My Math Forum Prime numbers in base 6

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 September 15th, 2018, 07:45 AM #1 Newbie   Joined: Jul 2018 From: Georgia Posts: 28 Thanks: 7 Prime numbers in base 6 I did an online search and found a couple of references, but nothing that pursued this in any depth. First of all, it seems that considering primes in base 6 would be a very useful teaching tool - an introduction to prime numbers - as it clearly illustrates both how 2 and 3 are the basis for the bulk of composite numbers (2/3rds of them) and it provides an obvious demonstration of the distribution of primes and twin primes. In base 6, every prime number must end in 1 or 5, and every twin prime must be a pairing of numbers ending in 5 and 1 consecutively. The first 6 primes >3 (all twins) in base 6 are: 5, 11, 15, 21, 25, and 31. 35 is also a prime, but 41 (25 decimal) is of course a composite. Here's a list of the first 16 composite numbers with least prime factors >=5 (Decimal value in parentheses). Double-checked these, but hope I didn't mess one up. 41 (25) 55 (35) 121 (49) 131 (55) 145 (65) 205 (77) 221 (85) 231 (91) 235 (95) 311 (115) 315 (119) 321 (121) 325 (125) 341 (133) 355 (143) 401 (145) I've tried hard to see if there's anything useful about looking at that pattern in base 6 and haven't come up with anything. Does anybody see anything worth pursuing? I'll also add that as a longtime assembler language programmer, I became quite accustomed to thinking in base 16. Thinking in base 6 is much more of a challenge.
 September 15th, 2018, 08:33 AM #2 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,258 Thanks: 929 Math Focus: Wibbly wobbly timey-wimey stuff. Base 6 works much better than base 10 because, as you say, there are only two cases for the unit place of the number. (There are, of course, 1, 3, 7, and 9 for base 10.) So it's more efficient to work with to find prime numbers. A slightly better, though notationally harder, case is to use base 2 * 3 * 5, then 2 * 3 * 5 * 7 and so on. The percentages of how the primes show up are better the higher the base you use. -Dan Thanks from Sebastian Garth
 September 15th, 2018, 08:14 PM #3 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,258 Thanks: 929 Math Focus: Wibbly wobbly timey-wimey stuff. I don't deserve the credit for the "thanks" here. There was a non-fiction science book by Isaac Asimov (I don't recall the title) in which he discussed this idea thoroughly. If I can recall the title I'll let you know. It's an excellent book. If you can still get his non-fiction series I'd recommend it; they are worth the read. -Dan
September 16th, 2018, 02:10 AM   #4
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 Originally Posted by topsquark Base 6 works much better than base 10 because, as you say, there are only two cases for the unit place of the number. (There are, of course, 1, 3, 7, and 9 for base 10.) So it's more efficient to work with to find prime numbers. A slightly better, though notationally harder, case is to use base 2 * 3 * 5, then 2 * 3 * 5 * 7 and so on. The percentages of how the primes show up are better the higher the base you use. -Dan
Yes, I've laid out length 30 and length 210 sets of numbers (7 rows of 30 each in the second case). I was primarily looking at twin primes in that case. Beyond that it gets a little tough to do. I think humans could grasp base 30 using letters as is done in hexadecimal. But you're only 'identifying' two more integers by least prime factor with the last digit of the number (5 and Q if I'm counting letters correctly) Still something worth looking at.

Is the book you mentioned in your second reply 'Adding a Dimension'? I think I read everything by Asimov that I could get my hands on as a kid (50+ years ago) and I vaguely recall that one.

September 16th, 2018, 07:24 AM   #5
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 Originally Posted by RichardJ Yes, I've laid out length 30 and length 210 sets of numbers (7 rows of 30 each in the second case). I was primarily looking at twin primes in that case. Beyond that it gets a little tough to do. I think humans could grasp base 30 using letters as is done in hexadecimal. But you're only 'identifying' two more integers by least prime factor with the last digit of the number (5 and Q if I'm counting letters correctly) Still something worth looking at. Is the book you mentioned in your second reply 'Adding a Dimension'? I think I read everything by Asimov that I could get my hands on as a kid (50+ years ago) and I vaguely recall that one.
I'm afraid I don't know. I looked through a list of all his books and couldn't find the one I was looking for. It used to be in my Dad's "library" that I sort of stole from him but I wound up leaving a good fraction of those with an actual library.

-Dan

September 16th, 2018, 11:38 AM   #6
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 Originally Posted by topsquark There was a non-fiction science book by Isaac Asimov (I don't recall the title) in which he discussed this idea thoroughly.
I couldn't find this.

September 16th, 2018, 01:09 PM   #7
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 Originally Posted by skipjack I couldn't find this.
The book I mentioned, "Adding a Dimension," was a collection of previously published articles. One of them was an article titled "One, ten, buckle my shoe" which discussed alternates to base 10.

September 16th, 2018, 05:31 PM   #8
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Quote:
 Originally Posted by RichardJ The book I mentioned, "Adding a Dimension," was a collection of previously published articles. One of them was an article titled "One, ten, buckle my shoe" which discussed alternates to base 10.
Hey! I think that was the one.

-Dan

September 16th, 2018, 07:44 PM   #9
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 Originally Posted by topsquark I don't deserve the credit for the "thanks" here. There was a non-fiction science book by Isaac Asimov (I don't recall the title) in which he discussed this idea thoroughly. If I can recall the title I'll let you know. It's an excellent book. If you can still get his non-fiction series I'd recommend it; they are worth the read.
Absolutely love Asimov but honestly haven't yet read any of his non-fiction stuff. Definitely on my list now though. Looks like the internet archive has a pretty nice library of his writings so I'll probably start there.

 November 21st, 2018, 01:02 AM #10 Newbie   Joined: Nov 2018 From: Tbilisi, Georgia Posts: 1 Thanks: 0 Seems prime numbers are the result of some mapping from some functional space to N. T. Dzigrashvili

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