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12 August 2005

Senary for study of prime numbers

Senary may be considered useful in the study of prime numbers since all primes other than 2 and 3 have 1 or 5 as the final digit:

2,3,5,11,15,21,25,31,35,45,51,101,105,111,115,125,....


This is actually a really interesting thing... so, just to clarify:
in base 6 (0,1,2,3,4,5,10,11,12,13,14,15,20,21,22,23,24,25,etc) [NOT BASE 10], all prime numbers except the number 2 and the number 3 end in 1 or 5.... thus:

BASE6 BASE10 PRIME
---------- ------------ -----------
5 5 yes
11 7 yes
15 11 yes
21 13 yes
25 17 yes

ok, so... if we are thinking of base-6 in terms of hexagons... let's say top side is 0, clockwise, top-right is 1, bottom-right is 2, bottom is 3, bottom-left is 4, top-left is 5....

now, in the first hexagon, all but the bottom-left corner is prime. if we have additional hexagons to the left (more digits) they would be set to 0 which is the top-side.

flip the digit to the left to the 1-position (top-right). now, the far-right digit is prime is and only if it is in the 1-position (top-right) or the 5-position(top-left)... ie: either side of 0.

flip the left digit to 2 (bottom right). again, only primes are if the right one is in the top-left or top-right position.

flip the left digit to 3 (bottom). again, only primes are if the right one is in the top-left or top-right position.

flip the left digit to 4 (bottom-right). now it changes. only prime is if the right is the top-left.

flip the left digit to its last position (5, top-right). now only prime is if the right is on the top-right.

ok, let's add another digit. we have 3 now...
if the far left is in the 1-position (tr), center in the 0-position (top), and the right is in the top-right(1) or top-left(5) position, we got primes.

far left is in 1 (tr), center in 1 (tr), we got primes on 1 (tr) and 5(tl)....

I think we have a easily visibly repeatable pattern here.
if the digit on the left of far right is on the right-half of the hex (including bottom or top), we get primes on TopRight and TopLeft. if the digit to the left of the far right is on the left side of the hex is on the left side (BL or TL), then we alternate (first TL, then TR).

I need to see this on a larger scale to see if the entire pattern repeats for each digit, or if it only relies on the right hand... I would think that the pattern would have to be bigger. Will have to try it to see.