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

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158 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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47 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 $\...
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2answers
363 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 \...
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1answer
128 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 $m$...
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1answer
194 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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4answers
471 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/...
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1answer
92 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 ...
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1answer
77 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 ...
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709 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 $...
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1answer
183 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
296 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. ...
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2answers
512 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 ...
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52 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 ...
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0answers
178 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 ...
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0answers
85 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 ...
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1answer
143 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
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1answer
166 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 ...
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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) = ...
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118 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 ...
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161 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 ...
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1answer
350 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 ...
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4answers
366 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 ...
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257 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 ...
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2answers
522 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 ...
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1answer
99 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_{...
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1answer
585 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 (...
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1answer
501 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)...
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113 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)\...
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1answer
516 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&...
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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 ...
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73 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(...
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1answer
384 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 \...
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2answers
156 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 $H/H^+$ ...
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1answer
112 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. ...
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1answer
231 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 ...
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1answer
170 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-...
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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 ...
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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 ...
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692 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 $37 \, 156 \, 667$ ...
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306 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 $n \in \mathbb{N}^*$. ...
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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
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1answer
196 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-...
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2answers
545 views

GRH and the rank of elliptic curves

I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up ...
3
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1answer
209 views

Norm of a Vector in a Number Field (or Order in a Number Field)

I am looking for a measurement, which gives a length of a vector in a number Field? Is there any way or definition for that. For the Maximal order, What if, I tried to define a map from Maximal order ...
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205 views

Hercules and the Hydra with time constraints

The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
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1answer
419 views

Seeking references for finding primes infinitely often

I've been pondering this weakened version of the finding primes problem for a while: Is there an algorithm which given $k$ outputs a prime $p > 2^k$ in time $F(\log_2(p))$? This differs from ...
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90 views

Finding short linear combinations in abelian groups

Let $M$ be a finitely generated abelian group. Assume we are given a presentation of $M$, that is \begin{equation*} M = \frac{\bigoplus_{i=1}^r \mathbf{Z}g_i}{\sum_{j=1}^s \mathbf{Z} r_j} \end{...
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0answers
236 views

Parity of number of primes

In https://arxiv.org/abs/1009.3956 is it shown there is a $c>0$ such that $\pi(x)\bmod2$ can be computed in $o(x^{\frac12})$ time (more precisely number of primes $\bmod 2$ for an interval of ...
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0answers
65 views

Transformation or correspondence between language and real number

As we know, formal language can be regarded as a set of strings of alphabet, and real number can be regarded as sequence generated by set of integers, for example, denominators of the simple continued ...
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0answers
78 views

Any proved connection between Roth theorem and hartmanis stearns conjecture?

Roth theorem classifies numbers into two classes, one is rational and transcendental, another is irrational algebraic numbers, by the number of solutions to the inequality (finite or infinite), and ...