April 26th, 2017, 08:10 AM  #1 
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5  Min Max range from one Prime to the next
Take the Prime you want and the number of Prime it is and you can find the Minimum and Maximum range between the chosen one and the next one. Prime / n of Prime = <x> Prime + (n * <x>) + n = a Pronic  (a * (n  1)) = b b / n = <y> <x> + <y> = z (1 + z) / 2 = <c> Min + 2 Max +(c * 2) EXAMPLE 3 / 2 = 1 3 + (2 * 1) + 2 = 1 6  (1 * (2  1)) = 5 5 / 2 = 2 1 + 2 = 3 (1 + 3) / 2 = 2 Min + 2 Max + (2*2) This means that after 3 it can only be 5 or 7 I'm pretty sure this has never been done before, and I think this is why the ancients considered Pronics to be so important. <n> this means round down. n this means absolute value Last edited by HawkI; April 26th, 2017 at 08:15 AM. 
April 26th, 2017, 10:22 AM  #2 
Senior Member Joined: Aug 2012 Posts: 1,262 Thanks: 295 
I didn't follow your notation or your idea but are you aware that the distance between consecutive primes is unbounded?

April 27th, 2017, 11:26 AM  #3  
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5 
Thank you for the feed back I have looked into this What Is the Meaning of Unbounded & Bounded in Math?  Sciencing Quote:
So far when ever I have tested the set of instructions, there has always been a finite set to choose from.  
April 27th, 2017, 11:46 AM  #4  
Senior Member Joined: Aug 2012 Posts: 1,262 Thanks: 295  Quote:
Does that affect your idea? As I say, I couldn't actually understand what you're trying to say.  
April 27th, 2017, 12:50 PM  #5 
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5 
Ok so what I'm trying to say is, between a chosen Prime and the next Prime my instructions does indeed seem to say there is a finite amount in min and max range between the chosen and the next Prime. I suppose it's relatively bounded consecutively 
April 27th, 2017, 12:55 PM  #6 
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5 
Imagine someone using this method doesn't know what the next Prime is, this method would be of great use. They give it for example 3 and they now know that it can only be 5 or 7. So if they were forced to guess then this would be a huge help.

April 27th, 2017, 12:58 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,700 Thanks: 2177 Math Focus: Mainly analysis and algebra 
Well, the gap between a prime $p$ and the next prime is between $1$ and $p!$, both of which are finite. But that's really obvious. Sure, your idea probably returns better bounds than that, but how much better? Is the range of order $p$ or $\sqrt{p}$ or what? Last edited by v8archie; April 27th, 2017 at 01:00 PM. 
April 27th, 2017, 02:18 PM  #8  
Senior Member Joined: Aug 2012 Posts: 1,262 Thanks: 295  Quote:
But can you just clarify your idea and notation? P(999) = 7907. What's the range for the next prime after that? I'm just trying to understand what you're saying. ps  p(9999) = 104723. What's the range for the next one? Not challenging your formula, just trying to understand it. Last edited by Maschke; April 27th, 2017 at 02:30 PM.  
April 28th, 2017, 11:52 AM  #9 
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5 
Oooh this should be exciting. 7907 / 999 = 7 7907 + (999 * 7) + 999 = 85 999,000  (85 * (998 ) ) = 914,170 914,170 / 999 = 915 7 + 915 = 922 923 / 2 = 461 Min + 2 Max +(461 * 2) I suppose in a mundane way all this tells us is that Pronics (number squared + number) also known as rectangle numbers are to do with the range of Primes. Probably why Ulams spiral EDIT: sorry, Sacks spiral (I always mix those two up) uses them so effectively. EDIT: the 1,000th Prime is 7919 Quote "Is the range of order p or √p or what?" I will look into this Last edited by HawkI; April 28th, 2017 at 11:59 AM. 
April 29th, 2017, 09:30 AM  #10 
Senior Member Joined: Mar 2015 From: England Posts: 177 Thanks: 5 
p(9999) = 104723 104723 / 9999 = 10 104723 + (9999 * 10) + 9999 = 5266 99,990,000  (5266 * 9998 ) = 47,340,532 47,340,532 / 9999 = 4734 10 + 4734 = 4744 4745 / 2 = 2372 min + 2 max + (2372 * 2) p(10,000) = 104729 

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