August 17th, 2017, 10:11 AM  #31  
Senior Member Joined: Aug 2012 Posts: 2,126 Thanks: 618  Quote:
Random could mean: * Nondeterministic. Not the output of any Turing machine. Or * Statistically random. Satisfies some particular technical definition of randomness, regardless of whether the sequence is deterministic or not. Or * Computationally inefficient. I've never heard "random" used that way but a couple of posters in this thread seem to be using it like that. What do YOU mean by random? Please be specific.  
August 17th, 2017, 10:34 AM  #32 
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24 
Sorry, I hope it was clear: The word "random" was wrongly used by Sautoy as "unpredictable" since he is just trying to phishing fish... This is the point I would like to underline with my concerning... The "book" is: "primes enigma" I don't know if there is an english version. 
August 17th, 2017, 10:40 AM  #33 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond 
With a good look I would think the primes are statistically random. You may be able to determine the "next prime" deterministically but it can't be inferred from the distribution of the previous primes, can it?

August 17th, 2017, 10:46 AM  #34 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond  That's not answering the question. You were asked what you mean by "random" in the OP.

August 17th, 2017, 01:35 PM  #35 
Senior Member Joined: Oct 2009 Posts: 684 Thanks: 222  However, given a range of consecutive numbers, I can give a close approximation to how many primes there are!

August 17th, 2017, 08:51 PM  #36 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2550 Math Focus: Mainly analysis and algebra  Marcus du Sautoy, being a highly eminent professor of mathematics and native English speaker, is unlikely to have used the word "random" wrongly. I'll trust his word over that of anybody on this forum. But he might not have been using it in a strictly technical sense if the book is aimed at a more popular audience than purely mathematicians.
Last edited by v8archie; August 17th, 2017 at 09:40 PM. 
August 17th, 2017, 09:10 PM  #37 
Senior Member Joined: Aug 2012 Posts: 2,126 Thanks: 618 
Oh he's a contemporary mathematician. Hadn't heard of him. Thanks all for the reference. https://en.wikipedia.org/wiki/Marcus_du_Sautoy So OP, what specific book and what specific page on that book contains the word random; and are there any definitions of what random means in a few of the preceding pages? Also note that his Wiki page says he's involved in popularization of math; so as someone noted, he might be waving his hands a little for a nontechnical audience. We do need to know the context of the quote. Last edited by Maschke; August 17th, 2017 at 09:12 PM. 
August 17th, 2017, 10:34 PM  #38  
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24  Quote:
Once again: I'm playing with "random" word too, since is clear that they are well fixed and sorted, but we can lay the trick, for example asking if the next prime will be right or left respect the previous medium behavior, and the collection of a large number of primes will said us it is close to fiftyfifty.... etc... But I hope the focus will be on the other result using complicate numbers on what nobody say a word...  
August 18th, 2017, 09:10 AM  #39 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 286 Thanks: 88 Math Focus: Number Theory, Algebraic Geometry  
August 18th, 2017, 10:53 AM  #40 
Senior Member Joined: Sep 2015 From: USA Posts: 2,262 Thanks: 1198 
A much better, imo, way to phrase the problem is "Is the spacing of adjacent primes a random variable?" As someone earlier in the thread noted once you satisfy the axioms of the natural numbers the primes and their separation are fixed. They aren't random. Given that one can ask. "Does the spacing of adjacent primes satisfy the properties of pseudorandom numbers?" The properties are pseudorandom numbers are well known. Another important question is "Is the implied distribution of the spacing of adjacent primes within a bounded subset of them independent of the location of the subset?" This is the equivalent to the ergodicity of a time series. Finally an important question is "Does the implied distribution of the spacing of adjacent primes within a subset of all primes converge as we let the subset grow to include all primes. Knowing the properties of the primes we know that the spacing of adjancent primes is bounded. (I think) Thus we have a finite series of values that we can trivially assign probabilities to to achieve a distribution. The other questions are less trivial. 

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primes, proof, random 
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