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Questions tagged [computational-number-theory]

Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.

1
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1answer
97 views

Factoring with partial information on gaps

If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...
3
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1answer
143 views

Number of lattice points on spheres with center not at the origin

Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\...
4
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0answers
152 views

Is there some computational evidence of the $pq$ analog of Serre's conjecture?

The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
-1
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1answer
123 views

summation of Euler totient function

Let $\phi(n)$ be the Euler totient function and let $2\leq k\in\mathbb{N}$. For $m\in\mathbb{N}$, are there any known results, upper bounds (tighter than just removing the coprimality) or ...
3
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1answer
86 views

Is coprimality in $NC$?

Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
2
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0answers
86 views

Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
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0answers
35 views

Computing with a vector subspace equipped with a prescribed basis over finite fields

Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]...
0
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1answer
253 views

Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...
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0answers
53 views

Can a rational symmetric matrix $A$ be diagonalized as $A = P \Lambda S$ for some $P, S $ in the general linear group?

Let $A$ be a $3 \times 3$ symmetric matrix with rational entries. Does there exists a unique pair of matrices $P, S \in \text{GL}_3(\mathbb{Z} )$ depending on $A$ such that $A = P \Lambda S$, where $\...
11
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2answers
366 views

Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...
2
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1answer
137 views

Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$

Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences. For a positive integer $m$...
4
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1answer
219 views

Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?

The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
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0answers
21 views

Uncertainty in semiprime factors

What is maximum $p\in(0,\frac12)$ such that if we know $2p\in(0,1)$ fraction of bits of $PQ$ with $P,Q$ primes it is possible to identify $p$ fraction of bits in each of $P,Q$ with certainty in ...
17
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4answers
485 views

In which cyclic cubic number fields does there exist this type of unit?

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$. Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...
1
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1answer
113 views

Taking pairwise coprime integers from prescribed sets

Given $m$ and $n$ in $\mathbb Z_{>0}$ what is the computational complexity of picking $n$ pairwise coprime integers each of $m$ bits when they exist? Given $m$ and $n$ in $\mathbb Z_{>0}$ what ...
1
vote
1answer
81 views

How many iterations the best biprime factoring method has to factor a number [closed]

I'm researching method of biprime number factoring. I have a biprime number 1012322327 * 1115382761 (19 decimal digits= 1129126872111204847). I'd like to know how many iterations (or trials) the best ...
10
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0answers
723 views

Euler's totient function and Riemann hypothesis

I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...
12
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1answer
187 views

Factoring polynomials over the abelian closure of the rationals

What algorithms are known to perform the following task? Input: a univariate polynomial over the rationals $f \in \mathbb{Q}[t]$. Output: the factorization of $f$ into irreducible factors over the ...
11
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1answer
304 views

$p$-adic sums of $p$ terms

My question is inspired by this riddle: Let $p \geq 5$ be prime, and let $$ 1 + \frac 1 2 + \frac 1 3 + \dots + \frac 1 {p-1} = \frac a b $$ where $a/b$ is the fraction expressed in lowest terms. ...
15
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2answers
526 views

Analogues of the Riemann zeta function that are more computationally tractable?

Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as ...
0
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0answers
56 views

Best method to compute sum of divisors of bounded evaluation of a bivariate quadratic?

Given a bivariate quadratic polynomial $g(u,v)\in\mathbb Z[u,v]$ and an integer $n$. How fast can we compute $\sum_{i=-\ell}^{\ell}\sum_{j=-\ell'}^{\ell'}\sigma_0(g(i,j))\bmod2$ where $\sigma_0$ is ...
0
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0answers
180 views

On number of squares by a quadratic

Given a bivariate degree $2$ polynomial $$m^2+(ax−by)^2−2x(ma+2br′)−2y(mb+2br)−4rr'\in\mathbb Z[x,y]$$ where $0<r,r'<\max(a,b)<rr'<m<ab$ with $\gcd(a,b)=1$ holds with max coefficient ...
0
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0answers
90 views

How to prove a Mersenne number is not pseudoprime to the base 3?

I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$ Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...
0
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1answer
146 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
4
votes
1answer
190 views

Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...
3
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1answer
73 views

Level sums, displacements: how to determine them efficiently?

Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$, $\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$. Define $$S(\delta,m) = ...
2
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0answers
122 views

lower bounding the absolute value of a determinant

In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...
3
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0answers
172 views

Factoring problem similar to $RSA$ structure that is possibly not $NP$ complete and not $coNP$ also?

Standard factoring problem $\Pi_1$ is 'Given integers $N$ and $M$ is there a factor $d\in[1,M]$ of $N$?'. This is in $NP$ since such a factor is the witness and in $coNP$ since one can check all the ...
1
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1answer
434 views

Efficient sum of squares decomposition

Sum of 4 squares decomposition is the well-known result. I'm interested only in negative/non-negative separation with focus on efficiency and large numbers. I'm looking for alternatives or extensions ...
6
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4answers
368 views

(Non)uniqueness of the common-factor graph

Let $S=\{x_1,\ldots,x_k\}$ be a set of $k$ distinct natural numbers, a subset of $\{1,\ldots,n\} = \mathbb{N}_{\le n}$. Define the common-factor graph $G(S)$ as the (undirected) graph with a node for ...
1
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0answers
258 views

approaching the border between absolute convergence and divergence of series

Let us consider absolute convergent series $\ell^{1^+}$ ordered under eventual dominance (mod finite) $<^*$. T. Bartoszynski proved that unbounded number ${\frak b}(\ell^{1^+}, <^*)$ equals ...
7
votes
2answers
531 views

How to compute Dedekind eta function efficiently?

According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic ...
1
vote
1answer
102 views

How to compute the Müller modular polynomials?

According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as $$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
18
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1answer
589 views

What computer program for automorphic forms

This question has its origins in this entertaining discussion on MO. There are many programs (CAS) and libraries that are able to handle algebraic expressions. These are both a verification tool for (...
4
votes
1answer
508 views

Conjecture on palindromic numbers

The conjecture is as follows: Let $n\in\mathbb{N}\setminus\{1\}$. Define $a(n)=2^n+1$ and the set: $$S(n) = \{ (a(n)^m+1)/2\ :\ m\in \mathbb{N}_0\}.$$ Then for all $c\in\mathbb{N}$, the number $(a(n)...
2
votes
0answers
117 views

Quick computation of a certain exponential sum

Is there a quick way to compute (somewhat accurately) for large $X$, the following exponential sum, where $\Lambda$ is the von Mangoldt function? $$\int_0^1 \bigg|\sum_{n\le X} \Lambda(n)e(n\alpha)\...
5
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1answer
535 views

Eulerian ordering of the integers modulo n

Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$. An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that: $$\forall i \le n \ \forall j&...
10
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1answer
428 views

Representing field elements in a computer

I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be ...
3
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0answers
76 views

Deterministic procedure to find irreducible polynomials

In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...
8
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1answer
387 views

simple conjecture on palindromes in base 10 [closed]

The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \...
6
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2answers
164 views

Computing the relative class group (with Galois action) of relatively large cyclotomic groups

For a cyclotomic field $K = \mathbb Q(\zeta_n)$, let $K^+$ be its maximal totally real subfield. We know that $H^+ = Cl(K^+)$ injects into $H = Cl(K)$. I am interested in computing the group $H/H^+$ ...
4
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1answer
114 views

Solving a system of linear equations over the integers

I have a matrix with integral entries $A$ and integer vector $b$, and want to determine if there is exactly one vector $x$ such that $Ax=b$. $A$ is rectangular, and I know there always is a solution. ...
4
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1answer
232 views

Combinatorial computational problem about 0-1 vectors and sampling algorithms

Let $M \in \{0,1\}^{m\times n}$, where $n\gg 1$ and $m\le n$. A procedure consisting of the following three steps is repeated $t\gg 1$ times: A row $\require{amsmath} \boldsymbol{r}$ of $M$ picked in ...
0
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1answer
172 views

Could the sequence A287326 be generalized in order to receive expansion of natural power n>3? [closed]

The sequence https://oeis.org/A287326 - is Binomial distributed triangular array, that shows us necessary items to expand perfect cube $n^3$. Summation of $n$-th row of Triangle A287326 from $0$ to $n-...
1
vote
1answer
99 views

How many integers $x$ satisfy that $x*p(x) \leq n$, where $p(x)$ means the largest prime factor of $x$?

I guess that the number of integers $x$ which satisfy the condition $x*p(x) \leq n$ is $O(n^{2/3})$ or $O(n^{3/4} / \ln n$), but I cannot prove it. I just write a program to count the number. The ...
0
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0answers
75 views

Shape Basis Generator of an Ideal

Let $R=K[x_1,...,x_n]$, for a field $K$, and $I$ is radical zero dimensional ideal of $R$. We define $s=\Sigma_{i=1}^{n}a_ix_i$, with $c_i\in K$, a shape basis generator of $I$, if $(s+I)$ generates ...
10
votes
2answers
695 views

Question on the 50th (known) Mersenne prime number

In a footnote to the list of known Mersenne prime numbers which can be found here, we read that the "ranking" therein is a provisional one since not all possible exponents between $37 \, 156 \, 667$ ...
6
votes
2answers
308 views

Is the nth-power-sum graph connected?

This post was inspired by the Square-Sum Problem presented in Numberphile by Matt Parker. He asked about Hamiltonianness for $n=2$, and we ask about connectedness for all $n \in \mathbb{N}^*$. ...
2
votes
1answer
207 views

A problem of divisibility

I came across the following problem. Find two integers, $u_{n}$ and $v_{n}$, such that $$a_{n}=4u_{n}v_{n}+(6n-1)v_{n}+(6n-1)u_{n}+8n^{2}-4n$$ divides $$b_{n}=-(2n-1)u_{n}v_{n}-(2n^{2}-3n)v_{n}-(2n^...
4
votes
1answer
203 views

Fastest deterministic factoring algorithm in subexponential space?

Strassen's factoring algorithm shows that $\text{FACTORING} \in \text{DTIME}(N^{\frac{1}{4}+o(1)})$, but if I'm not mistaken in my analysis it also uses a similar amount of space. By making a trade-...