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I've passed by this article:
http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/
and this paper:
http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf
They say that there is a formula that when I give it (n) then it returns nth prime number. Where in other articles they say that no formula discovered so far that does such thing.
If the formula exists indeed, then why from time to time they discover the largest prime number known ever, It would be very simple using the formula to find a larger one.
I just want to ensure that such formula exists or not.
Conceptually it is very simple to test that a given number n is a prime number: just check for all smaller numbers 'm' (larger than 1) whether 'm' divides 'n' without remainder. If
such an 'm' exists 'n' is not a prime number.
Then, to find the k-th prime number you just iterate this procedure until you found the k-th number which is a prime. So yes, such a formula exists.
But, executing the above procedure is very inefficient. So even having this formula (and in real cases you would use more intelligent variants), it can take literally ages before you get an answer. And that is why more efficient variants and tricks are used to find large prime numbers.
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An input sequence is given. Each stage of the iteration finds another sequence by calculating difference between n-i and n-i-1 number. We continue the process and at the end of the last iteration (iteration: n-1) we find only 1 number. What is the mathematical formulation for finding the last number as shown in the image?
Basically, the mathematical formulation is finding the n-1'th derivative of the degree-n-1 polynomial passing through all points (i,arr[i]). That derivative is guaranteed to be a constant. This is equivalent to the coefficient of the term with exponent n-1, divided by (n-1)!.
This method is a special case of what is known as Neville's Algorithm.
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In formal, does there exist such that for all ?
No, $\pi$ and thus $\pi/2$ are irrational, thus the (additive) equivalence classes of the integers modulo $2\pi$ are dense in $\Bbb R$ and thus approach infinitesimally, but never reach $\pi/2$.
The fundamental fact is that for any given number x the set of numbers {mx+n : m,n integer} is either
an arithmetic sequence {mr : r integer} which implies and is equivalent to x as a multiple of r being rational, or
dense in the real numbers, which by the first case happens for all irrational x.
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Well is it possible to find private key x for this equation y=g^x mod p of course big integers if you have p ,g, y, q?.
What method can be used if there is method to find it out? ..........Note:These are big Integers
This is called the discrete logarithm problem. You seem to be interested in the prime field special case of this problem.
For properly chosen fields with sufficiently large p this is infeasible. I expect this to be reasonably cheap (100$ or so) for 512 bit p and extremely expensive at 1024 bit p. Going beyond that it quicky becomes infeasible even for state level adversaries.
For some fields it's much cheaper. For example solving DL in binary fields (not prime fields as in your example) produced quite a few recent papers. For example Discrete logarithm in GF(2^809) with FFS and On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in F_2^1971.
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My question deals with the fracdiff.sim function in R (in the fracdiff package) for which the help document, just like for arima.sim, is not really clear concerning initial values.
It's ok that stationary processes do not depend on their initial values when time grows, but my aim is to see in my simulations the return of my long memory process (fitted with arfima) to its mean.
Therefore, I need to input at least the p final values of my in-sample process (and eventually q innovations) if it is ARFIMA(p,d,q). In other words, I would like to set the burn-in period's length to 0 and give starting values instead.
Nevertheless, I'm currently not able to do this. I know that fracdiff.sim makes it possible for the user to chose the length of a burning period (which leads to the stationnary behavior) and the mean of the simulated process (it is simulated and then translated to make the means match). There is also a condition: the length of the burn-in period must be >= p+q. What I suppose is that there is something to do with the innov argument but I'm really not sure.
This idea is inspired by the arima.sim function which has a start.innov argument. However, even if my aim was to simulate an ARMA(p,q), I'm not sure of the exact use of this argument (the help is quite poor) : must we input only q innovations ? put with them the p last values of the in-sample process ? In which order ?
To sum up, I want to simulate ARFIMA processes starting from a specific value and having a specific mean in order to see the return to the mean and not only the long term behavior. I fund beginnings of solutions for arima.sim on the internet but nobody clearly answered, and if the solution uses start.innov, how to solve the problem for ARFIMA processes (fracdiff.sim doesn't have the start.innov argument) ?
Hopping I have been clear enough,
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Which is more random?
rand()
OR
rand() + rand()
OR
rand() * rand()
Just how can one determine this? I mean. This is really puzzling me! One feels that they may all be equally random, but how can one be absolutely sure?!
Anyone?
The concept of being "more random" doesn't really make sense. Your three methods give different distributions of random numbers. I can illustrate this in Matlab. First define a function f that, when called, gives you an array of 10,000 random numbers:
f = #() rand(10000,1);
Now look at the distribution of your three methods.
Your first method, hist(f()) gives a uniform distribution:
Your second method hist(f() + f()) gives a distribution which is peaked in the centre:
Your third method hist(f() .* f()) gives a distribution where numbers close to zero are more likely:
As to amount of entropy, I guess, is comparable.
If you need more entropy (randomness) than you have currently have, use cryptographically strong random generators.
Why they are comparable --- because if attacker could guess next pseudorandom value returned by
rand()
it would not be significally harder for him to guess next
rand()*rand()
Nevertheless argument about different distributions is important and valid!