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**Posts by LIHJ**

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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?

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

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

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