My Math Forum  

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

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
September 24th, 2007, 05:31 PM   #1
Senior Member
 
Joined: Dec 2006

Posts: 1,111
Thanks: 0

An unusual prime number question

Suppose that we took all the prime numbers (in whatever base we were working with) and wrote them all down in a row, with a decimal point at the beginning. In other words:

0.23571113171923...

Would the resulting number be irrational (and thus non-constructable)? Would it also be transcendental?
Infinity is offline  
 
September 24th, 2007, 07:36 PM   #2
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
This is the Copeland–Erdős constant.

It is base-10 normal, and as such irrational. I don't think its transcendence has been proved, but surely no one doubts it -- think about the consequences of showing that it's algebraic!
CRGreathouse is offline  
September 25th, 2007, 05:28 AM   #3
Senior Member
 
Joined: Dec 2006

Posts: 1,111
Thanks: 0

Quote:
think about the consequences of showing that it's algebraic!
Wow, I didn't realize that somebody has already tried that! (Then again, people have tried just about everything relating to prime numbers.) If it were shown to be an algebraic constant, would that mean that the prime numbers could be generated somehow using an algebraic formula? If it were proven to be transcendental, would that mean that the prime number sequence was non-algebraic, and thus that there is no formula for prime numbers?
Infinity is offline  
September 25th, 2007, 05:46 AM   #4
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
Originally Posted by Infinity
Wow, I didn't realize that somebody has already tried that! (Then again, people have tried just about everything relating to prime numbers.) If it were shown to be an algebraic constant, would that mean that the prime numbers could be generated somehow using an algebraic formula? If it were proven to be transcendental, would that mean that the prime number sequence was non-algebraic, and thus that there is no formula for prime numbers?
Yes and no. Clearly if it was algebraic one could reverse-engineer it to get the primes. But there are many, many formulas for the prime numbers, and if this isn't one there are still plenty of others. How else could I list 2, 3, 5, 7, 11, ...? There are even a number of closed form formulas for the primes, using just sigma notation and basic functions.
CRGreathouse is offline  
September 25th, 2007, 07:51 AM   #5
Senior Member
 
Joined: Dec 2006

Posts: 1,111
Thanks: 0

Quote:
How else could I list 2, 3, 5, 7, 11, ...?
Oh, I see. So you need the primes to get the constant, so there's no point in reverse engineering the constant to get the primes. When people talk about the magic holy grail formula for prime numbers, are they typically referring to a formula whose output could predict the next n prime numbers, in order, and also whose output consisted of nothing but prime numbers?
Infinity is offline  
September 25th, 2007, 08:02 AM   #6
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
Originally Posted by Infinity
Oh, I see. So you need the primes to get the constant, so there's no point in reverse engineering the constant to get the primes.
That's not what I said, but it's true enough.

Quote:
Originally Posted by Infinity
When people talk about the magic holy grail formula for prime numbers, are they typically referring to a formula whose output could predict the next n prime numbers, in order, and also whose output consisted of nothing but prime numbers?
There are functions of that sort already. Most of them are too slow to be useful.
CRGreathouse is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
number, prime, question, unusual



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
prime number nukem4111 Number Theory 4 October 7th, 2013 12:29 PM
prime number fantom.1040 Algebra 2 June 29th, 2011 04:46 PM
Prime Number xfaisalx Number Theory 15 July 6th, 2010 04:32 AM
Prime number Brock Algebra 1 December 13th, 2008 05:50 AM
Looking for a certain Prime number dancer42 Number Theory 5 March 18th, 2008 03:42 PM





Copyright © 2018 My Math Forum. All rights reserved.