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Questions in number theory

12
votes
1answer
150 views

Base-k representations of the co-domain of a polynomial - is it context-free?

In chapter 4 of Jeffrey Shallit's A Second Course in Automata Theory the following problem is listed as open: Let $p(n)$ be a polynomial with rational coefficients such that $p(n) \in \mathbb{N}$ for ...
4
votes
1answer
99 views

Complexity of counting integer roots of multivariate polynomials in a polyhedron?

Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total ...
0
votes
0answers
51 views

Modular and multiplication operations

Does modular operation reduce to integer multiplication integer multiplication reduce to modular operation in fully linear time and/or in logarithmic time on linear number of processors?
14
votes
1answer
1k views

Deciding whether an interval contains a prime number

What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
2
votes
0answers
176 views

Primality in $NC$ hierarchy?

AKS primality testing solves whether a given integer is prime in $P$. AKS algorithm is following: Input: integer n > 1. Check if $n$ is a perfect power: if $n = a^b$ for integers $a > 1$ and $b &...
5
votes
0answers
132 views

How hard is it to generate a set of relatively prime numbers between two given bounds?

Informal Question How hard is it to generate a set of relatively prime numbers between two given bounds? Decision Problem Given $a$, $b$, and $k \in \mathbb{N}$. Does there exist a set $S \...
2
votes
1answer
115 views

Is there any time efficient way of achieving the result of FKS hashing lemma?

FKS hashing lemma states. Given a set of $n-$bit numbers $\{x_1,x_2,\dots,x_k\}$ there exist a prime $p$ of $O(\log n + \log k)$-bit such that $x_i$ mod $p \neq $ $x_j$ mod $p$ if $x_i \neq x_j$...
0
votes
1answer
69 views

Does this pairwise independent random process have expected max load $\sqrt{n}$?

This is an extension to the question about balls into bins: Example of pairwise independent random process with expected max load $\sqrt{n}$ . There the following question is asked and answered in ...
1
vote
1answer
190 views

Relation between transcendental numbers and computational complexity?

Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number ...
6
votes
0answers
284 views

Is there any algorithm outputing $e$ in real time?

The Hartmanis-Stearns Conjecture says that a number computed in real time by a Turing Machine is either rational or transcendental. We know that there is some transcendental (Liouville) number that ...
9
votes
1answer
308 views

Algorithm to compute distance between powers

Given coprime $a, b$, can you quickly compute $$ \min_{x, y > 0} |a^x - b^y| $$ Here $x, y$ are integers. Obviously taking $x = y = 0$ gives an uninteresting answer; in general how close can these ...
5
votes
1answer
1k views

Is Hartmanis-Stearns conjecture settled by this article?

The paper "On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited" by Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec https://arxiv.org/abs/1601....
3
votes
0answers
44 views

Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ is irreducible what is the best technique to factor such ...
8
votes
0answers
190 views

Algorithms to generate consecutive primes

The prime number theorem, states that the "average length" of the gap between a prime $p$ and the next prime is ln(p). I am looking for (preferably deterministic efficient) an algorithm that generates ...
2
votes
1answer
152 views

Is there a fast algorithm to quickly evaluate $a^{b^c}$ mod $n$?

I need to quickly evaluate $a^{b^c} \mod n$ where $c$ is pretty big. Using the usual repeated squaring trick, this can be performed in $O(\log(b^c)) = O(c)$ time. In my problem, $c$ is huge, (say, $&...
1
vote
0answers
55 views

Computing the class number using the prime factorization of the discriminant

i was wondering if there is a way to use the prime factorization of the discriminant $d$ when computing the class number $h(d)$. E.g., assume you have an integer $n = pq$ with $p \equiv 1\pmod{4}$ and ...
11
votes
1answer
226 views

Comparing two products of lists of integers?

Suppose I have two lists of positive integers of bounded manitude, and I take the product of all elements of each list. What's the best way to determine which product is larger? Of course I can ...
21
votes
5answers
657 views

Easy problems with hard counting versions

Wikipedia provides examples of problems where the counting version is hard, whereas the decision version is easy. Some of these are counting perfect matchings, counting the number of solutions to $2$-...
3
votes
1answer
100 views

Factoring assuming smoothness of some numbers

I have came across a lot of factorization methods and most of them seem to assume smoothness of some numbers. For example When $p-1$ is smooth When $|E(\mathbb{F}_p)|$ is smooth. (Elliptic curve ...
1
vote
0answers
88 views

Counting points on curves

It is known (see "Counting curves and their projections" (free version) by von zur Gathen, Karpinski, and Shparlinski) that the problem of finding the number of $\mathbb{F}_q$-rational points on a ...
8
votes
0answers
136 views

Recognition of a primitive root

Adleman and McCurley published a paper in 1994 called "Open problems in number theoretic complexity, II" (http://ww.cstheory.com/papers/open.ps.gz) Problem 18 of this list of open problems is about ...
17
votes
0answers
263 views

Does Factoring have a Statistical Zero Knowledge Proof?

The title should be pretty self-explanatory, but to be more precise, consider the decision version of factoring, which is given input $(x,k)$, where $x$ and $k$ are binary encodings of integers, to ...
8
votes
0answers
158 views

Simplified lattices

Consider the following question: Let $N$ be some large prime number, and suppose we are given $n$ uniformly independent samples $g_i$ from $0...,N-1$. Think of $N$ as being exponentially large in $n$...
3
votes
0answers
102 views

Number theoretic problems complete for $\mathsf{RL}$

Are there number theoretic problems (such as those related to $\mathsf{gcd}$) that are in $\mathsf{RL}$? Can these also be $\mathsf{RL}$-complete problems (is there any $\mathsf{RL}$-complete ...
-2
votes
1answer
130 views

A curious statement in an old blog

In http://blog.computationalcomplexity.org/2009/08/finding-primes.html, a statement is added which reads "Oddly enough we would usually prefer a probabilistic over the deterministic method to find ...
-3
votes
1answer
269 views

Consequences of polynomial time algorithm to variant of integer factorization

Given $N,U,V\in\Bbb N$ is there $n\in[U,V]\cap\Bbb N$ such that $n|N$ is $\mathsf{NP}$-complete modulo Cramer's conjecture on prime gaps is shown in An NP-complete variant of factoring. So supposing ...
12
votes
1answer
382 views

Which complexity class does this number theory problem belong to?

'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by=c$' is $\mathsf{NP}$-complete. Which complexity class does 'Given $a,b,c\in\Bbb N$, is there $x,y\in\Bbb N$, $ax^2+by^2=c$' belong to?
18
votes
1answer
772 views

Why does Odlyzko improvement of Shor's Algorithm reduces the number of trials to $O(1)$

In his 1995 paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Peter W. Shor discusses an improvement on the order-finding part of his ...
3
votes
1answer
251 views

Two rectangles whose sum of areas is given

Given is $\mathcal{P} \in \mathbb{Z}$. Are sought two rectangles which edges have integer, positive value and sum of their areas is $\mathcal{P}$. Find this two rectangles, if you know that sum of ...
3
votes
0answers
126 views

Is there a PPAD algorithm for computing primes that sum to even numbers?

Goldbach's conjecture states that every even number greater than 2 can be expressed as the sum of 2 primes. I'm interested in this function problem: Given an even natural number n greater than 2, ...
0
votes
0answers
70 views

Finding exact value with a quotients of products of random values

Sorry for the haphazard title: really not sure what this should be called Suppose we have a set of $z$ random values $S = r_1, \dots, r_z$ drawn from $\mathbb{Z}_N$ (where $N$ is some large prime). ...
6
votes
1answer
221 views

Integer factorization using polynomial whose roots are prime factors

Let $n$ be a square-free positive integer, let $n=p_{1}p_{2}\ldots p_{k}$ be the prime factorization of $n$ into $k$ distinct primes $p_{i}$. For such $n$, define $F_{n}(x)\triangleq\prod_{i=1}^{k}(x-...
8
votes
2answers
762 views

Is square removal easier than factoring?

It seems to me that the square removal task can be reduced to the factoring task, but that there is no way to reduce factoring to square removal. Is there a way to make this "feeling" more precise, i....
6
votes
1answer
423 views

Downward self-reducibility of factorization

Is integer factorization downward self-reducible? Is anything known about this?
4
votes
1answer
133 views

Solving a system of sums-of-powers polynomials

What is the complexity of calculating the values of the integers $x_i$, where $0 \leq x_1 < x_2 < \dots < x_k < n$, given only the values $s_m = \sum_{i=1}^k x_i^m$? for $1 \leq m \leq k$? ...
2
votes
0answers
234 views

Sieve Methods for Twin Primes - How to extract algorithm from formula

I am reading Cojocaru and Murty's Introduction to Sieve Methods and their Applications. They wait until Chapter 5 to discuss the Sieve of Eratosthenes for finding primes - and their version of it is ...
2
votes
1answer
196 views

n irrational number whose digits are pseudo-random: conceptual mismatch?

Are there irrational numbers whose digits are considered pseudo-random? Both concepts seem to be mismatched as a pseudo-random number generator typically is periodic and therefore generates a ...
6
votes
0answers
106 views

Unique factorization representation and complexity

Suppose that $N = p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ with $p_i$ prime and $a_i \geq 1$. Given a representation of the factorization of $N$ and an integer $m$ (using alphabet $\Sigma = \{0,1,,\}$): ...
-1
votes
1answer
753 views

What is the most efficient algorithm to generate a sequence of prime numbers?

I know about algorithms like Sieve of Eratosthenes and Sieve of Atkin for generating prime numbers. I would like to know what is the most efficient known algorithm to generate the sequence of $k$ ...
6
votes
1answer
335 views

Complexity of factorial exponent over composite moduli

I know that computing factorial modulo a composite number has no fast algorithm and showing non-polylogarithmic lower bound in BSS model for factorial would separate P from NP in that model. Given $a\...
11
votes
3answers
1k views

Using the de Bruijn sequence to find the $\lceil\log_2 v \rceil$ of an integer $v$

Sean Anderson published bit twiddling hacks containing the Eric Cole's algorithm to find the $\lceil\log_2 v \rceil$ of an $N$-bit integer $v$ in $O(\lg(N))$ operations with multiply and lookup. The ...
3
votes
1answer
85 views

Complexity of higher order residues

Let $P$ be a prime. Given a number $a$, what is the computational complexity in establishing if $a$ is a cubic or higher order residue modulo $P$? Are there any good algorithms?
1
vote
1answer
180 views

What algorithms do you know for beltway reconstruction? [closed]

I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem? Beltway Reconstruction Problem: Assume there is a set of ...
6
votes
2answers
170 views

Difference Sets

Suppose we have a set $$P=\{p_1,p_2,...,p_K\}$$ where $$1\leq p_k\leq N , k=1,...,K \qquad \& \quad p_k \in \mathbb{N} $$ and $p_k$'s are distinct. We calculate the differences as: $$d=p_i-p_j\mod ...
3
votes
1answer
688 views

What NP-complete problems are most similar to integer factoring?

The exact complexity of factoring integers (the decision problem) is a major open question in TCS (with important implications, especially in cryptography because of the RSA algorithm), and is widely ...
19
votes
4answers
630 views

How to obtain the unknown values $a_i,b_j$ given an unordered list of $a_i-b_j\mod N$?

Can anyone help me with the following problem? I want to find some values $a_i,b_j$ (mod $N$) where $i=1,2,…,K, j=1,2,…,K $ (for example $K=6$), given a list of $K^2$ values that correspond to the ...
12
votes
2answers
1k views

What's the most efficient algorithm for Divisibility?

What is the most efficient (in time complexity) algorithm known nowadays for the Divisibity Decision Problem: given two integers, say $a$ and $b$, does $a$ divide $b$? Let it be clear that what I ask ...
13
votes
1answer
668 views

Is there a quantum NC algorithm for computing GCD?

From the comments on one of my questions on MathOverflow I get the feeling that the question regarding GCD being in $\mathsf{NC}$ vs. $\mathsf{P}$ is akin to the question regarding Integer ...
25
votes
2answers
892 views

Complexity of factoring in number fields

What is known about the computational complexity of factoring integers in general number fields? More specifically: Over the integers we represent integers via their binary expansions. What is the ...
1
vote
0answers
127 views

Computational Complexity of RESTRICTED primality testing

Input: Any number $n \in \mathbb{Z}^+$ that can be represented in the form of $n = 2^a + b,\ |b|= c $. output: YES if $n$ is prime , else NO . Now, length of binary input is $\log(a) + O(1)$ which ...