Certainly, many researchers and mathematicians are trying to study primes over the set of natural numbers. We try to use natural numbers for indexing prime numbers. From this, the first question which arises in my mind is can we actually index primes with natural numbers and find a closed form expression for predicting the nth prime?
I believe the answer is No. The reason is, while plotting Y vs X graph, there exists a trend iff X is independent of Y. Like while doing time series analysis, your time axis is independent of parameter of Y axis. For the case of primes, I don’t think this is possible. One might prove in future that primes are the basis for the set of natural numbers, however this is beyond scope of mathematics at present, because firstly natural numbers set don’t form a vector space. Because of which some other way must be devised to prove that primes are basis for the natural numbers. If it is somehow proven, since span elements of basis form the set itself, and elements are linearly independent from each other, it becomes reasonable to argue that since primes itself form natural numbers, thus if we use natural numbers which is more or less a positional number system, can never have a closed form expression for predicting the nth prime number.
If this can be progressed further and if someone finds a new set of which primes are not basis, the set will have no linear mapping with set of natural numbers.
Any comments on the discussion are appreciable.
Misleading Answers
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I didn't understand why someone posts answers that are not related to the question. Such answers make confusion about the main goal of the RG platform!
I read the mentioned articles by Muhammad Ali, unfortunately, the articles present different topics that are not related to the question!!
I think our time is precious. Misleading answers cause confusion and wasting our time. So, I ask to stop this style of answers seeking RG points regardless of any academic value!!
Dear Aziz Lokhandwala
The attached article includes a closed formula for the nth prime.
The efficiency of the algorithm is accepted within a suitable value for n.
See
http://m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf
Best regards
Issam Kaddoura Thank you sir. Although, sir, here I’ve presented my concern on a closed form linear or non linear equation, I’ve referred and cited your article in my previous works. The work of yours basically holds on the fact that every prime can be of +-1 form when divided by 6.So, sir maybe it can be called as a prime sieve with probably much higher efficiency. I do appreciate that sir, but sir that is not a closed form expression what we say. let me try to explain what I am trying to say. For example you have 1234...as a sequence of numbers with common difference 1. We know that it’s closed form expression is ’n’. But sir did you required to sieve out this set from some other set? This example might be misleading to you. Let me try to give a more better version of it. Suppose you’ve and arithmetic progression with first term as 1 and common difference 2. Here did you need to sieve out odds over set of natural numbers. The reason is no. Now if we consider primes over set of natural numbers, first thing is they are like compression and rarefactions of a sound waves in a medium which means that they are denser over one interval and sparse over other unlike the evens/odds over natural numbers. Same goes with Fibonacci numbers too, you may think of them also. People tried to find this ”Prime Wave->(Compression waves as analogy I am taking)” relations over natural numbers and they‘ve succeeded too like we all know about Mr. Riemann’s work. But, sir don’t you think is reasonable to question when and what type of set we must use in order to position elements of some other set.
Thank you,
Sincerly,
Aziz Lokhandwala
Aziz Lokhandwala
Closed formulas can be written in the form f(n), the only variable is n,
and the article includes such a formula.
f(1) provides the first prime.
f(2) provides the second prime.
f(n) provides the the nth prime.
Indeed the formula is a closed formula for all primes according to the official definition of closed-form. You can see page 3754, formulas 4.5 and 4.6 are closed formulas, only substitute n to find your request within fractions of a second for small primes. The included S function is simple, and you can substitute it directly.
By the way, the article cited by (15) researcher.
Regards
Hello.
Here is my solution :
The set N-P or N is the set of natural numbers and P the set of even numbers can be partitioned into 3 classes two by two disjoint , 2 of these classes give all prime numbers and composite numbers. These composite numbers can appear in an infinite triangle composed of matrices. Each of the numbers of these matrices (ending in 1, 3, 7, 9) can be obtained from a finite number of patterns (formulas). These patterns can be codified in programs operating in parallel (massively parallel). We submit a number to each of these programs, if it does not belong to the infinite triangle of composite numbers ( déterminist algorithm) then it is prime. If it belongs to the triangle of compound numbers, we can recursively determine ( déterminist algorithm) the prime numbers that compose it.
Sorry, I can not give you more details because my solution is not yet submitted for publication.
I can only say that Gauss closed this path of research to researchers. Too bad ... He did not demonstrate a thorough and nuanced observation of prime numbers in this research path.
Finding an explicit closed formula that generates primes is one of the most challenging unsolved problems.
Even great mathematicians failed to construct such a formula.
It is extraordinary for somebody to claim that he had found such a formula. One can hear hundreds of such claims every day.!!
It is a matter of telling the truth and the academic reputation!!
It is true that there is not a formula but patterns of formula obtained from a theorem and the infinite triangle of composed numbers...
Well, there's actually a perfect ordering of the prime numbers, that is, an analytic function that assigns for each integer in the naturals, its prime, though it's very complicated, it involves the prime zeta function. I put together a paper that explains how that is done, head to my page to read it, it's called "An exact formula for the prime counting function".
Now I am in the process of proposing an integral representation for that function and should be publishing it soon, though still not sure if that will be feasible.
PS-formulas for prime numbers exist by the bucket, though all of them are convoluted and impractical.
If you're trying to create prime formulas, make sure that you create simple ones with simple arithmetic patterns, because only then can they be useful, interesting and new.
Other that these formulas will only be curiosities.
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