Well, here is the proof why any whole number ending in digit #9 is less likely to be prime versus whole numbers ending in digit #3 or digit #7 in base10. Assuming you have read the article, R U Ready?
1. 3*7=7*3=21. 3*9=9*3=27. 7*9=9*7=63. So, giving these each a “count of 1” versus counting them both ways, gives one ending in #1, one ending in #7, and one ending in #3, respectively.
2. The other possibilities are 3 * 3 = 9, 7*7 = 49, and 9*9=81. This gives 2 ending in #9 and one ending in #1.
3. Therefore, per the approach of not “double-counting” forwards and backwards, the totals are two ending in #1 and two ending in #9 versus just one ending in #7 and #3, respectively.
4. Therefore, whole numbers ending in #9 are less likely to be prime versus those ending in either #7 or #3, respectively.
~~
BK
* If somebody wants to prove that incorrect, please do.
I might as well go on record with the post above linked - I'll link it below cause it was here I posed this on 101822 1430:
~~~
OK, so I did a search using the term "geometry" and this was the top article so this is where I'm gonna try in my mind to take the hexagonal puzzle in imagination mostly out to the 4th tile. Wish me luck because this is all a mind game just now mainly, but later if I'm still alive, I'll take the hexagon tiles I have and put it to a real-world test. Probably just moments after posting this.
~~~~here goes~~~~~~
1. First tile is placed - there is one shape only distinct.
~
2. Second tile is placed - there is still only one shape distinct, because two tile edges connecting on hexagons have no differentiation without consideration for orientation. Basically, the two of them together will be the same regardless of which two sides initially connect no matter perspective 2-dimensionally - the shape is singular even with two pieces. Can you visualize this? I'm not talking trash - really.
~
3. As explained elsewhere, and per the logic expressed already, there are 3 possible shapes without consideration for orientation and in a 2-dimensional plane or field when 3 hexagonal shapes are connected - side by side. One of the three is three in a row so that they form a line (***). The second of the three is three combined so that they form the beginnings of a circle so-to-speak - they are all matched together in nice symmetry with each other (*$*). The third is when one of the many options to place the 3rd tile after two already been placed off to the side of either starting tile not being either the straight line option (#1) nor the all packed in there option (#2), but at the end of the day, there is only one discrete shape and it doesn't matter really which side gets connected even if there are 4 options for this on the two previous played tiles having 2 each....by the way, for the 3rd tile placement into a discrete shape there are two options for #1 above and two options for #2 above as well, but for the 3rd shape there are 4 options and so likely its occurrence might be higher all other things being equal.
~
4. Herein lies the formula already I suspect, but it wouldn't be difficult to extend this out to the fourth hexagonal tile placements, just walk through it per each of the 3 options indicated above. For each of the 3 there will then be several options I suspect when the 4th tile is added. This is where doing it physically is helpful because when unbiased a mind sometimes has a hard time coming to conclusion with respect to redundance without orientation - it helps to get the senses involved.
~
5. If you can do it for 4, you can do the same thing for 5. I think the solution beckons and I'll figure it out myself if need be.
~
Peace and I hope all are doing well and how about them damn Buffalo Bills?
Nobody wants to play them and now they get a week off to relish, but I hope they don't relish too much.
The season is young.
Peace, Poem of the Day, 101822 1430
BK
~~~
If you got the solution - please don't tell me - I want to figure it out on my own.
You know what the funny thing is - I know I could just do some searching using the current mathematical tools easy and ask the question - what digit is most likely to end a prime number?
Or, better yet, I could take the computers they have nowadays and run the numbers up as high as they could go and then look at the data evident....of course there will be trends there as well....
and then - wa-la -
the code is broken once again but other codes remain and anybody understands a triangle or a 2-dimensional image, must realize there is endless code.....
still, in this season, can't deny - seems time for some code to be broken.
I got ideas on primes and hexagon shapes - I like to share...
Phew - well above my pay-grade Ken but I do know Stonehenge and drive past it often. It is indeed an enigma and contains secrets we know not of. The ancients keep their secrets close it seems.
This is what is on my mind…..
* ??? *
Well, here is the proof why any whole number ending in digit #9 is less likely to be prime versus whole numbers ending in digit #3 or digit #7 in base10. Assuming you have read the article, R U Ready?
1. 3*7=7*3=21. 3*9=9*3=27. 7*9=9*7=63. So, giving these each a “count of 1” versus counting them both ways, gives one ending in #1, one ending in #7, and one ending in #3, respectively.
2. The other possibilities are 3 * 3 = 9, 7*7 = 49, and 9*9=81. This gives 2 ending in #9 and one ending in #1.
3. Therefore, per the approach of not “double-counting” forwards and backwards, the totals are two ending in #1 and two ending in #9 versus just one ending in #7 and #3, respectively.
4. Therefore, whole numbers ending in #9 are less likely to be prime versus those ending in either #7 or #3, respectively.
~~
BK
* If somebody wants to prove that incorrect, please do.
I might as well go on record with the post above linked - I'll link it below cause it was here I posed this on 101822 1430:
~~~
OK, so I did a search using the term "geometry" and this was the top article so this is where I'm gonna try in my mind to take the hexagonal puzzle in imagination mostly out to the 4th tile. Wish me luck because this is all a mind game just now mainly, but later if I'm still alive, I'll take the hexagon tiles I have and put it to a real-world test. Probably just moments after posting this.
~~~~here goes~~~~~~
1. First tile is placed - there is one shape only distinct.
~
2. Second tile is placed - there is still only one shape distinct, because two tile edges connecting on hexagons have no differentiation without consideration for orientation. Basically, the two of them together will be the same regardless of which two sides initially connect no matter perspective 2-dimensionally - the shape is singular even with two pieces. Can you visualize this? I'm not talking trash - really.
~
3. As explained elsewhere, and per the logic expressed already, there are 3 possible shapes without consideration for orientation and in a 2-dimensional plane or field when 3 hexagonal shapes are connected - side by side. One of the three is three in a row so that they form a line (***). The second of the three is three combined so that they form the beginnings of a circle so-to-speak - they are all matched together in nice symmetry with each other (*$*). The third is when one of the many options to place the 3rd tile after two already been placed off to the side of either starting tile not being either the straight line option (#1) nor the all packed in there option (#2), but at the end of the day, there is only one discrete shape and it doesn't matter really which side gets connected even if there are 4 options for this on the two previous played tiles having 2 each....by the way, for the 3rd tile placement into a discrete shape there are two options for #1 above and two options for #2 above as well, but for the 3rd shape there are 4 options and so likely its occurrence might be higher all other things being equal.
~
4. Herein lies the formula already I suspect, but it wouldn't be difficult to extend this out to the fourth hexagonal tile placements, just walk through it per each of the 3 options indicated above. For each of the 3 there will then be several options I suspect when the 4th tile is added. This is where doing it physically is helpful because when unbiased a mind sometimes has a hard time coming to conclusion with respect to redundance without orientation - it helps to get the senses involved.
~
5. If you can do it for 4, you can do the same thing for 5. I think the solution beckons and I'll figure it out myself if need be.
~
Peace and I hope all are doing well and how about them damn Buffalo Bills?
Nobody wants to play them and now they get a week off to relish, but I hope they don't relish too much.
The season is young.
Peace, Poem of the Day, 101822 1430
BK
~~~
If you got the solution - please don't tell me - I want to figure it out on my own.
Oh yeah - here is the link:
https://www.sott.net/article/157471-Stonehenge-builders-had-geometry-skills-to-rival-Pythagoras
You know what the funny thing is - I know I could just do some searching using the current mathematical tools easy and ask the question - what digit is most likely to end a prime number?
Or, better yet, I could take the computers they have nowadays and run the numbers up as high as they could go and then look at the data evident....of course there will be trends there as well....
and then - wa-la -
the code is broken once again but other codes remain and anybody understands a triangle or a 2-dimensional image, must realize there is endless code.....
still, in this season, can't deny - seems time for some code to be broken.
I got ideas on primes and hexagon shapes - I like to share...
you got a problem with that?
If so, it ain't my problem - tis yours!
Phew - well above my pay-grade Ken but I do know Stonehenge and drive past it often. It is indeed an enigma and contains secrets we know not of. The ancients keep their secrets close it seems.
https://www.english-heritage.org.uk/visit/inspire-me/blog/blog-posts/30-things-you-might-not-know-about-stonehenge/