The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
86 views

parametrizing a conic in $F_p$ [closed]

Let $F_p$ be a finite field and $p\equiv 3 \pmod 4$, and $a,c$ are non-square elements in $F_p$. I want to parametrize the conic: $$cy^2=-3x^2-2ax-16a$$ ($-1$ and $3$ are non-squares in this field ...
1
vote
0answers
40 views

Complexity of $\mathsf{gcd}(a,b)\bmod N$

Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$. My query is given $N,a,b$ where $a,b$ is $n$-bits ...
7
votes
0answers
207 views

Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
4
votes
1answer
347 views

On the mixed sum of three k-th powers

Let the set $S_k=\{\pm x^k \pm y^k \pm z^k \ \vert \ x,y,z \in \mathbb{Z} \}$. Note that the signs are independently positive or negative. First of all $S_2 = \mathbb{Z}$ because (see the answers ...
5
votes
1answer
260 views

The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$ I am not really sure quite where to start here as I am ...
10
votes
2answers
467 views

Why going to number fields in number field sieve help beat quadratic sieve?

To factor an $n$ bit integer number field sieve roughly takes $$e^{c{(\ln\ln n)^{\frac23}}({\ln n})^{\frac13}}$$ time while quadratic sieve takes $$e^{c{(\ln\ln n)^{\frac12}}({\ln n})^{\frac12}}$$ ...
10
votes
2answers
485 views

Computing millions of coefficients of non self-dual modular forms

To test some conjectures made by some colleagues, I need to compute millions of coefficients of non self-dual modular forms, preferably in low weight (say 2 or 3). A form such as this. For elliptic ...
3
votes
1answer
123 views

Equivalence between Diffie Hellman and Discrete Log

For which non-trivial groups, do we know that the Diffie Hellman problem and the Discrete Log are equivalent? Is there any group for which we suspect them to be different? Could there be a finite ...
2
votes
0answers
197 views

Avoiding Chinese Remainder Theorem

Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ ...
5
votes
0answers
120 views

On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive. What other rings $\mathcal{O}$ can we use instead of ...
2
votes
0answers
80 views

Computing the density of a set of multiples

Erdős and his coauthors often wrote about problems relating to the densities of sets of multiples. I have a computational question about the same topic. I have a finite* set $A=a_1<\cdots<a_r$ ...
0
votes
0answers
45 views

Finding spanning vector sets

Let $V$ be the set of all vectors over the non-negative integers. For any two subsets $S$ and $T$ of $V$, define $S + T$ to include: All vectors in $S$ All vectors in $T$ All vectors that can be ...
7
votes
2answers
512 views

On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
2
votes
1answer
195 views

A quadrant of residues

Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$ ...
7
votes
1answer
223 views

Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...
0
votes
1answer
191 views

A three variable linear diophantine promise problem

Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find ...
2
votes
1answer
191 views

On Pell's equation

A post was made (Reduction from factoring to solving Pell equation) seeking clarification to solving $$x^2-Dy^2=1$$ to factoring when $D>0$. An answer was posted stating that to factor $N$, it ...
1
vote
1answer
73 views

Runge-Kutta convergence [closed]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
0
votes
0answers
55 views

lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$. What is the lattice ...
5
votes
1answer
300 views

Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider $$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$ the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then ...
5
votes
0answers
295 views

Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes: $f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is prime} \\ \lfloor n/2 \rfloor & \text{if} \;n ...
10
votes
4answers
507 views

Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...
2
votes
0answers
117 views

algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$ If $K$ is a number field, let $\delta(K)$ denote ...
2
votes
0answers
103 views

Comparing the size of two sums

Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements. Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. I am working on a research project, where I bounded a ...
1
vote
1answer
234 views

Error term for prime harmonic

What is known about the asymptotic behavior of $$ f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1? $$ Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or ...
2
votes
1answer
253 views

Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something. Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$, i.e. $\gamma_0\sim 14.134...$. 1) what is ...
4
votes
0answers
176 views

Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either returns true or false, we say that $\mathcal{A}$ works for ...
7
votes
0answers
235 views

On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
1
vote
1answer
159 views

What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?

I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where ...
2
votes
1answer
233 views

Computing all “suboptimal” rational approximations to $\pi/2$

I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy $$ n \epsilon(n)^2 \leq \tau $$ where $\tau$ is a known ...
3
votes
0answers
174 views

Fourier expansions of newforms at width-1 cusps

Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label $\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
1
vote
0answers
200 views

Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...
2
votes
3answers
247 views

How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...
3
votes
0answers
149 views

Algorithm to compute a common denominator of a finite set of rational numbers

Let $x_1,\dots,x_n \in \mathbb{Q} \cap (0,1)$ be fixed, but unknown, and assume that we know a number $K \in \mathbb{N}$ such that there is a number $N \in \{1,\dots,K\}$ such that $x_1 N,\dots,x_n N ...
18
votes
0answers
991 views

Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...
2
votes
1answer
110 views

Growth of the truncation of the integral multiples of an irrational number

Let $[a]$ denote the integral part of a real number $a$. Let $a$ be an irrational number and $b$ a real number greater than $1$. Consider the sequence $(b^n(na-[na]))$ with $n$ running on the ...
1
vote
2answers
327 views

Computational complexity of solution of Pell equation and more

What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity? And more,could ...
0
votes
1answer
106 views

Size of approximate solution of an integer relation

Let $X_1$, ..., $X_n$ be a list of real numbers. Consider an integer relation equation $A_1 X_1 + \ldots + A_n X_n = 0$ where $A_1$, ..., $A_n$ are unknown integers. Suppose somehow we are not so ...
3
votes
1answer
209 views

Calculating (n ^ fibonacci(k)) MOD m for a large value of k

The value of $k$ can be very large indeed (up to $10^{12}$). Is there an efficient way to calculate the output? Edit : 'm' is a prime number.
1
vote
1answer
275 views

Number Field Sieve for factorization with non-monic non-linear polynomial. Can't understand calculating ideal valuations

There is a paper "Factoring integers with the number field sieve" (download it here, for example). I can't understand how they reason the correctness of computing ideal valuations in the case of ...
2
votes
0answers
86 views

Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that $$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...
5
votes
2answers
377 views

Minkowski successive minima inequality for a lattice base?

Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors. The Minkowski successive minima inequality says ...
1
vote
1answer
308 views

Valid Difference Sets

Suppose $$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$ We calculate the differences as: $$ d=p_i-p_j\mod N,\quad i\ne j $$ Now let $a_d$ denote the number of occurrence of $d$ (for $d = 0, 1, 2, ...
2
votes
1answer
363 views

Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$

In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$. In each case the x coordinates are ...
0
votes
1answer
202 views

Spreading-out integers via multiplication

Let $a_1,...,a_n\in [0,m]$ be a set of $n$ positive integers, where $n<<m$, $m=poly(n)$. One can assume $m$ is prime. Is there an efficient, possibly randomized, way to find an integer ...
2
votes
2answers
284 views

On Cubic Non-Residues Modulo a Prime [closed]

What is a good test for identifying cubic non-residues/residues and higher power non-residues/residues modulo a prime $R$ in terms of computational complexity? Given $M$ and $N$, is there a good way ...
6
votes
3answers
907 views

Square Root Algorithm

I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?
2
votes
0answers
379 views

Textbooks on Algorithmic Number Theory

I am looking for a good textbook suitable for graduate or advanced undergraduate students who want to explore algorithmic number theory. Specifically, algorithms for primality testing, and factoring ...
0
votes
0answers
189 views

Computational Ring Theory

I have tried to understand and program CGT algorithms though I am a beginner still. But I never get to hear Computational Ring Theory. Even GAP largely supports Groups Theory. Is there some initiative ...
4
votes
2answers
425 views

Average involving the Euler phi function

Does $$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$ converges or not when $N$ goes to infinity?