Counting the primes

One of the key missing elements of the prime number sequence is the ability to identify the th  term in the sequence. That works two ways – can we calculate the  th  term and if we have a prime number can we identify its position in the sequence?

This blog has shown how it is possible to lay out specific composite numbers in a series of arrays or helices. Each array/helix is related to a specific prime number and the composite numbers in each array/helix are prime multiples of that specific prime number. The composite array/helix can be translated into a secondary array/helix by dividing each composite by the specific prime number. The secondary array is rich in prime numbers and the composites that fill the gaps in the secondary array are all multiples of the specific prime number and all larger prime numbers (i.e. multiples of smaller primes are excluded). The important thing to bear in mind is that ALL primes larger than the specific prime number are included AND that the primes are arranged in an array. With a little careful transfer of “lines” (i.e. rows or columns depending on how you have arranged them) a “line of symmetry” at p#/2 runs through the middle of each secondary array/helix.

The next thing to bear in mind is the number of positions in the each array/helix. This is crucial to counting primes and identifying the  th  term. The number is the same for both primary (i.e. composite) and secondary (i.e. prime rich) array.

So the array associated with prime number 5 has a mere 2 positions. As we have seen this leads to the well known 6n+/-1 formula.

The array associated with prime number 7 has 8 positions. Due to the absence of composites in the secondary array (i.e. the first composite appearing is 49) we can count the primes:

The array – associated with prime number 7, the fourth prime number – starts with 11, which is the fifth prime number (i.e. +1 from 4 to 5). We know there are 8 lines in the array so the first prime number in the second line will be the 13th prime number. Based on the more advanced/larger arrays we cannot predict exactly what the 13th prime number will be but we do know that the NEXT prime number will be the 13th. In this case the next prime is + 5#=30 i.e. 11 + 30 = 41.

Moving on we can see that adding  pi   to pi-1 #   gives a good way of counting the primes as we know the number of primes separating pi    from pi  + pi-1 #  .

Looking at the first prime in the second row of each array/helix:

13th prime is 41

48th prime is 223

345th prime is 2327……except that 2327 is not prime: 2333 is the 345th prime so you will need to see the table to see what it provides

3,250 etc….

42,333

646,031

12,283,533

300,369,798

8,028,643,012

259,488,750,746

9,414,916,809,097

Meanwhile the series 13, 48, 345, 3250, 42333 is not registered on OEIS so maybe no one has got here yet?

Meanwhile I have found out that Microsoft Excel starts to be of little use if you need more than around 15 significant digits so this is as far as I can go:

The 9,414,916,809,097th prime number is the first prime number after 304,250,263,527,257.

I can make a stab at both the

362,597,750,396,746th prime number

and the 1,539,772,852,781,310th prime number

but excel being excel and I have also reached the limit of primes.utm.edu

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2 Responses to Counting the primes

1. Deciheximal says:

I found this site after experimenting with drawing various representations of the primes within the distribution of numbers (Klauber Triangle, Ulam Spiral, and more), trying to find some new way to show them. When I started staggering the center point of a “triangle” at n+(n+5), I discovered the symmetry. And then when searching such symmetry, I found this page! I’m so glad that you’re investigating this. There’s a lot that I don’t understand, but I’m working on it. As an amateur, I don’t recognize the # symbol, but I expect that when you write 5#, you’re referring to starting at zero, adding 5, then 10 to that sum, then 15 to that sum, then 20 – what I referred to as n+(n+5) above? (The sequence goes 5, 15, 30, 50, 75, etc.)

I charted these numbers on the Sacks Spiral and drew lines between them – with my limited tools, it looks like these make angles of 30 degrees each turn, though perhaps a little wider for for 5-10-15.

Do you know if the numbers where symmetry fails to be prime (57, 77, 129, 141, 143, 169…) or their would-be counterpart primes (43, 73, 139, 137 151,191…) have any similar properties?

Also, I’ve noticed that the primes near the 5# seem to fall in a bit of a sin wave pattern, though this could just be an illusion that falls apart once I’ve plotted further out.

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• 5# = 5 x 3 x 2
7# = 7 x 5 x 3 x 2

There are a multitude of patterns to be found in the primes……..explaining and understanding them all in a comprehensive simple-ish system is the challenge

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