The (double) Primorial sieve & Prime spirals |
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In 2008 a project was started in which a prime number was placed on the short side of a primitive Pythagorean triangle.

In 2013 preliminary results of the (double) Primorial sieve gave an opening to unravel the Ulam spiral and the opportunity to test the usefulness of the (double) primorial sieve.

The final product of the unraveling of the Ulam spiral.pdf was completed mid 2016.

The year 2016 was further used to fully document the (double) primorial sieve, see (double) Primorial sieve.pdf.

The (double) Primorial sieve consists of an infinite set of Primorial sieves, with the forth Primorial sieve the first sieve with all the proporties of the (double) Primorial sieves.

The base of the *n*-th Primorial sieve is built from the previous sieve and is *p*(*n*) times bigger.

The *n*-th Primorial sieve filters all natural numbers divisible by *p*(*n*), with *p*(*n*) the *n*-th prime number.

The principles of the (double) Primorial sieve give a platfom to further study regularities within prime numbers.

It led for instance to a possible explanation of the (9, 1) last digit preverence of prime numbers.

The picture above shows some of the 48 struts of the forth Primorial sieve.

A strut is co-prime to the prime numbers {2, 3, 5, 7} of the forth Primorial, but can still be a composite number. See for instance strut 39. Possible prime numbers > *p*4 (with *p*4 = 7) are only found above the struts of the Primorial sieve.

The picture below shows that prime numbers < 10^9 are evenly distributed above the struts.

The columns above the struts are mirrored down the middle of the base of the Primorial sieve.

In the second picture the black lines belong to the *f*(*a*) = 6*a* + 1 function, while the gray lines belong to the *
f*(

Dividing prime numbers into two groups, based on

Double Primorial Sieve: The brief

The (double) Primorial sieve is a platform to further examine the gap between prime numbers.

It also provides a possible answer to the last digit conundrum that favours the (9, 1) combination, see the (double) Primorial sieve.pdf.

The picture below shows the distribution of the prime gap for all prime numbers < 10^9.

Dividing prime numbers into two groups, based on *f*(*a*) = 6*a* ± 1, supports the conjecture that twin primes are not related by a common denominator.