My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Thanks Tree1Thanks
  • 1 Post By topsquark
Reply
 
LinkBack Thread Tools Display Modes
September 15th, 2018, 07:45 AM   #1
Newbie
 
Joined: Jul 2018
From: Georgia

Posts: 10
Thanks: 2

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.
RichardJ is online now  
 
September 15th, 2018, 08:33 AM   #2
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 1,879
Thanks: 761

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
topsquark is offline  
September 15th, 2018, 08:14 PM   #3
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 1,879
Thanks: 761

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
topsquark is offline  
September 16th, 2018, 02:10 AM   #4
Newbie
 
Joined: Jul 2018
From: Georgia

Posts: 10
Thanks: 2

Quote:
Originally Posted by topsquark View Post
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.
RichardJ is online now  
September 16th, 2018, 07:24 AM   #5
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 1,879
Thanks: 761

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by RichardJ View Post
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
topsquark is offline  
September 16th, 2018, 11:38 AM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 19,508
Thanks: 1741

Quote:
Originally Posted by topsquark View Post
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.
skipjack is offline  
September 16th, 2018, 01:09 PM   #7
Newbie
 
Joined: Jul 2018
From: Georgia

Posts: 10
Thanks: 2

Quote:
Originally Posted by skipjack View Post
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.
RichardJ is online now  
September 16th, 2018, 05:31 PM   #8
Math Team
 
topsquark's Avatar
 
Joined: May 2013
From: The Astral plane

Posts: 1,879
Thanks: 761

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by RichardJ View Post
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
topsquark is offline  
September 16th, 2018, 07:44 PM   #9
Member
 
Joined: Jul 2010

Posts: 81
Thanks: 1

Quote:
Originally Posted by topsquark View Post
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.
Sebastian Garth is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
base, numbers, prime



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
prime numbers and composite numbers shaimaa saif Algebra 10 November 17th, 2015 08:02 AM
The paradox between prime numbers and natural numbers. Eureka Number Theory 4 November 3rd, 2012 03:51 AM
A complex base for a set of numbers goedelite Real Analysis 2 August 14th, 2012 08:41 PM
Finding smallest n : (base^n + base^(n-1) + ... base^1) > x momesana Algebra 4 December 3rd, 2009 06:13 PM
How do i calculate base numbers on a scientific calculator? geissap Algebra 3 June 18th, 2009 12:21 PM





Copyright © 2018 My Math Forum. All rights reserved.