin prime problem. Essentially for every possible twin prime, there are four possibilities

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 approach presumes that something special happens for a twin prime to exist at higher values. This alternate 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.

The other point to consider is that twin primes are inherently useless. They are no more significant than primes separated by 4, or a group 1 composite separated from a group 5 composite. Are there an infinite number of group 1 composites separated by 2 from a group 5 composites? Such questions all just as meaningless.