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-   -   Min Max range from one Prime to the next (http://mymathforum.com/number-theory/340217-min-max-range-one-prime-next.html)

 HawkI April 26th, 2017 08:10 AM

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

 Maschke April 26th, 2017 10:22 AM

I didn't follow your notation or your idea but are you aware that the distance between consecutive primes is unbounded?

 HawkI April 27th, 2017 11:26 AM

Thank you for the feed back I have looked into this

What Is the Meaning of Unbounded & Bounded in Math? | Sciencing

Quote:
 Originally Posted by HawkI (Post 568498) Sets You can also have a bounded and unbounded set of numbers. This definition is much simpler, but remains similar in meaning to the previous two. A bounded set is a set of numbers that has an upper and a lower bound. For example, the interval [2,401) is a bounded set, because it has a finite value at both ends. Also, you could have a bounded set of numbers like this: {1,1/2,1/3,1/4...}, An unbounded set would have the opposite characteristics; its upper and/or lower bounds would not be finite.

So far when ever I have tested the set of instructions, there has always been a finite set to choose from.

 Maschke April 27th, 2017 11:46 AM

Quote:
 Originally Posted by HawkI (Post 568499) So far when ever I have tested the set of instructions, there has always been a finite set to choose from.
Do you understand that I can find gaps between successive primes of 10 thousand, 10 million, 10 billion, 10 trillion, 10 gazillion? And that there is no upper bound on how high that number can go?

Does that affect your idea? As I say, I couldn't actually understand what you're trying to say.

 HawkI April 27th, 2017 12:50 PM

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

 HawkI April 27th, 2017 12:55 PM

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.

 v8archie April 27th, 2017 12:58 PM

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?

 Maschke April 27th, 2017 02:18 PM

Quote:
 Originally Posted by HawkI (Post 568506) 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.
Of course it's finite, the distance between any two natural numbers is finite.

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.

 HawkI April 28th, 2017 11:52 AM

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

 HawkI April 29th, 2017 09:30 AM

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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