Questions on testing and computing prime numbers.

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1
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0answers
38 views

Internals of Prime function [duplicate]

Does anyone knows how exactly Prime works ? For example how Prime[1000000000] is calculated ? The only information I found was ...
6
votes
2answers
370 views

Memory management and speed for Fast GCD

Let's say that we have some $300\,\text{K}$ digits (arbitrary function) and want to trial factor with $100{,}000{,}000$ first prime numbers. ...
1
vote
1answer
312 views

How to represent very long integers with at-most-8-digit(base-10) short integers with * and +

I need an algorithm in Mathematica, which can represent any very long integer, prime or not, into the the product operation "*" and sum "+" forms by short integers (not more than 8 decimal digits). ...
8
votes
4answers
358 views

Generate PrimePower counting function

Is there a way to generate a counting function for prime powers - i.e. to create a similar function to PrimePi, but including prime powers. The following will, of ...
2
votes
0answers
124 views

Double sum over primes [duplicate]

Can anyone tell me the value of the sum $$\sum_{p\in \mathcal{P}}\sum_{n=1}^{\infty}\frac{\log (p^n)}{2}\left[\,p^{-n}-\psi\left(\frac{p^n+2}{2}\right)+\psi\left(\frac{p^n+1}{2}\right)\right].$$ $\...
12
votes
5answers
809 views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
12
votes
5answers
2k views

How can I simulate this animation of checking for prime numbers?

I can only implement a very simple one, I want to make my code look like that. Any better ideas? ...
13
votes
3answers
595 views

Next highly composite number?

R language has this function 'nextn' (link) which computes the next highly composite number greater than a given one, which is used to find the optimal padding size for the subsequent FFT operation. ...
4
votes
1answer
526 views

How does Mathematica calculate the nth prime?

When I enter Prime[2000000000000], the two-trillionth prime, Mathematica gives 61427839512211 for the answer after several ...
5
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3answers
297 views

How could I implement the equivalent of NextPrime

I would like to know what an implementation of the function NextPrime would look like if it were implemented in Mathematica's core language.
9
votes
3answers
604 views

Fast Sieve Implementation

I'm going to write a code to compete in performance with NewPGen. The goal is to sieve numbers that have small factors. Mine is very slow in compare with NewPGen. Here's my code: ...
15
votes
2answers
495 views

Why does iterating Prime in reverse order require much more time?

Say I would like to display the $10$ greatest primes that are less than $10^5$. I could do the following: ...
13
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2answers
452 views

Is there a way to use functions like Prime[n] within Solve[]?

I'm trying to see if a number can be written as the sum of two prime numbers. Ideally, I would like to use Solve[ Prime[n] + Prime[m] == 100, {n, m}] But that ...
5
votes
1answer
520 views

Time approximation of decrypting RSA algorithm

I've written a function that encrypts a text using the RSA algorithm. It then decrypts it using prime factorization, and takes the time it took to decrypt it and puts it in a vector together with the ...
8
votes
1answer
220 views

Testing for primality in quadratic rings?

Testing for primality in $\mathbb{Z}[\sqrt{-1}]$ in Mathematica is easy: PrimeQ[n, GaussianIntegers -> True] But how can I test for primality in, say, $\...
1
vote
1answer
506 views

Equivalent of Python's “all” function in Mathematica

The Python function def isPrime(n): return all(n % i for i in xrange(2, n)) checks if a number is a prime number by using ...
40
votes
10answers
2k views

Happy 2K prime question

This being the Q number 2K in the site, and this being the day we got the confirmation of mathematica.se graduating soon, I think a celebration question is in order. So... What is the fastest way to ...
52
votes
2answers
4k views

What is so special about Prime?

When we try to evaluate Prime on big numbers (e.g. 10^13) we encounter the following issue: ...
20
votes
4answers
909 views

Why does Mathematica claim there is no even prime?

I wonder if this is a bug, or if I'm misunderstanding something: Exists[n, EvenQ[n] && PrimeQ[n]] // Resolve (* ==> False *) So if I interpret this ...