Nested Hexagons

LIHP · 2016-01-22 22:30:39

Sat 160123 1030hrs GMT
I found this interesting relationship concerning PRIME NUMBERS derived from NESTED HEXAGONS. By that I mean the 1st nested hexagon contains H=7 hexagons and the 2nd one contains H=19 from the equation H=3n2+3n+1 where n=1,2... etc. While PRIME NUMBERS get further and further apart with increasing number values I found that for up to n=200 (covering ~120,000) the algorithm has about a 1:3 hit rate for PRIME NUMBERS.

Not being a mathematician, I have no idea whether this observation is significant or not but if you want to see the results I would be quite happy to e-mail a pdf to you.

Regards Phil

Relentless · 2016-01-22 23:05:55

Hi Whether the observation is significant or not, I think it is interesting. I have generated my own results. For the sake of clarity, the formula is:

There is a relation between the numbers. The difference between two successive numbers is always six more than the difference between the previous two.

Last edited by Relentless (2016-01-23 00:13:22)

LIHJ · 2016-01-22 23:35:20

Hi Relentless
Thank you for converting the squared term to superscript but there is no minus sign in the equation

Relentless · 2016-01-22 23:52:17

Oh, I see what has happened. My formula includes 1 in the sequence, whereas yours starts with 7.

It seems that all the composite numbers in the sequence only have prime factors. Furthermore, those prime factors, and the terms of the sequence, must be of the form 6x + 1 where x is a positive integer. If we could list the numbers that have only prime factors of the form 6x + 1, and list the prime numbers of the form 6x + 1, we might find a similar ratio of prime numbers and the question would be answered.

But that creates a whole new question of why nesting hexagons makes numbers with no composite factors.

Last edited by Relentless (2016-01-23 00:25:36)

Relentless · 2016-01-23 04:21:38

Just for the fun of it, I collected some stats.

Of the first 200 nested hexagons (not counting H=1):
71 are prime (35.5%)
56 have lowest prime factor of 7 (28%)
22 have lowest of 13 (11%)
11 have lowest of 19 (5.5%)
8 have lowest of 31 (4%)
7 have lowest of 37 (3.5%)
4 have lowest of 43 (2%)
4 have lowest of 61 (2%)
All others have less than 4

If we check for depths other than 200, we find that the proportions for any given factor are almost identical (as if they may be cyclic). But at greater depths, the proportion of prime numbers becomes diluted because of the greater number of factors to be represented. For example, in the first 50 nested hexagons, there are 26 primes (52%) because the factors above 31 do not appear and the factors 31 and below maintain their original proportions very closely.

LIHJ · 2016-01-23 05:27:06

Hi Relentless
I have not attempted your statistics but we agree in the number of primes up to 200 hexagons. I too have been busy expanding the number of hexagons thinking that the hit ratio might converge on a couple of well known constants! I completed 400 hexagons (range of 481,201) and the prime 'hit ratio' became 1/3.228 (provided I have not missed any of the primes). So, for hexagon numbers (n) I have have these primes (P):
n P
050 026
100 042
150 053
200 071
250 084
300 096
350 111
400 124

Additionally, I assembled this list of primes from https://www.mathsisfun.com/numbers/prime-number-lists.html

Totals Primes    Range
9592 9592 000,002-100,000
17984 8392    100,000-200,000
25997 8013    200,000-300,000
33860 7863    300,000-400,000
41538 7678    400,000-500,000
49098 7560    500,000-600,000
56544 7446    600,000-700,000
63951 7407    700,000-800,000
71275 7324 800,000-900,000
78498 7223 900,000-1,000,000

Relentless · 2016-01-23 05:39:49

Hello! I look forward to seeing the results of further hexagons, as I won't be attempting to go much further myself. It is an interesting question what the proportion of primes converges to. Note that the proportion of numbers with a particular lowest factor seems to stay the same, but more factors keep appearing; therefore, the proportion for new factors must be subtracted from the proportion of primes. So in order for the proportion of primes to converge instead of becoming very small, the proportion of new factors must become increasingly small.

I believe we have a reasonable answer to the original question, however. There are so many primes because all of the numbers must satisfy 6x + 1 and may not have any composite factors (besides themselves) or factors that do not satisfy 6x + 1. And there are probably so many primes among whole numbers that meet these conditions.

bobbym · 2016-01-23 06:32:22

Hi LIHJ;

Can you explain what this means?

9592 9592 000,002-100,000

Relentless · 2016-01-23 06:41:00

Hi,

The primes column shows how many prime numbers there are in the given range. The totals column is just a running count of these numbers (i.e., the total number of primes below the upper bound of the range).

So, there are 9592 prime numbers up to 100,000. There are 7223 primes between 900,000 and 1,000,000. But there are 78,498 primes up to 1,000,000.

Last edited by Relentless (2016-01-23 06:43:04)

bobbym · 2016-01-23 06:51:18

Seems that the number of primes among those hex numbers gets smaller and smaller. Tested up to 4 000 000. Convergence seems to be heading to 0, just as expected.

Relentless · 2016-01-23 06:53:18

LIHJ,

If you are interested in primes (I do not think their frequency in the whole numbers in general is directly relevant to this problem), I have just learned of three useful approximations that have to do with them.

1. A good approximation for the number of primes less than or equal to x is:

2. A good approximation for the nth prime number is:

3. A good approximation for the probability of a whole number x being prime is:

Last edited by Relentless (2016-01-23 06:54:20)

Relentless · 2016-01-23 06:57:40

Hi bobbym,

Thank you for confirming. That must mean there is a frequent supply of new prime 6x+1 factors.

Can you offer any insight into why nested hexagons come in numbers with no composite factors and everything must obey 6x+1? The latter should be simple, but I don't get the composite factors thing yet.

Last edited by Relentless (2016-01-23 06:58:40)

bobbym · 2016-01-23 07:10:15

RIPOSTP:

When you have lots of experimental data the best way to go is using EM. The first rule in EM when you have the sequence as you do here is to take a look at the compendium of sequences. This allows you to learn quickly what everybody in the whole world knows about this sequence. The OEIS is wiser than the Oracle at Delphi, put together by N. J. A. Sloane. A person of great intelligence and more importantly, great integrity. Next to that, my insights will be kaboobly doo. So I send you there...see if what you seek is currently known.

http://oeis.org/

Put in 1, 7, 19, 37, 61, 91, 127, 169, 217, 271

After that you might be led here:

https://en.wikipedia.org/wiki/Centered_hexagonal_number

http://mathworld.wolfram.com/HexNumber.html

LIHJ · 2016-01-23 10:48:13

I seemed to started a snowball! Bobbym, your drawing is excellent to illustrate the 'nested hexagons'. Meanwhile, I have analysed 1000 'nested hexagons', which contains 3,003,001 elemental hexagons so that there are 254 prime numbers and the 'hit ratio' is 1/4. Clearly, there is a trend for the denominator to grow from my original estimate of !/3 (for 200 hexagons) up to 1/4 for 1000 hexagons. 4 is a long way from infinity and, not being a mathematician, I would not know how to prove that ultimate convergence will be zero.

EXCEL is wonderful! I have used my equation H=3n2+3n+1 to calculate all the hexagons in the NESTED HEXAGONS and then this algorithm to identify the PRIME NUMBERS found in the H columns:

=IF(B9=2,"Prime",IF(AND(MOD(B9,ROW(INDIRECT("2:"&ROUNDUP(SQRT(B9),0))))<>0),"Prime","Not Prime"))

Place e.g. the algorithm in a cell opposite B9 and write in B9 and then CRL + SHIFT + ENTER in the address box to activate it.

Scroll down to include all the H cells and the list will read off the PRIME cells.

See: http://www.excelexchange.com/prime_number_test.html

I have competed 1000 NESTED HEXAGONS in pretty quick time with 3,003,001 elemental hexagons. The algorithm works up to 268,435,455 – I am NOT going to the limit!!

The PRIME NUMBER (P) hit ratio is steady diminishing and has reached ~1/4, having generated 254 P’s for 1000 NESTED HEXAGONS.

This exercise was fun but of what use?

Math Is Fun Forum

#1 2016-01-22 22:30:39

Nested Hexagons

#2 2016-01-22 23:05:55

Re: Nested Hexagons

#3 2016-01-22 23:35:20

Re: Nested Hexagons

#4 2016-01-22 23:52:17

Re: Nested Hexagons

#5 2016-01-23 04:21:38

Re: Nested Hexagons

#6 2016-01-23 05:27:06

Re: Nested Hexagons

#7 2016-01-23 05:39:49

Re: Nested Hexagons

#8 2016-01-23 06:32:22

Re: Nested Hexagons

#9 2016-01-23 06:41:00

Re: Nested Hexagons

#10 2016-01-23 06:51:18

Re: Nested Hexagons

#11 2016-01-23 06:53:18

Re: Nested Hexagons

#12 2016-01-23 06:57:40

Re: Nested Hexagons

#13 2016-01-23 07:10:15

Re: Nested Hexagons

#14 2016-01-23 10:48:13

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