August 15th, 2017, 06:26 PM  #21  
Senior Member Joined: May 2016 From: USA Posts: 904 Thanks: 359  Quote:
Perhaps the question is whether the distribution of primes is close to random in some statistical sense, but what sense is not specified nor is "close enough." Some interpreted the question to mean whether there was a deterministic way to determine whether a number is prime, and of course there is. In the sense that if it impossible to determine whether a given number is prime, primality is random, no one asserts that primes are random so the original question is absurd. I thought that the question made sense if it was asking whether a closed form formula to determine what the nth prime was in ascending order. If such a form exists, then primality is not "random." I have now been told both that such a formula has been found and that such a formula has not been found. Which of those answers is correct strikes me as an unambiguous question and certainly not meaningless although whether it is interesting is a matter of subjective taste.  
August 15th, 2017, 08:39 PM  #22 
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24 
The worrd "random" was used several time in Sautoy books. It stinks me lot, and push me to start my own investigation on them, finally finding the 3 algos to have back a prime: $z=n!/n^2$ Than the correct number of primes within a certain $x$ (starting from 5 because for $4$ the previous formula gives a false value), $$ Pi(x) = \left( \sum_{n = 5}^{\ P} (\lfloor(n!/n^2)\rfloor\lfloor(n!/n^2)\rfloor)+1/3 \right) +2 $$ and the next prime. I already posted them several years ago here, but someone argue that they are just algos, not clean formulas or functions etc. etc... But now I discover that applying a certain function on the Primes (only) produce a very similar output than if we apply it on the Integers, and this, I think, will confirm that they cannot be randomly placed since if so a similar distribution will be a very improbable (just to be carefull) case. And since this pattern is respected with various exponents of such function it confirm our suspect. But from here to write a theorem, than a math proof is another story.... 
August 16th, 2017, 04:26 AM  #23 
Global Moderator Joined: Dec 2006 Posts: 18,595 Thanks: 1493 
It's hard to prove something that neither makes sense nor works at all.

August 16th, 2017, 05:13 AM  #24 
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24  To what ? Why ? The trick is this: Wrote Integers from n,m=1 to n,m=1000 as Complicate Number $M_2$, and as Complicate Number $M_3$: $n= (\lfloor(n^{1/2})\rfloor)^2+(n (\lfloor(n^{1/2})\rfloor)^2)$ ; $m= (\lfloor(m^{1/3})\rfloor)^3+(m (\lfloor(m^{1/3})\rfloor)^3)$ where $Rest_{2,n}=(n (\lfloor(n^{1/2})\rfloor)^2)$ $Rest_{3,m}=(m (\lfloor(m^{1/3})\rfloor)^3)$ So you have: $1= 1^2+0$ ; $1= 1^3+0$ ... $1000= 31^2+39$ ; $1000= 10^3+0$ Than Sum: $MagicSum=Rest_{2,n}+ Rest_{3,m}$ and plot $n, MagicSum$ Now do the same using $n,m=1,2,3,5,7,11,13,17...... \pi_{1000}$ Put on a graph and jump on the chair... Ciao Stefano Last edited by complicatemodulus; August 16th, 2017 at 05:18 AM. 
August 16th, 2017, 08:08 AM  #25 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,087 Thanks: 700 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Wow! If there really is a formula, that's AMAZING That means that the evaluation of the nth prime is actually a Pproblem... the dream of proving if $\displaystyle P = NP$ is a bit closer! It's a pity that there's a "mod" in there that makes it computationally inefficient. If it were computationally efficient, then existing encryption algorithms based on large prime numbers would become obsolete. Last edited by Benit13; August 16th, 2017 at 08:18 AM. 
August 16th, 2017, 08:24 AM  #26  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 119 Thanks: 38 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
 
August 16th, 2017, 08:30 AM  #27  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,087 Thanks: 700 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
https://en.wikipedia.org/wiki/Primality_test Interesting stuff!  
August 16th, 2017, 08:48 AM  #28  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,087 Thanks: 700 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
By using "deterministic" I was referring to Pproblems in complexity theory. Pproblems are solved by algorithms that take some set of input parameters, perform a known set of finite operations on them and then obtain a set of output parameters without doing any guesswork. A computer algorithm that implements the solutions to Pproblems has no steps that perform "if" conditions of any kind. Consequently, it should be possible to determine precisely how many steps the algorithm is going to perform without actually performing any of the steps. NPproblems are nondeterministic in that the number of steps is unknown, even for a specific scenario under investigation (it doesn't mean that it can't be estimated or that the number of steps is unconstrained... just that the actual number of steps is unknown until the algorithm is actually initiated). They usually require some form of estimation or guesswork followed by conditions that determine improvements. Classic examples of NPproblems are the bucketfilling problem and travelling Salesman problem. All (afaik...) numerical methods for approximating solutions based on some form of convergence criterion are NP problems. Techniques for solving optimization problems can also be in NP (e.g. the genetic algorithm), but not all of them (e.g. differentiating a quadratic and setting it to zero to get a stationary point is in P). Basic algorithms that implement the sieve of Eratosthenes are NP algorithms because the number of steps made is known only when you actually perform the algorithm... it requires constructing the sequence of primes by iterating over all numbers less than the target number and then deciding whether that number should be in the sequence or not by testing for primality which, if iterating over a set of divisors, is an NPproblem. However... there are tests for primality in P and algorithms now exist for evaluating the primes which is in P... that's amazing! Last edited by Benit13; August 16th, 2017 at 09:24 AM.  
August 16th, 2017, 11:53 PM  #29 
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24 
Here the interesting fact: If primes was "randomly" placed the Magic Sum will have an high probablility to have a fully different behaviour respect to the complete distribution of all the Integers. Here the Magic Sum for All the Integers (1 to 1000) Here the Magic Sum for PRIMES JUST (1 to $\pi_{1000}$) The reason can be that, as we know, the distribution of the Primes is "Quasi" Smooth Rising distribution, but the Math truth is the unknown very deep point to be investigated. (of course some easy concerning on the Class of the Rest can be done, pls see yellow ones) MORE: Forgotting the Primes If for example you'll plot R2R4 (for all Intgers), so the Magic Sum for Square and Fourth power (^4), or (R2R6) (for all Intgers), you'll see that the distribution is more "Regular". When some post ago I talk of "Turbolence", or Interferences in Physics, I'm referring to this Math phenomenon. We all imagine Physics phenomenon are consequances of continous variables, but in case of a scalar variable (and I suspect are ALL scalar), we have to take count of this effect. So while is very easy to predict smooth waves summations, it is not soo easy to predict all the effects of this kind of scalar summation. One example is the electron's jump: till it do not receive/loose enough energy it continue to lay on the same orbit (so in the example associated with an Integer Root value), but continuing to push/suck energy, the Delta Energy is accumulated in another "inthernal" variable, here represented as the Rest. When the Rest, or the Sum of the Rests Rise to an high enough level, the Two Hand Clock take the Rest to Zero, and the Integer Hand make one step (foreward / backward). It happens elsewhere in the universe.... Little O.T. but I think very very interesting.... I hope... Ciao Stefano Last edited by complicatemodulus; August 16th, 2017 at 11:57 PM. 
August 17th, 2017, 02:13 AM  #30 
Banned Camp Joined: Dec 2012 Posts: 1,028 Thanks: 24 
Edit: Sorry here some corrections to the formula to start the counting from n=1: The Number of Primes $Pi(x)$ within a certain $P$: $$ Pi(x) = \left( \sum_{n = 1}^{\ P} \lfloor([((n1)!/n)\lfloor((n1)!/n)\rfloor)+1/3] \rfloor \right) $$ Asap the ones to the one to computate the next prime. Last edited by complicatemodulus; August 17th, 2017 at 02:52 AM. 

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