Research Suggestions: Prime Number Distribution

[Nebula]

I have been doing some research concerning the prime number distribution, and basically have come to a point in which I'm not sure what direction to take. I'll breifly explain my work in hopes that someone may have some suggestions, or some conjectures which may spark new directions.

My work has mainly been a statistical analysis of the distribution and the corresponding prime gaps. A prime gap is the difference between to consecutive primes. Now what I have done so far is to compute the average prime gap over certain intervals. For instance a have computed the average prime gap for the frst thousand primes, the second thousand primes and so on. Now we have a sampling distribution of means, and so we can linearly regress the means to form an equation. Also from the data we can compute confidence intervals about the slope of our linear regression line. Now it just so happens that the confidence interval includes zero, so we cannot rule out the possibility that there is no relation, a slope of zero. It would appear that the average value for prime gaps taken in samples of one thousand progress in linear fashion. But who is to say that this does not exhibit an exponetial trend somewhere down the number line when prime numbers become increasingly sparse. Another thing to consider is the fact that I'm dealing with an infinite population, which obviously makes things a bit difficult.

What I'm looking for is perhaps a new direction. Maybe there is somthing I over looked, or a new relationship I can explore, maybe somthing analytical or somthing to help solidify arguements. If you have any questions please ask, and any ideas, suggestions or comments would be appreaciated. Thanks.

 

[matt grime]

The distibution of the primes is well known to be related to the RIemann Zeta function, and there is some indication mathematical physics (random matrix theory) might shed some light on it.

Of course you must be aware that given any number n there is sequence of n-1 consecutive composite numbers, n!+2, n!+3,... hence if you're conjecture is true about long term trends, then it would have to take this into account, which might say something about the distribution of primes either side of this range. There is of course a well known upper bound on the difference between two primes, with several improvements, how do they relate to your conjecture?

For instance, for every n greater than1 there is a prime between n and 2n, so exponential is the worst growth possible for the average gaps. This is Bertrand's Postulate; there are better results, probably by Erdos

 

[tacman]

One potential direction you could take with your research on prime number distribution is to explore the relationship between prime numbers and other mathematical concepts. For example, you could investigate the relationship between prime numbers and the Fibonacci sequence or the Golden Ratio. You could also look at the distribution of prime numbers in different number systems, such as binary or hexadecimal. Additionally, you could analyze the distribution of prime numbers in different mathematical structures, such as graphs or matrices. These approaches may provide new insights and potentially uncover patterns or relationships that could help explain the distribution of prime numbers. It may also be worth considering consulting with other mathematicians or attending conferences to discuss your research and gather new ideas and perspectives. Overall, the key is to continue exploring different avenues and considering different perspectives in order to gain a deeper understanding of prime number distribution. Good luck with your research!