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). Doublechecked 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: 2,162 Thanks: 879 Math Focus: Wibbly wobbly timeywimey 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 
September 15th, 2018, 08:14 PM  #3 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,162 Thanks: 879 Math Focus: Wibbly wobbly timeywimey stuff. 
I don't deserve the credit for the "thanks" here. There was a nonfiction 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 nonfiction series I'd recommend it; they are worth the read. Dan 
September 16th, 2018, 02:10 AM  #4  
Newbie Joined: Jul 2018 From: Georgia Posts: 28 Thanks: 7  Quote:
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  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,162 Thanks: 879 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
September 16th, 2018, 11:38 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,654 Thanks: 2087  
September 16th, 2018, 01:09 PM  #7 
Newbie Joined: Jul 2018 From: Georgia Posts: 28 Thanks: 7  
September 16th, 2018, 05:31 PM  #8 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,162 Thanks: 879 Math Focus: Wibbly wobbly timeywimey stuff.  
September 16th, 2018, 07:44 PM  #9  
Member Joined: Jul 2010 Posts: 83 Thanks: 2  Quote:
 
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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