All Questions
Tagged with integer-programming nt.number-theory
19 questions
1
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0
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67
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System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
- 25.4k
1
vote
2
answers
192
views
Direct algorithm for an integer program
Let $p$ be a prime and let $h_1,h_2\in\{1,2,\dots,p-1\}$ be integers.
Is there any direct algorithm to solve for following in polynomial in $\log p$ time?
$$\min (x_1-x_2)^2$$
$$x_1,x_2,k\in\mathbb Z$$...
- 13.9k
2
votes
0
answers
221
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Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
- 13.9k
0
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0
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122
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Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?
There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$.
Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
- 13.9k
4
votes
0
answers
104
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Questions in number theory related to $NC$ and $P$-completeness
Given $a,b\in\mathbb N$ find $\operatorname{GCD}(a,b)$.
Given $a,b,c\in\mathbb N$ find $x,y\in\mathbb Z$ such that $ax+by=c$.
Euclidean algorithm solves both.
My question is if either 1 or 2 is in ...
- 13.9k
1
vote
2
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204
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Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs
This might be related to counting hamiltonian cycles.
@Peter Taylor gave negative result about the one dimensional case, but we believe his attack is
not directly applicable to this question.
Given ...
- 25.4k
2
votes
1
answer
179
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Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
- 25.4k
0
votes
1
answer
87
views
Constructing an integer with small residues for two distinct primes in polynomial time
Given two primes $p,q\in[T,2T]$, how many integers $m$ of size $O(T^{3/2+\epsilon})$ are there such that the residues $m\bmod p$ and $m\bmod q$ are both $O(polylog(T))$? Looking for an answer
Is it ...
- 13.9k
6
votes
0
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410
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Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$
Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying ...
- 4,164
2
votes
1
answer
913
views
Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way [closed]
I need to emulate this sequence for a program: http://oeis.org/A025302
Stuff that I've taken into account:
After finding the prime divisors of a number. I take any divisor as p and apply the ...
- 123
3
votes
2
answers
608
views
Minimal solution of simultaneous congruences
I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that:
$$x\equiv a_1\mod p_1$$
$$\vdots$$
$$x\equiv a_n\mod p_n$$
where every ...
- 31
2
votes
1
answer
366
views
System of congruences
I have a system of $n$ congruences.
the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form:
$(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq m}}^n{\frac{x_iy_m}{p_ip_m}}+\...
- 21
2
votes
0
answers
154
views
Listing all Lattice Points in a Box
Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
- 173
1
vote
2
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752
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basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
- 51
0
votes
0
answers
104
views
Linear system with many solutions from a finite set
Basically I am looking for a linear system with
many solutions from a finite set.
Choose a finite set of rationals $S$ and fix
positive integer $k$.
Let $A$ be a linear system with $n$ variables $...
- 25.4k
2
votes
0
answers
163
views
existence of lattice point in polytope
This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
- 501
2
votes
3
answers
753
views
Reference Request for Integer factorization with LP/ILP
Have there been attempts to factor integers with Linear Programming?
Searching the internet suggests that for Integer Factorization only Number Theoretic algorithms, like sieves, are taken into ...
- 13.2k
4
votes
3
answers
2k
views
Lattice points close to a line
Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point $\boldsymbol{p}\in\mathbb{Z}^2$ within a distance $\epsilon>0$ of the ...
- 561
10
votes
1
answer
595
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Condition for existence of certain lattice points on polytopes
Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.
I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:
...
- 30.5k