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Questions tagged [primes]

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1answer
39 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...,\...
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1answer
28 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 $q$ ...
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0answers
64 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
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1answer
59 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
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1answer
82 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
130 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
30 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
66 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
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2answers
89 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
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1answer
397 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 ...
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2answers
73 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
1k views

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?
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2answers
152 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
30 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 ...
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4answers
190 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 ...
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2answers
121 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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2answers
171 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 ...
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1answer
154 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
49 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
86 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 ...
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1answer
110 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 ...
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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 ...
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1answer
55 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: $k_1 * a_1 + b_1 = x$ $k_2 * a_2 + b_2 = x$ $\dots$ $k_n * a_n + b_n = x$ ...
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0answers
56 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 $r$...
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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 ...
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0answers
489 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 ...
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2answers
115 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
74 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 ...
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3answers
413 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
317 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 ...
1
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1answer
150 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 ...
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1answer
101 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 ...
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2answers
619 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 = ...
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1answer
697 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
519 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 ...
1
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1answer
226 views

I have a problem understanding a solution for number theory problem from SPOJ [closed]

Problem statement can be found here or down below. The solution which I'm trying to understand can be found here or down below. Problem Statement. Peter wants to generate some prime numbers for his ...
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0answers
51 views

Reference Request: Factorization of PseudoPrimes?

Is there any literature/survey/papers/books regarding the factorization of Strong PseudoPrimes (wrt. to a given base). I am aware of the fact that weak Pseudo Primes can be factorized in Polynomial ...
1
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1answer
85 views

Is the prime factorization problem not an instance of the change making problem?

When using as the set of coins all logarithms of the prime numbers or numbers in general, and when using the logarithm of the number to be factored. The problem is just finding the logarithms that can ...
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3answers
2k views

Complexity of finding factors of a number

I have come up with two simple methods for finding all the factors of a number $n$. The first is trial division: For every integer up to $\sqrt{n}$, try to divide by $d$, and if the remainder is $0$ ...
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1answer
661 views

How to compute Jacobi symbol efficiently?

How do I compute the Jacobi symbol $(N|A)$ efficiently? In particular, for every odd $N, A$, define the Jacobi symbol $(A|N)$ as $\prod_i Q_{p_i}(A)$ where $p_1, \dots , p_k$ are all the (not ...
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2answers
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developing a Turing Machine that checks for powers of 2

I want to write a Turing machine which checks for unary powers of 2 but without the use 0s, only accepting as input a series of 1s and dashes. I do not know of a sequence of states which would allow ...
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2answers
1k views

Why is factoring large integers considered difficult?

I read somewhere that the most efficient algorithm found can compute the factors in $O(\exp((64/9 \cdot b)^{1/3} \cdot (\log b)^{2/3})$ time, but the code I wrote is $O(n)$ or possibly $O(n \log n)$ ...
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1answer
710 views

What is the average-case complexity of trial division?

The trial division algorithm for checking if a number $N$ is prime works by trying to divide $N$ by all integers in the range 2, 3, ..., $\lfloor \sqrt{n} \rfloor$. If any of them cleanly divide $N$, ...
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2answers
2k views

What is the time complexity of checking if a number is prime?

Could some one please explain how to get the time complexity of checking if a number is prime? I'm really confused as to if it is $O(\sqrt{n})$ or $O(n^2)$. I iterate from $i=2$ to $\sqrt{n}$ and ...
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1answer
725 views

Algorithm for generating coprime number sequences?

Does anyone know of an algorithm to generate a set of numbers of size $N$ which are all co-prime to eachother? Ideally I'm looking for something that has random access abilities so i could ask for ...
3
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1answer
151 views

Worst-case prime sieve

The Sieve of Eratosthenes bothers me because you have to specify an upper bound before you begin the algorithm. Is there a prime sieve that doesn't require this? More Formally: Is it possible to ...
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1answer
985 views

Is determining if there is a prime in an interval known to be in P or NP-complete?

I saw from this post on stackoverflow that there are some relatively fast algorithms for sieving an interval of numbers to see if there is a prime in that interval. However, does this mean that the ...
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5answers
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Data compression using prime numbers

I have recently stumbled upon the following interesting article which claims to efficiently compress random data sets by always more than 50%, regardless of the type and format of the data. Basically ...
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3answers
197 views

Complexity-theoretic difficult of checking the value of $\pi(x)$?

The prime-counting function, demoted $\pi(x)$, is defined as the number of prime numbers less than or equal to $x$. We can define a decision problem from $\pi(x)$ as follows: Given two numbers $x$ ...