Questions tagged [nt.number-theory]
Questions in number theory
101
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Is the following equitable factoring problem $NP$-hard or in $P$?
Consider the following factoring problem:
Given an integer $r$ and another integer $N$ along with all of its $n$ number of prime factors and their corresponding multiplicities $\{p_i,e_i\}_{i=1}^n$, ...
- 12.5k
0
votes
1
answer
131
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What is a known sequence for which being constant is not provable?
My question concerns the property of being constant for computable functions ${\mathbb N}\to \{0,1\}$, within any common framework $T$ strong enough to include Heyting arithmetic (and of course not ...
- 505
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0
answers
57
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Number of composite factors as a function of the number of bits of an integer
Is there a standard formula to calculate the number of composite factors using the number of bits of an integer?
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2
answers
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Decidability of diophantine equations over {=, +, gcd}
It is well-known that polynomial diophantine equations are undecidable (Hilbert's 10th problem): that is, given a quantifier-free formula over the language $\{=, +, \cdot, 1\}$ (of equality, addition, ...
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2
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253
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Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?
Since $e$ is transcendental, the function $f:\mathbb Z_{\geq 0}^{n+1}\to \mathbb R$ is injective,
$$ f(\underset{\text{Integers}\ \geq\ 0}{\underbrace{a_0,a_1,\ldots, a_n}}) = a_0 + a_1e+a_2e^2+\cdots ...
- 1,704
9
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1
answer
468
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Formalizing the "no formula for primes" intuition
I was trying to formalize the intuition is that there is no formula for primes, and this is my best attempt:
Conjecture: There is no $O(n^2)$ expected time randomized algorithm to generate $\ge n$-...
- 3,211
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0
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37
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Is this a proof that diophantine equation solutions can't be bounded by power towers?
From this 2017 paper on upper bounds for solutions to diophantine equations:
Conjecture 1. If a system of equations S ⊆ Bn has exactly one solution
in positive integers x1, . . . , xn , then x1, ....
1
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1
answer
192
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Analytic Number theory in TCS [closed]
Are there any applications of analytic number theory in TCS?
- 31
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81
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is factoring harder than deciding if all prime factors lie in a particular residue class?
Let $n$ be a large positive integer. Suppose I want to know if all the prime factors of $n$ are congruent to, say, 3 mod 8. Is this any easier than just factoring $n$?
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1
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Is prime-counting function #P-complete?
Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
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Complexity of planted root of a system of quadratic homogeneous polynomials?
Given homogeneous degree $2$ randomly chosen polynomials $f_1,\dots,f_{m}$ in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ each with only monomials $x_iy_j$ with condition that the system $f_1=\dots=f_{m}=...
- 12.5k
14
votes
4
answers
609
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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 ...
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5
votes
1
answer
170
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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 ...
- 12.5k
14
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1
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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 ...
3
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answers
245
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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 &...
- 12.5k
5
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0
answers
142
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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 \...
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1
answer
142
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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$...
- 103
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1
answer
84
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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 ...
- 3,910
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1
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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 ...
- 1,009
6
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0
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315
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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 ...
- 1,009
10
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1
answer
341
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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 ...
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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....
- 1,009
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votes
0
answers
46
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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 ...
- 12.5k
8
votes
0
answers
284
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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 ...
- 20.5k
6
votes
2
answers
284
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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, $&...
- 261
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80
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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 ...
- 615
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716
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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 ...
- 213
21
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5
answers
792
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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 $2$-...
- 12.5k
2
votes
1
answer
112
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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 ...
- 243
1
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0
answers
89
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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 ...
- 1,519
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161
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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 ...
- 521
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347
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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 ...
- 3,698
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0
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172
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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 $n$...
- 1,214
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0
answers
109
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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 ...
user34945
-2
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1
answer
144
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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 ...
- 12.5k
-3
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1
answer
391
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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 ...
- 12.5k
12
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1
answer
420
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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?
user34945
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1
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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
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1
answer
279
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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 ...
- 149
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0
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136
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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, ...
- 5,070
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votes
0
answers
72
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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).
...
- 139
6
votes
1
answer
274
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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-...
- 717
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votes
2
answers
798
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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....
- 3,013
8
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1
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727
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Downward self-reducibility of factorization
Is integer factorization downward self-reducible? Is anything known about this?
- 717
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1
answer
140
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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 $1 \leq m \leq k$?
...
- 11k
2
votes
0
answers
262
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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 ...
- 578
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votes
1
answer
272
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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 ...
- 29
6
votes
0
answers
107
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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 $\Sigma = \{0,1,,\}$):
...
- 22.3k
-1
votes
1
answer
1k
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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 $k$ ...
- 89
6
votes
1
answer
339
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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\...
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