Questions tagged [computational-number-theory]
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
477 questions
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Computing coefficients of theta functions associated to quadratic forms
If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
- 323
2
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0
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45
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Reordering entries of integer symmetric matrix via linear combinations into a symmetric matrix with all its eigenvalues positive with det condition
Suppose we have a symmetric matrix $M\in\operatorname{Sym}{M}_{n}(\mathbb{Z})$ having some negative eigenvalues. Are there algorithms filling the entries of a (possibly) bigger symmetric matrix $M'\in\...
- 573
5
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1
answer
237
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On a result of Euler on pseudoprimes
In several sources (for instance on page 58 of the first ed. of Crandall & Pomerance book on prime numbers or at the end of this paper by J. H. Jaroma), I have seen a result that goes like this:
...
- 87
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0
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Is this factorization problem in EXP?
Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...
- 13.9k
6
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4
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552
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(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 ...
- 151k
11
votes
1
answer
278
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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 ...
- 333
1
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0
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583
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Langlands program and complexity theory
Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now.
One of the motivations I imagined for the Langlands program was for ...
- 59
4
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1
answer
756
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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)...
1
vote
1
answer
150
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Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)$ time
Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time?
Geometrically this means we have to sort through a positive proportion of the ...
- 21
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245
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Hilbert 10th problem for genus 2 equations
Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...
- 7,182
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0
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313
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A question on infinite arithmetic progressions
I was working on a problem that consisted of deciding if the language a finite automaton (the alphabet of which is $\{0,1\}$ and the words accepted are binary encoded positive integers) contains an ...
- 51
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1
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858
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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 ...
- 319
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4
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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/...
-2
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1
answer
494
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Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $?
I have tried to get representations of some integers as sum of three cubic of the form $x^3+(k*10^n)^3+z^3$ with $k$ is integer and $n$ is a postive integer, I took this example : $(48807585839879)^3-(...
11
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4
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Computational complexity of finding the smallest number with n factors
Given $n \in \mathbb{N}$, suppose we seek the smallest number $f(n)$ with
at least $n$ distinct factors,
excluding $1$ and $n$.
For example, for $n=6$, $f(6)=24$,
because $24$ has the $6$ distinct ...
- 151k
3
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1
answer
167
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Is factorial computation known to be in a class smaller than $FEXP$?
Functional version of the counting hierarchy is $FCH$. It is an open problem whether there a sequence of $poly(log(n))$ number of $+,\times$ operations utilizing the assistance of $O(1)$ number of ...
- 13.9k
0
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1
answer
607
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Method to solve modular quadratic polynomial [duplicate]
If $q$ is a prime what is the best method to compute roots of a quadratic polynomial $f(x)\equiv0\bmod q^2$ which is of form $x^2+bx+c\equiv0\bmod q^2$ where $b^2-4c\equiv0\bmod q$ and $gcd(b,q)=1$ ...
- 13.9k
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Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form.
OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I ...
- 41.4k
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3
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How good is Kamenetsky's formula for the number of digits in n-factorial?
In Number of digits in n!, now closed, there was a mention of Dmitry Kamenetsky's formula, $[\bigl(\log(2\pi n)/2+n(\log n-\log e)\bigr)/\log 10]+1$, for the number of decimal digits in $n$-factorial. ...
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2
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648
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A Pell like equation
If one takes in general $(\star)\, \,x^2-dy^2=C$ where $d$, $C$ in $\mathbb{N}$.
Taking $d=w^2p^2+p$ with $w\in \mathbb{Q}\ge 1$ and $p\in \mathbb{Z}$ which is verified (explained later), for the ...
- 785
1
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0
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84
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How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
1
vote
0
answers
50
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Algorithm to compute S-units in imaginary quadratic number field
What efficient algorithms are there to compute the $S$-units of a given imaginary quadratic field $K$, where $S$ is a finite set of non-archimedean primes?
Computing $S$-units are implemented in ...
- 577
1
vote
1
answer
217
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Computing Lucas sequence for large n
I've been trying to write a test function for Fibonacci pseudo-primes with large $n$. Fibonacci pseudoprimes are composite numbers such that $V_n(P,Q) \equiv P \mod n$ for $P>0$ and $Q =\pm 1$, ...
- 11
1
vote
0
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136
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Can PARI compute class numbers without factoring the discriminant?
When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
- 125
11
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2
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410
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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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1
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328
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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 $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...
3
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1
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272
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Connecting different ways of constructing cubic extensions of $\mathbb{Q}$
There are at least two ways to construct cyclic cubic extensions of $\mathbb{Q}$ as explained below. (A third one is given in the answer to an earlier question).
Given $A, B, C$ integers with $A\neq ...
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8
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1
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893
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Is it possible to find a (nonsquare) integer which is a quadratic residues modulo a given infinite list of primes?
I'm wondering if it's possible, given a prime p and an infinite list of primes $q_1$, $q_2$, ... to find an integer d which (1) is not a square mod p, but (2) is a square mod $q_i$ for all i. Always, ...
- 121
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0
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813
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Borderline Collatz-like problems
The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
0
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0
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135
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On a deterministic primes search problem
I feel the following problem might be resolved already. But I could not find any related answers.
If $p_1,p_2,\dots,p_t$ are primes where $2\leq t=o(\log n)$ is there a prime within $$\prod_{i=1}^...
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2
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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)?
$$(-1)^n\cdot(\pi - ...
- 335
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1
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Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...
- 34.4k
5
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0
answers
180
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Certificate for computation of ideal class group
Is there a known way of producing a certificate that can be used to more quickly verify that an ideal class group of a number field was computed correctly? More formally, I would like to know if there'...
- 1,856
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3
answers
1k
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Explicit formula for elementary symmetric sum
For $k\ge1$, $j\ge1$, Let $$e_k(j)=\sum_{1\le i_1<...<i_k\le j}i_1\cdot\cdot\cdot i_k.$$ We know that $e_k(j)$ is a polynomial in $j$ with coefficients depending on $k$. I am curious about ...
- 113
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1
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250
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If the coefficient of the polynomial positive
I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$
$$\bar{S}(k)=\...
- 685
3
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1
answer
116
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Can we construct composite Fermat pseudoprimes to integral algebraic bases?
Let $0\neq \beta\in\overline{\mathbb{Z}}$ and let $n$ be a positive integer coprime to $N_{\mathbb{Q}(\beta)/\mathbb{Q}}(\beta)$. Say that $n$ is a Fermat pseudoprime to base $\beta$ if
$$\beta^{n^{[\...
- 458
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1
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488
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How to simulate Poisson point process
How to simulate a process $S_t=\sum_{0\leq s\leq t}\Delta_s,$ where $\Delta_s$ is a Poisson point process with values in $(0,\infty)$ and with characteristic measure $\Pi(dx)=\frac{\alpha}{\Gamma(1-\...
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1
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Is there a short proof for the permutation invariance of this combinatorial map?
Consider a positive integer $n$ and integers $(c_i)_{1\le i \le 4}$, with $1 \le c_i \le n$. Conside the map:
$$f_n: (c_1,c_2,c_3,c_4) \mapsto \delta_{c_1,c_2}\delta_{c_3,c_4} - \# \{ |2n+1-2|x||, \ x ...
19
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1
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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 (...
- 5,424
18
votes
1
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607
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Complexity of a Fibonacci numbers discrete log variation
In my work I encountered the following
FIBMOD PROBLEM:
Given $k,m$ in binary, decide if there exists $n$ such that
$\, F_n = k \,$ (mod $m$). Here $F_n$ is a Fibonacci number.
This is a variation ...
- 17.1k
2
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0
answers
207
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Sum of all primes below $n$ without listing all primes below $n$
Asymptotically there is around $\frac{n}{\ln n}$ primes below a given integer $n$. Thus $\frac{n}{\ln n}$ is a lower bound for the time complexity of any algorithm that at some point finds each prime ...
- 21
6
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1
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243
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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 ...
user6671
5
votes
0
answers
197
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Ramsey Numbers for Integers
Erdos defined $f(n)$ to be the minimum $r$ such that there is an $r$-coloring of the positive integers less than $n$, wherein $n$ cannot be written as the sum of distinct monochromatic integers. ...
- 141
17
votes
2
answers
1k
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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 ...
- 82.7k
18
votes
2
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6k
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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 ...
- 1,625
3
votes
0
answers
113
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Next smooth number
I want to find the next $n \in \mathbb{N}$ such that
$$s < n = \prod_{p_i \in \mathbb{P}_B} {p_i}^{a_i}$$
Where $\mathbb{P}_B$ is the set of primes not greater than $B$
I know that we can generate ...
- 131
5
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2
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932
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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 ...
4
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0
answers
213
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What is the complexity class of this problem without Cramer's conjecture?
The problem 'Given $0<a<b$ is there a prime in the interval $[a,b]$?' is in $\mathsf{NP}$. If we assume Cramer's conjecture the problem is in $\mathsf{P}$ since if $b-a>(\log a)^{2+\epsilon}$ ...
- 13.9k
1
vote
1
answer
92
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What are the complexity classes of these problems about divisibility and coprimality?
The problems
'Given $0<a<b$ and a prime $p<a$ is there an integer $\ell\in[a,b]$ such that $p|\ell$?'
'Given $0<a<b$ and an integer $q\not\in[a,b]$ is there an integer $\ell\in[a,b]$ ...
- 13.9k
2
votes
2
answers
2k
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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 ...
