Questions tagged [primes]
The primes tag has no usage guidance.
85
questions
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Fastest algorithm for finding the number of primes in a range
Is there an algorithm for finding the number of primes in a given range $[N, M)$ that works in time linear to $M-N$? For context, $N$ and $M$ can go up to $10^{10}$, but the distance between N and M ...
1
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0answers
85 views
how did the authors of the AKS-Paper come up with the upper bound for r? and what does the multiplicative order have to do with anything?
I have been recently reading the paper "PRIMES is in P", but unfortunately a lot the steps were skipped, which led to confusion. My main problem is with the upper bound on r which was not explained at ...
-1
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0answers
25 views
Difference between NP Intermediate and NP Complete [duplicate]
Assuming P ≠ NP
How do you determine whether a problem belongs in NP Intermediate or NP Complete?
Why does integer factorization belong in NP Intermediate, but the knapsack problem belongs in NP ...
0
votes
3answers
44 views
Primality testing algorithm
Say, I would like to check a hypothesis concerning primes. Something like "there exists a prime between $n$ and $2n$ for every choice of $n$". I would like to run a code in MATLAB for choices of $n$ ...
3
votes
1answer
53 views
Efficiently prime factorising an integer with an oracle
Suppose you have a program one_factor(N) that, given an n-digit binary number, N, returns ...
0
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1answer
33 views
What computational model supports arbitrarily sized integers?
I want to do some research, but I don't think it's important the number of bits it takes to represent the integer input and arithmetic on the abstract machine. So what is the model that addresses ...
0
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1answer
62 views
How much time (in hours) will it take to check if the number with 20 binary digits is the prime number?
How much time (in hours) will it take to check if the number with 20 binary digits is the prime number, in problem it's mentioned that for number with 10 digits it took 1 hour it's also said that the ...
3
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1answer
46 views
Prime checking and factorization with just bit cheking
I've read about some methods of prime factorization like here and here. However, I'm wondering: what can we do in prime factorization with just some bit manipulation and without other variables/...
1
vote
1answer
55 views
Is SEMIPRIME in P?
The title says it all: is there a deterministic polynomial time algorithm that tests for semiprimality? (A number $N$ is a semiprime if it is the product of two primes.) I don't understand the ''...
1
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2answers
58 views
Pumping Lemma with Prime Number [closed]
$\text {Could someone please help me with this proof: }$
$L:=\left\{a^{n} d^{m} b^{k} | n, m, k \in \mathbb{N} \wedge m \text { is a prime number}\right\}$
$\text {Maybe we can say, that } w=a^{n}d^{...
3
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1answer
22 views
Solve SUBSET SUM for Reciprocals of Primes
Let $p_1, ..., p_n$ distinct prime numbers with $P = \prod_{i=1}^{n}{p_i}$ and $A=(a_1, ..., a_n)$ with $a_i = P/p_i$.
Problem
Show the SUBSET SUM problem $(A, \alpha)$ can be solved in polynomial (...
0
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1answer
259 views
Time complexity of Sieve of Eratosthenes [closed]
Wikipedia states that the Sieve of Eratosthenes runs in time $O(n\log\log n)$. Why is that so?
1
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1answer
78 views
Is binomial(n, k)=0 (mod n) when gcd(n, k)=1? When gcd(n, k)>1?
I was programming the AKS primality test, and I have two questions.
Is it correct that if gcd(n, k)=1, then binomial(n, k)=0 (mod n)?
Is it correct that if gcd(n, k)>1, then binomial(n, k)>0 (mod n)?...
0
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0answers
22 views
Is the Time Complexity of Trial Division Exponential? [duplicate]
I know that there is already a similar question asked on here, but after reading the wiki page on trial division, I am confused, and the other answer doesn't help. The wiki page states that when doing ...
1
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1answer
102 views
What is the complexity of this prime trial division algorithms?
I have two algorithms. What are their time complexities?
The first algorithm checks the modulo of all odd integers from $3,5,...\sqrt{n}$.
The second algorithm generates a list of prime from $2,3...,\...
0
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1answer
32 views
Coprimes satisfying a pair
We know that number of coprimes less than a number can be found using Euler's totient function.
But if there are two numbers $p$ and $q$ and we need to find number of numbers less than $q$ ...
2
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0answers
135 views
Min no.of operations required to convert an array to which it should contain elements of equal frequency
I have come across this tricky problem. An array of N elements should be converted to another array within k operations such ...
2
votes
1answer
188 views
Complexity of brute force primality test in the number of digits
I'm wondering how to express the complexity of a brute force primality testing algorithm in the number of digits the number under test has. The brute force algorithm just checks whether $n$ is prime ...
2
votes
1answer
88 views
Listing all prime numbers less than an integer N
I am trying to solve this problem listing all prime numbers less than an integer but using the smallest amount of memory.
Is it possible to solve this problem using a smaller amount of memory?
<...
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2answers
179 views
Randomly Choosing a N-Bit Prime
I've been studying some number theory, and I came across this problem:
Lagrange’s prime number theorem states that as N increases, the number of primes less than $N$ is $Θ(N/ log(N))$.
...
1
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1answer
59 views
The time complexity of the Wikipedia version of Pollards $(p-1)$ algorithm
I am trying to understand the runtime of Pollard's $(p-1)$-algorithm as presented on Wikipedia. There the author writes that it takes $\mathcal{O}(B\log B\log^2n)$ time, but I do not see why.
Here ...
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3answers
76 views
Determining number of elements divisible by at least one prime of set
I have an array ($|A|\leq 10^6$) of numbers ($A_i\leq10^6$) and a set of prime numbers. I have to find the count of the elements in the array that are divisible by at least one of the numbers in the ...
2
votes
2answers
129 views
Logarithmic run time for calculating prime numbers?
Here's the function I'm currently analyzing. I know it's not the most optimal but I'm not understanding the $\theta()$ of this algorithm. I've been told that it's not actually $\theta(n)$ but instead ...
3
votes
1answer
1k views
Why is the complexity of factorial a function of n?
When we compute the complexity of calculating factorial of a number $n$ why is it in terms of $n$ instead of the number of the number of bits occupied by the number of bits occupied by $n$ (like we do ...
0
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2answers
142 views
Is finding all primes less than n, doable in polynomial time?
Bear in mind I'm almost a complete noob at complexity theory.
I was reading about how AKS Primality shows that numbers of size n can be shown to be prime or composite in polynomial time. Given that, ...
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3answers
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Algorithm for checking if a list of integers is pairwise coprime
Are thre any efficient algorithms for checking if a list of integers is pairwise coprime, or would a more general algorithm be the best option available?
3
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2answers
182 views
Dijkstra's Notes on Structured Programming - concerning the program to compute first 1000 prime numbers
In his "Notes on Structured Programming" essay, E. W. Dijkstra gives an example of a program that computes the first 1000 primes (Section 9. "A First Example of Step-Wise Program Composition").
The ...
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1answer
35 views
Show a sequence of distinct Primes number is O(log n)
Suppose I have a sequence:
$$n = \prod_{i=1}^{r(n)} p_i^{d_i}$$
for some primes $p_1 < p_2 < \dots < p_{r(n)}$, and each $d_i \geq 1$ an integer. The function $r(n)$ denotes the number of ...
1
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4answers
398 views
What is the big-$\Omega$ complexity of Fermat's Little Theorem?
Fermat's Little Theorem states that if an integer $n$ is prime them
$$
a^n \equiv a \pmod n \hspace{10mm} (*)
$$
for any $a \in \mathbb{N}$
My question is, is it correct to say that testing $(*)$ for ...
2
votes
2answers
132 views
What is the complexity of checking whether an integer $n \geq 2$ is expressible in the form $a^b$ where $a, b \in \mathbb{N}$?
I am currently studying the paper Primes is in P and have a question regarding 5 section of this paper. Line 1 of the algorithm (on page 3) requires the following operation to be performed
...
2
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3answers
399 views
Primality testing: Why is dividing a number $n$ by every integer between 2 and $\sqrt{n}$ an inefficient test?
In the paper "PRIMES is in P" the following is said (page 1):
Let PRIMES denote the set of all prime numbers. The definition of prime numbers already gives a way of determining if a number $n$ is ...
4
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1answer
160 views
How to compute all primes between upto $n$ in time $O(n)$ time?
Suppose that I want to compute all the prime numbers between 2 and $n$. The natural way or most obvious way to do so is given below. Let $A$ is an array contain the numbers from $1$ to $n$.
For $j=2$ ...
2
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1answer
50 views
Time complexity of finding an integer between $x$ and $2x$
Consider receiving as input $x\in\mathbb N$ and computing some (any) prime $p\in[x,2x]$.
What is the complexity of the above problem?
A natural way to approach this problem is to generate random ...
1
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1answer
92 views
What should I do if I think I wrote an algorithm that is _generally_ faster than Atkin Sieve? [closed]
I've been having some fun with prime numbers. A few months ago I sat down to see if I could write something that could compete with Atkin Sieve and ended up with an algorithm that, on my local tests ...
1
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1answer
121 views
Is Mersenne primality testing necessarily EXPTIME?
The computational complexity of primality testing is usually specified in relation to the bit length of the number being tested.
However, Mersenne numbers have the special property that the ...
1
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1answer
108 views
Is this a Fruitful Primality Testing scheme?
Today I had an insight into an alternative deterministic algorithm for testing the primality of a number. I want to know if this algorithm is useful, and worth pursuing. I'll describe the idea behind ...
1
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1answer
73 views
Algorithm which finds the maxmal solution that satisfies the following constraints
I have $a_1, a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ and an upper bound $U$ and $n$ linear equations of the form:
$k_1 * a_1 + b_1 = x$
$k_2 * a_2 + b_2 = x$
$\dots$
$k_n * a_n + b_n = x$
...
2
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0answers
80 views
AKS primality test- Lemma 4.3, not following
I'm reading the AKS primality test paper as it is found here. I'm confused about a statement in Lemma 4.3:
"Note that $(r, n)$ cannot be divisible by all the prime divisors of $r$ since otherwise $r$...
0
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0answers
31 views
Why some state that Primes is in NP? [duplicate]
Why some books state that Primes is a NP problem if, as a decidibility problem, it can be solved in polynomial time?
A simple example:
A number can has its primality tested by dividing it by all ...
0
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0answers
582 views
Design a Turing Machine Checking if power is prime
I have to design a Turing Machine to do the following, but I don't really know where to start with this question. Any help would be very much appreciated.
I should design a Turing Machine accepting ...
1
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2answers
129 views
minimize sum of primes with a lower bound on product
I can't quite figure out an algorithm for this:
Given some integer n, what subset of the primes (so no repeats) would yield the lowest possible sum if their product is at least n?
Example: 6 -> 2*3,
...
0
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1answer
85 views
Given a prime power, is it possible to efficiently compute the prime
Suppose that I am given as input the number $p^i$ for some prime $p$ and some positive integer $i$. I wish to find $p$ and $i$.
Is there an algorithm to do this that works in time polynomial in the ...
7
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3answers
564 views
More details about the Baillie–PSW test
It's clear that Mathematica uses the Baillie–PSW test for its PrimeQ function (which tests primality), and as I read in the Mathematica documentation, it starts with trial division, then base 2 ...
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3answers
2k views
Hash function to hash 6-digit positive integers
Let UID denote a unique identifier. UID's are represented as 6-digit positive integers.
I want to insert a collection of UID's in a hash table with $M$ buckets, where $M$ is a prime number (for ...
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2answers
516 views
Efficiently computing the smallest integer with n divisors
In order to tackle this problem I first observed that
$$\phi(p_1^{e_1} \space p_2^{e_2} \cdots \space p_k^{e_k}) = (e_1 + 1)(e_2 + 1)\cdots(e_k +1)$$
Where $\phi(m)$ is the number of (not ...
2
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1answer
173 views
Average runtime of naive primality test
The naive prime test goes something like this:
is_prime(n):
for(i=2; i<=sqrt(n); ++i):
if n mod i == 0 :
return false
return true
If $n$ is ...
1
vote
1answer
127 views
Modifications to speed up the AKS primality proving
It is clear that AKS primality proving is the newest one, but as the results show it is not the fastest one.
When I try the 9 digits long prime number it consume about 6 minutes to give you the ...
5
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2answers
767 views
Express a number as a sum of consecutive primes (How many ways?)
I found this problem on codeforces in an ACM archive.
Given a number $n$, find the number possible of ways of expressing that number as a sum of consecutive primes. Example :
Given $n = 41$:
$41 = ...
0
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1answer
988 views
Is the given language decidable for turing machine?
I am going through undecidability of TM and found this question
$L=\left \{ \left \langle M \right \rangle |M\ is\ TM \ and \ number\ of\ strings\ in\ the\ language\ \ is\ prime\right \}$
I think it ...
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3answers
663 views
Determine nth prime number in O(?)
If f(n) is the problem to determine the nth prime number, how fast can this be done, i.e.
What is the fastest known algorithm to find the nth prime number?
What are lower bounds for the time ...
