I wrote a function, GetCumulativePrimeSpacing[i], in Mathematica to return the interference pattern created by all primes up to the ith prime. This sequence works for finding primes up to the next prime squared. I have obtained some interesting results.
Some interesting results are that the Length and Total of this sequence are related to the Primorial[i]. The count of 2s and 4s in the sequence is equal to the product of primes as well.
I have some ideas on what to do with these results but I'd like to hear thoughts from the experts first.
Any help would be appreciated.
Thank you.
I wish i could understand it better. Am a physicist with a lifelong love for
mathematics, so i appreciate this wonderful area to work in.. Have read a lot
of stuff about euler and Ramanuthan, ,Riemann conjecture, etc. , so, hopefully
you will learn something here..Good luck! ken
Thanks Ken, I can hopefully glean some things from these data patterns.
This might be important for my research (Phd research). I did not get there yet, but once I do, I will need to find, as fast as possible, the first prime of a certain form. Now the first thought I had about this was, well, I will just have to go over every number of that form, starting from the first and test for each one whether it is a prime or not. But if you know and can prove for sure what the prime gaps are (proved and not only expected) than I would know that certain number ranges surely won't contain primes (this would have to be available as a proof in a paper or something like that) and I would be able to bypass many of my prime tests and go directly into the range of the first prime of the form I am looking for. Prime testing is already polinomial (AKS primality test - Agrawal's algorithm), but this would be interesting anyway because it could make my search a lot faster.
Even if it is just expected (and not proved) prime gaps, it could be interesting to me, provided that this "expectation" was something that you could extract out of a deterministic formula so that if you repeat it, it delivers always the same result. In this situation I would conform to find the first prime over an "expected interval of occurrence of primes". Since I will use this for modelling purposes, I would loose on the modelling side, because I would miss the natural idea of Next or First prime of a certain form, but I would eventually gain on the efficiency side, by making my search a lot faster. So, depending on this balance, even "expected" against "proved" prime gaps, might be useful to me.
Well, I have thought of this before too. But i believe that any gaps are dependent upon all the primes smaller than that number in question. Of course with the proviso, that
any given prime does not make its presence known until the square of that prime.
So, this is what makes the primes act so randomly in appearance (because if they depend upon them all, previously, then the gaps etc. become ever more complicated and seemingly random..
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