Questions tagged [computational-number-theory]
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
446
questions
6
votes
2
answers
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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
1
answer
351
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
1
answer
385
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
1
answer
223
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 $N=poly(n)...
2
votes
2
answers
494
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 ...
5
votes
3
answers
2k
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
0
answers
706
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
0
answers
433
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
2
answers
504
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?
4
votes
1
answer
336
views
Hejhal's algorithm and computational methods for non-classical Maass wave forms
Hejhal's algorithm [1] was a little gadget invented in the 90's for calculating the Hecke eigenvalues and Fourier coefficients of Maass wave forms. Later, Booker, Strombergsson, and Venkatesh (BSV) [2]...
11
votes
1
answer
532
views
When adding a constant makes a multivariate polynomial reducible?
Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?
...
1
vote
2
answers
321
views
Transformation of a bivariate polynomial into a homogeneous one
For a given a bivariate polynomial $P(x,y)$ with rational coefficients:
Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...
0
votes
2
answers
734
views
On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...
8
votes
1
answer
1k
views
Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets?
Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Sigma_1^2$ sets, or, ...
2
votes
0
answers
300
views
Large numbers in small systems
Can we ever know the sum of the first $10^{10^{100}}$ digits of $\pi$?
Can we calgulate the $n$th digit of $\pi$ when the Kolmogorov-complexity of $n$ is larger than the complexity of the calculating ...
13
votes
3
answers
1k
views
Is there a composite number that satisfies these conditions?
We know that if $q=4k+3$ ($q$ is a prime), then $(a+bi)^q=a-bi \pmod q$ for every Gaussian integer $a+bi$. Now consider a composite number $N=4k+3$ that satisfies this condition for the case $a+bi=3+...
8
votes
3
answers
2k
views
Numerical evaluation of the Petersson product of elliptic modular forms
It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
4
votes
4
answers
761
views
Proving a least prime factor
Suppose that I find a small prime factor $p$ dividing a large number $n$ and I wish to prove that it is the least prime dividing $n$. There are two obvious approaches: either factor $n/p$, or divide $...
6
votes
1
answer
454
views
Solving equations in a subset of rational numbers
Let $S$ be a set of all positive rational numbers $x$ such that $2x^2 - 1$ is a square, excluding $x=1$.
I am interested in computing as many as possible solutions in $S$ to either the following ...
1
vote
1
answer
210
views
Counting modular squares in an interval
For an integer $m$, let $S^m_{x_0,x_1} = \{ t | x_0 ≤ t ≤ x_1 $ and $t$ is a square modulo $m \}$. Let $S^m_x$ = $S^m_{0,x}$.
Determining whether the sets $S^m_x$ are empty is easy (1 is always a ...
3
votes
2
answers
1k
views
Integer partition and sum of squares
Hello,
The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)
For all integers $n\geq 2$ denote by $\mathcal{P}...
4
votes
1
answer
363
views
Most orthogonal lattice basis
Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...
0
votes
1
answer
178
views
Efficiency in deriving differences of divisor pairs
I have a computational problem where I need to derive the differences in divisor pairs in as few cpu cycles as possible.
In particular I am interested in divisors of numbers of the form $x^3+3*x^2*y+...
5
votes
2
answers
815
views
12 descent scripts for pari/gp
I'm looking around for scripts to facilitate 12 descent on Mordell curves, preferably in Pari/gp.
I understand that Magma implements this feature, but unfortunately this software isn't available to ...
18
votes
2
answers
6k
views
Efficient computation of integer representation as a sum of three squares
Recently I've been studying the problem of integer representation as sum of three squares. Most of the articles that I've found study the function $r_m(n)$ which counts the number of representations ...
11
votes
1
answer
360
views
Does this notion of pseudoprime relative to a matrix appear in the literature?
Let $M$ be a square matrix with integer entries. Then Fermat's little theorem for matrices holds:
$$\text{tr}(M^p) \equiv \text{tr}(M) \bmod p.$$
This follows by an examination of the action of the ...
14
votes
2
answers
1k
views
Saying things rapidly about integer factorisations
Let $N$ be a positive integer. Thanks to the Miller-Rabin test and the work of Agrawal, Kayal and Saxena, these days people have much much faster algorithms for testing whether $N$ is prime or ...
8
votes
1
answer
1k
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Recent Fast Multiplication Algorithms for Large Integers
The STOC 2008 paper "Fast Integer Multiplication using Modular Arithmetic" by De et al
http://arxiv.org/abs/0801.1416 shows how to use $p$-adic numbers instead of $\mathbb C$ used in Furer's ...
2
votes
1
answer
199
views
Is there a security analysis of the GQ digital signature scheme?
I'm doing summer cryptography research and I am have been looking for a security analysis of the Guillou-Quisquater (GQ) digital signature scheme, but I have been unable to find one.
Since this is not ...
8
votes
5
answers
4k
views
Computational number theory
I am interested in learning computational number theory and doing some computer experiments.
Which sort of number theory problems can be solved by using computers? For example, is it possible to ...
0
votes
2
answers
3k
views
Fibonacci Numbers Modulo m [closed]
In the paper "Fibonacci Series Modulo m" by D.D. Wall (found here), there is a table in the Appendix listing values for the function $k(p)$. This function is defined as the period of the Fibonacci ...
1
vote
0
answers
122
views
What is the largest computed summatory liouville interval ?
I am interested to know the largest computed summatory liouville interval, an implementation of which is detailed in Section 4.1 of [1].
The wikipedia page [2] for the function charts summatory ...
2
votes
0
answers
255
views
Efficient counting of Egyptian fractions with bounded denominators
I was amazed to discover that sequence http://oeis.org/A020473 in the OEIS has almost four hundred terms computed.
I wonder how one can get that far? E.g., how one can compute A020473(100)?
P.S. ...
11
votes
3
answers
2k
views
Mertens' function in time $O(\sqrt x)$
This MathOverflow question seems to indicate that the state of the art in computing
$$
M(x)=\sum_{n\le x}\mu(n)
$$
takes time $\Theta(n^{2/3}(\log\log n)^{1/3}),$ which matches my understanding. ...
15
votes
2
answers
1k
views
Sum of $\sum_{k=1}^nd(k^2)$
There is a literature dealing with
$$
\sum_{k\le x}d(f(k))
$$
where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which ...
11
votes
2
answers
752
views
Interesting result on the Euler-Maschroni constant - what is the background?
Today I entered the following expression in maple:
$$a_i = H_{10^i} - ln(10^i) - \gamma$$
Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant.
When I computed $a_n$ ...
6
votes
2
answers
2k
views
Computing the fixed field of an automorphism of a function field
Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(...
2
votes
0
answers
362
views
Rational integer solutions of a linear Diophantine equation of cyclotomic integers
I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.
In particular, suppose my lattice vectors are $a$ and $b$ and I want to know ...
9
votes
2
answers
761
views
Complexity of Membership-Testing for finite abelian groups
Consider the following abelian-subgroup membership-testing problem.
Inputs:
A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with arbitrary-...
39
votes
1
answer
6k
views
How to calculate [10^10^10^10^10^-10^10]?
How to find an integer part of $10^{10^{10^{10^{10^{-10^{10}}}}}}$? It looks like it is slightly above $10^{10^{10}}$.
1
vote
0
answers
428
views
Witt rings and prime number generator?
Let $p$ be a fixed prime number. We define the ring of Witt vectors $W(R)$ for any commutative ring $R$ as follows:
For every ring morphism $R \rightarrow R'$ the induced morphism $W(R) \rightarrow ...
3
votes
1
answer
300
views
Hermit H-machines
I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H.
Is there a ...
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
8
votes
3
answers
1k
views
Effective detection of CM modular forms
Say $f$ is a newform of weight $k$ and level $\Gamma_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th ...
4
votes
0
answers
245
views
Algorithm/denominators of elements of a rational affine space
I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
4
votes
0
answers
159
views
Range of the least witness function
Let W(n) be a function from the positive odd composite numbers to the least positive b such that n is not a b-strong pseudoprime. W(n) exists for all numbers in its domain and its range is unbounded. ...
5
votes
0
answers
285
views
Lower bound for p-adic distance between roots
Let $f$ be a formal power series with coefficients in the ring of integers of a finite extension of ${\mathbb Q}_p$. Is there a simple algorithm to compute a positive lower bound for $|\alpha - \beta|...
1
vote
0
answers
107
views
Why do subspaces of the space of Global holomorphic differentials of a function field correspond to its subfields
I'm asking this question as a follow up to the Felipe Voloch's answer to this question:
Subfields of a function field
which you can read it here:
Subfields of a function field
(I just didn't have ...
1
vote
1
answer
317
views
Number of biquadrates mod n
Is there an explicit formula for the number of fourth powers mod n?
Finch & Sebah [1] give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for ...
8
votes
1
answer
715
views
Finding colinear points in F_q^n
Forgive me if this is well known, it's not really my field, but it's a problem I've run across and thought about a bit.
Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $n\ge2$, and let $A,...