Questions tagged [computational-number-theory]
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
476 questions
11
votes
1
answer
278
views
4-cliques of pythagorean triples graph and its connectivity
Let natural numbers $a, b > 2$ be adjacent if $|a^2 - b^2|$ is a square number. One can find a 3-clique.
For example 153, 185, 697. The questions are: does there exist a 4-clique? Is this graph ...
- 23.7k
1
vote
0
answers
192
views
Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
69
votes
1
answer
4k
views
Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...
3
votes
0
answers
73
views
Reference: Asymptotic bit-complexity of algebraic operations and transcendental functions
This question is a reference request. Does anyone know of a reference that lists the asymptotic bit-complexity of algebraic operations and transcendental functions implemented on a Turing machine that ...
- 2,183
9
votes
1
answer
858
views
Prove that there are no composite integers $n=am+1$ such that $m \ | \ \phi(n)$
Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a<p$ and $ m \ | \ \phi(n)$ then $n$ is prime.
This question is a ...
- 34.3k
2
votes
0
answers
95
views
Geometry for an odd perfect number? (Second question)
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{R})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
- 3,064
7
votes
1
answer
1k
views
Expressing primes $p\equiv 1 \pmod 3$ in the form $p = x^2 + xy + y^2$
Fermat famously showed that the only primes $p$ of the form $x^2 + y^2$ are the primes such that $p \equiv 1 \mod{4}$. Furthermore, we now know “effective” versions of Fermat's theorem, i.e. given a ...
- 44.4k
0
votes
0
answers
110
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Common integer roots of polynomials
I have two polynomials of form
$$f_1(w,x)=M_1$$
$$f_2(y,z)=M_2$$
and I have two polynomials of form
$$g_1(w,x,y,z)=M_3$$
$$g_2(w,x,y,z)=M_4$$
where $f_1,f_2,g_1,g_2\in\mathbb Z[w,x,y,z]$ and $M_1,M_2,...
- 13.9k
3
votes
1
answer
137
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Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
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6
votes
1
answer
242
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Inductively computing Mersenne primes / perfect numbers?
For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$.
Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$.
Set $\hat{\phi}(1) = \{x_1\}$ and ...
- 34.3k
3
votes
0
answers
81
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I have a question on the definition of 'good' primes in the paper of Cohen and Martinet
I'm reading the paper of Cohen and Martinet 'Etude heuristique des groups de classes'.
In the section 6, for an central idempotent $e$ of $\mathbb{Q}[\Gamma]$ and a prime $p$, the 'goodness' of $p$ is ...
- 1,013
2
votes
1
answer
152
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When can we decompose a multivariable p-adic power series into product of single variable power series?
Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?
For example, consider the following power series in $n$ variables:...
- 2,472
8
votes
1
answer
893
views
Is it possible to find a (nonsquare) integer which is a quadratic residues modulo a given infinite list of primes?
I'm wondering if it's possible, given a prime p and an infinite list of primes $q_1$, $q_2$, ... to find an integer d which (1) is not a square mod p, but (2) is a square mod $q_i$ for all i. Always, ...
- 121
5
votes
0
answers
339
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About a diophantine equation from group theory
Is there any set of odd primes $\{p_1, p_2,..., p_k\}$ and natural numbers $a_1,..., a_k$ such that the following equation satisfied:
$${p_1^{2a_1+1}+1 \over p_1+1}\times ....\times {p_k^{2a_k+1}+1 \...
- 63.9k
2
votes
0
answers
287
views
Best known primality test for the whole intervals of integers up to $10^{20}$ — like the sieve of Eratosthenes
What are the best known primality test(s) for the whole intervals of integers up to $N=10^{20}$ ? "Best" means "have minimal amortized time per tested integer".
That is, the ...
- 121
8
votes
1
answer
328
views
On the density map of the abundancy index
Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...
5
votes
3
answers
2k
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Goldbach conjecture and other problems in additive combinatorics
The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...
- 3,098
2
votes
1
answer
145
views
Estimation of a sum involving Stirling's number of second kind and binomial coefficient
Let $S(n, j)$ be Stirling's number of second kind. Let $p\in [0,1]$ and $m \in N$.
Bound from above the following sum:
$$
\sum_{j=0}^m S(n,j) {m \choose j}\, j! \, p^j
$$
- 5,342
1
vote
1
answer
218
views
On a quadratic diophantine equation
Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form
$$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$
$$\|(a,b,c,d,e,f)...
- 1,826
3
votes
0
answers
110
views
Minimum of a product of polynomial evaluated at primitive roots of unity, given that the value of the polynomial at the same lies on unit circle
This is something that came out of working on a problem:
Let $m$ be an odd positive integer and $f \in \mathbb Q[x]$ be a polynomial of degree less than $m$. With $\zeta_m$ denoting a primitive root ...
- 739
2
votes
0
answers
93
views
Elementary Iwasawa module
Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?
- 21
1
vote
2
answers
119
views
For a given value of $n$ and $m$, find $\text{fib}(n)$ $\text{mod } m$ where $n$ is very huge. (Pisano Period) [closed]
Input
Integers $'n'$ (up to $10^{14}$) and $'m'$(up to $10^3$)
Output
$\text{Fib}(n)$ $\text{modulo}$ $m$
My questions
For example : Why $\text{fib}(n=2015)$ $\text{mod}$ $3$ is equivalent to $\...
- 47.1k
2
votes
0
answers
139
views
Double Diophantine approximation
Let $0 < \alpha < 1$. For any $n$ there is a closest lower Diophantine approximation $\max p / q \leq \alpha$ with integer $0 \leq p < q \leq n$. It can be found efficiently, e.g., with Stern-...
- 5,590
11
votes
1
answer
3k
views
Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
- 12.9k
2
votes
0
answers
152
views
On the smallest solution of a linear congruence
I have the following question. First, consider the following congruence for primes $p\geq 5$:
$24x\equiv -1\;(\mbox{mod}\;p)$.
The smallest $x$, that is, $1\leq x\leq p-1$ for which the above ...
- 21
6
votes
0
answers
91
views
Computing all eta quotients of given weight and level
I have written a rather naive program for finding all holomorphic eta quotients of
given weight and level (and varying character). When the level has few divisors it is
very fast, but incredibly slow ...
- 13.1k
1
vote
0
answers
77
views
Digit summation of squared numbers
In olympiad teaching period, we have a session that students must try to design a good problem for others. Many times we arrive to good questions but sometimes there are some challenges. In one of our ...
- 4,784
4
votes
0
answers
447
views
The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd
This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here.
We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
3
votes
1
answer
266
views
Calculating greatest common divisor series: $\gcd(1,x)+\gcd(2,x)+\gcd(3,x)+....+\gcd(x,x)$ [closed]
How to compute the value of $$[\gcd(1,x)+\gcd(2,x)+\gcd(3,x)+....+\gcd(x,x)]$$ efficiently?
When x can be as large as million.
- 4,713
13
votes
0
answers
1k
views
Why am I unable to find primes of the form $(9n)!+n!+1$?
See also Math StackExchange: Is there a prime of the form $(9n)!+n!+1$?
Recently, user Peter from Math StackExchange asked for a prime of the form $(9n)!+n!+1$ (where $n$ is some natural number).
...
6
votes
0
answers
290
views
Effective bounds for Fermat's Last Theorem
Suppose $n>2$. By Fermat's Last Theorem, we know that $a^{n}+b^{n}=c^{n}$
has no non-trivial solutions. Can we quantify it more?
More specifically, given $a,b,c,n\in\mathbb{N}$ with $n>2$ and $...
- 583
2
votes
1
answer
89
views
Partitioning integers into two parts and exploring relationships with positional numeral systems
I asked this question in Mathematics StackExchange (link) about a month ago, but I have received no answer. It is about the following problem:
Problem:
Are there sets $A,B$ of integers such that $A\...
CommunityBot
- 1
6
votes
0
answers
260
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Brief history of primality testing theory after 2002?
Its clear that there is about 15 years (2004-2019) after the publication of AKS primality testing in 2002 and its modifications in 2003-2004. AS result, is there any development happened in this ...
- 199
3
votes
0
answers
189
views
Largest observed value of $S(t)$
Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation.
What is the largest observed value of $S(t)$ today?
Here is a quote from a ...
- 499
4
votes
0
answers
248
views
All digits of $2^n$ are even if and only if $n=1,2,3,6,11$ [closed]
All digits of $2^n$ are even if and only if $n=1,2,3,6,11$.
For example,
$2^1=2,2^2=4,2^3=8,2^4=16,2^5=32,2^6=64,\ldots,2^{11}=2048,2^{12}=4096$.
Do you know a proof of this fact or some related ...
3
votes
1
answer
227
views
How can we justify the use of Example 5,4 (of Cohen, Lenstra) assuming their heuristics
In these days, I'm studying Cohen-Lenstra heuristics to understand the paper of Rene Schoof "Class Numbers of Real Cyclotomic Fields of Prime Conductor".
On page 932 of Schoof's paper, there is a ...
- 13k
0
votes
1
answer
165
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About another potential characterization of normal numbers
Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ...
- 24.8k
2
votes
0
answers
99
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A problem in modular roots
We have three mutually coprime integers $r,t,M$ where $M\asymp K^{\frac12-2\epsilon}$ and $r,t\asymp K^{\frac14+\epsilon}$ holds with some fixed $\epsilon>0$ and $K>0$ is a large parameter. ...
- 13.9k
4
votes
0
answers
189
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How to find a CM point with the image in the elliptic curve under modular parametrization given
everyone! Let $E:y^2+y=x^3-61$ be the minimal model of the elliptic curve 243b. How can I find the CM point $\tau$ in $X_0(243)$ such that $\tau$ maps to the point $(3\sqrt[3]{3},4)$ under the modular ...
- 37k
7
votes
0
answers
274
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $...
- 5,470
5
votes
0
answers
179
views
Finding a presentation matrix with low dimension
Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...
1
vote
1
answer
189
views
Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
- 25.4k
2
votes
0
answers
137
views
Compare my software's representation of exponential numbers and 0?
Suppose I have a real number
$$
x=\sum_{i=1}^n a_i e^{\lambda_i}
$$
where $a_i,\lambda_i$s are complex algebraic numbers.
Is there an algorithm to determine whether it is greater than 0 or less than ...
CommunityBot
- 1
5
votes
2
answers
932
views
How can I find explicit examples of maximal orders of quaternion algebras that are not isomorphic?
I know that there exist algorithms that will construct maximal orders of a quaternion algebra over, say, $\mathbb{Q}$. However, the implemented algorithms that I know of require that you input an ...
- 6,039
4
votes
2
answers
301
views
Quadratic diophantine equations and geometry of numbers
Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...
- 5,382
0
votes
0
answers
83
views
Generating the digits in a base system by repeated multiplication of a number
The first 15 terms of the sequence {a_i} = 2^i are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. All of the digits in base-10, i.e. {0, ...
4
votes
1
answer
324
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 ...
CommunityBot
- 1
8
votes
1
answer
1k
views
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 ...
- 325
2
votes
0
answers
78
views
Accelerating convergence of a product by multiplying by zeta values: history?
Let $R(s_1,\dotsc,s_n) = \prod_p r(p^{-s_1},\dotsc,p^{-s_n})$, where
$r$ is a rational function on $n$ variables. Say we want to compute the value of $R(s_1,\dotsc,s_n)$ for some choice of $s_1,\dotsc,...
- 545
2
votes
2
answers
413
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 ...
- 121