Would being able to limit the average difference between two primes to a narrow region be useful at all?
Yes, if you include “overthrowing all of human mathematics right down to their foundations” counts as “useful”. The difference between successive primes gets larger without bound as you go to larger numbers, and so there is no average.
I suppose it would also be useful to change the ratio of the circumference to the diameter to an even 3.0000
Well I have set up an excel spreadsheet and calculated the average difference for all the consecutive primes like 5-7; 7-11; 11-13 etc. for the firs 19459 primes. I then set up another column to calculate the lower boundary for this average and another for the upper boundary of this average. The accuracy level for the lower boundary is 99.1461% on average and the upper boundary has an accuracy of 99.742% on average. So you can predict where the average gap for the first n primes will fall. This is not effective early on and is more effective for large values of n.
You should read the Wikipedia page on Prime Gaps.
You misunderstand the prime gaps as you rightly point out increase but they increase in a logarithmic fashion. Therefore the upper and lower boundaries of this average also increase in a logarithmic fashion. They are not constant due to the fact that prime gaps tend to increase.
There is an average for the first n prime gaps that can be calculated. To do this for the first billion primes is not practical therefore an upper and lower boundary clearly defines a region where this average must lie. This is far more practical as it is two calculations.
I did that is what gave me the idea to see how the average changes and why I say the increase in the average prime difference for the first n primes is logarithmic due to the fact that the rate at which the differences increase is logarithmic in nature.
I thought that said that the average did rise without bound, but I admit I ddin’t read it carefully. I think you’re right that Chronos’s conclusion doesn’t follow just from the gaps rising without bound.
I may not have been clear enough originally. I couldn’t tell you if the maximum gaps between two primes rise without bound. from what I’ve read the maximum gap rises in a logarithmic fashion though. The average though does rise with the boundaries.
It’s all kind of related. We know the gaps between primes tends to increase logarithmically.
There are probably several ways to consider “average”, but at any rate, as per Chronos’s link, there are existing upper and lower bounds on prime gaps, which should give equivalent information as any proposed alternate bounds, which is what you are essentially proposing.
It doesn’t state it directly, but as there exist both upper and lower bounds that behave logarithmically, it follows that any average of such bounds will also be logarithmic.
That’s not to say that some kind of new formulation couldn’t be interesting, but it wouldn’t necessarily give us new insight, either.
If you know how many primes are in a range (which you do here, because you know there are 19459 primes in the first X integers, where X is the value of the largest prime plus one), then can’t you calculate directly the average gap between them with only that info?
I don’t know what the 19459th prime is, but let’s say it’s 1,000,000. So out of the first 1,000,000 integers, 19459 of them are primes. The total gap is (1000000 - 19459), so divide that by 19458 to get the average gap of 50.39269195.
Is that what you’re doing?
I think you mean “Zenbeam’s link”, there. I didn’t link to anything.
You misunderstand if you take the difference of all the first 19459 primes then you can calculate the average. So you create a list of the first 19458 differences. But what I am saying is to do this can take an incredibly long time. So instead you create a region where this average must lie.
So for 19459 the upper boundary is 11.28 approximately and the lower boundary is 11.16 approximately. So I can’t tell what the actual average immediately but I can tell you the average lies between 11.28 and 11.126. those two values are calculated using only logs you can do it on your calculator in under a minute.
No that is not what I am saying. I know there are three well known ones that you see on Wikipedia that define the rate at which the maximum gap between primes increases. This isn’t that.
To avoid further confusion I’m going to state as simply as possible. Let’s say we take the first 1 000 000 prime gaps. Now I know of no way currently that will tell what the average of all these prime gaps will be. But you can find the region where this average will lie.
So for the first 1000 000 prime gaps the maximum the average of the all the first 1000 000 prime gaps can be is 15.78 approximately. The lowest value that the average of the first 1000 000 prime gaps can be is 15.61 approximately.
This region is small and the average will lie in between these two values and if someone finds otherwise I would die of shock. The upper bound is ineffective for very small primes due to technical reasons but when you start reaching larger values then it becomes effective.
It is all about establishing a region where the average of all the prime gaps will lie. Now as you tend to infinity so does the average and so do the upper and lower boundaries as well they are not static they change they are for a specific average after a specific number of prime gaps.
I can calculate it exactly with only pencil and paper in under a minute.
I’ll show you - what is the value of the 19459th prime number?
That is the genius I don’t need to know any prime numbers to give you those values. It doesn’t rely on primes. Name a number and I will give you the boundary. I timed myself I can calculate the upper and lower boundary in under 30 seconds with just my calculator or any scientific calculator for that matter.
Nevermind, I found it - it’s 218,171. So out of the first 218,171 integers, 19,459 of them are primes. The range between 2 and 218,171 is divided up into 19458 gaps, so the average gap is:
(218171 - 2) / 19458 = 11.21230342
That’s not an estimate, it’s exact.
ETA: I posted this before I read Armstrongm’s most recent post.
But does anyone know if this has any practical use though?
That’s very good you will find it lies in the region I predicted it would as well. For those type of primes you can attain an exact value but for the first trillion primes? then an estimation may be useful.
OK, so your method would then let me know that the value of the 19459th prime is between 219,488 (corresponding to 11.28 avg) and 217,153 (corresponding to 11.16). The actual value is 218,171.
How certain is it? This one was right in the middle of your range, but for other nth prime numbers, is it guaranteed to be in your range, or just highly likely?