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## #1 Re: Guestbook » Nested Hexagons » 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?

## #2 Re: Guestbook » Nested Hexagons » 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

## #3 Re: Guestbook » Nested Hexagons » 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