A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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What is an ideal lattice?

Can anyone describe in layman's terms what an ideal lattice is? I've seen them mentioned in many places, but haven't found a good definition of what exactly they are, nor any good terms to know where ...
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Index of a sublattice

Assume that $\Lambda \subset \mathbb{Z}^2$ is an integer lattice of rank $2$ and define for each integer $k$, $$\Lambda(k):=\Big\{(x_1,x_2) \in \Lambda: k\mid (x_1,x_2)\Big\}.$$ What is the index of ...
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An inequality related to lattice points 'around' a circle

Take a circle of radius $r$ with centre at the origin such that $r^2=N_1^2+N_2^2$ for $N_1,N_2\in\mathbb{N}$. Consider a lattice coordinate $(a,b)$ such that $a\in(-r,-2)$ and define $b$ to be the ...
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Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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Finding total number of lattice points on a circle and one of the coordinate values

I was wondering if it was possible to quickly find the total amount of lattice points on a circle, given its equation (origin and radius), and I was also a bit confused on how to find a particular ...
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Maximize the product of pairwise distances between particles on a lattice

I am trying to solve the following problem : Consider an $N\times N$ square lattice ($N$ even integer, we can assume that N is large) on the complex plane, with lattice sites at position $j+ik$, $j$ ...
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Number of Lattice points in the intersection between an N-Sphere and an N-Cube

I have a hyper-Cube lattice with coordinates between 0 and 10, therefore 11^N lattice points. I take one of this point, C, and then another point P at distance D from C. I need to know a good ...
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Compute a reduced basis for the lattice with basis matrix

For an exercise I need to compute a reduced basis for the lattice with basis matrix: \begin{pmatrix} 1 & 0 \\ 1414 & 1000 \\ \end{pmatrix} I found the algorithm for ...
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Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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Finding the number of lattice paths

Find the number of lattice path of length $2n$ that starts on $(0, 0)$ such that for all the points $(x, y)$ in the path, $x < y$. So pretty much all the points besides the origin are strictly ...
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Representation of integers by sum of squares with linear constraints of a special form

I would like to know what integers $d$ can be written as a sum $d=\sum_{i=1}^N\sum_{j=1}^M a_{ij}^2 $ with $a_{ij} \in \mathbb{Z}$ and where the row and column sums of $a_{ij}$ are fixed $\sum_{i=1}^N ...
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Short Primitive Vectors in a Lattice in $\mathbb{Z}^2$

Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$. I want to find an order-of-magnitude bound on the shortest ...
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How can I visualize ideals on a ring of integers of imaginary quadratic fields?

If I were to visualize the ideal $(2, 3+3i)$ of $Z[i]$ on the complex plane, I would find a gcd of $2$ and $3+3i$ (for example $1+i$) and the ideal $(2, 3+3i)$ is identical to $(1+i)Z[i]$, which forms ...
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Set of $K > N$ integer-valued vectors so that any N-subset is a base for $\mathbb{R}^N$

Be $N$, $K$ natural numbers so that $K > N > 1$. Be $V$ a set of $K$ vectors in $\mathbb{R}^N$: $V = \{v_1, \ldots, v_K \in \mathbb{R}^N\}$. First of all, I need to find a $V$ so that, for any ...
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Complete toric varieties with given codimension of the singular locus

Let $N\cong \mathbb{Z}^n$ be a lattice and $\Delta\subseteq N_\mathbb{R}$ be a fan such that $X=X(\Delta)$ is a complete and simplicial (i.e. $\mathbb{Q}-$factorial) toric variety of dimension $n$ ...
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how to find the number of integer coordinates in the interior of triangle

How to find the number of integer coordinates in the interior of the triangle with vertices(0,0) (0,21) (21,0).
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Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$

I plotted $\Big\{ (n, n \sqrt{2} \, \mathrm{mod} \,1) \;\Big| -50 \leq n \leq 50 \Big\}$ and even though the $n \sqrt{2}$ is a line, the pattern that emerges is a lattice. What is the basis of this ...
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How many edges, faces, cells in a $2\times 2 \times 2 \times 2$ hyper cubic lattice?

If I have a $2\times 2\times 2\times 2$ hyper cubic lattice, how many corners, edges, faces, and cells will it be composed of? E.g. the 4D analogue of figure below. Assume the faces within the figure ...
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Creating n-dimensional lattices from lower dimensional parts

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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Farthest point on parallelogram lattice

On points arranged in a parallelogram lattice, like on the image in this Wikipedia article, how to calculate the maximal distance any point on the plane may have to its closest point from the lattice. ...
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Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
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n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite ...
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Why are there only so many Bravais Lattices?

I am in doubt as to why there are exactly five 2d Bravais lattices? For example, I could take the square lattice and place a lattice point at the midpoint on every side of each square. Shouldn't ...
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Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
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Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
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Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
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Is there a connection between lattices in the sense of orders and lattices in the sense of groups?

I'm wondering about this for some time now - is there a intuitive connection between those concepts or have they been named the same by chance? In particular, my interest in this question was ...
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Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
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Are lattice approximations of lines always tile-able patterns of lattice points?

I'm interested in sets of the form: $L = \left\{(\left\lfloor x \right\rfloor, \left\lfloor y\right\rfloor) \mid ax + bx + c = 0; x, y \in \mathbf{R}\right\}$. I want to know if we can always write ...
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Finding the quotient ring $\mathbb{Z}[i]/(4+i)$

Find the quotient ring $\mathbb{Z}[i]/(4+i)$ by identifying elements with the lattice points in the square generated by $4+i$. I know that $N(4+i) = 17$. Therefore, $4+i$ is irreducible. Now ...
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Define all points in the affine integral lattice.

Define all points in the affine integral lattice $\mathcal{L}=\{(x,y,z,t) : x+y+z+t=5$ and $x-z \equiv 0$ (mod $12$)$\} \subset \mathbb{Z}^4$. This is a question from a practice exam I have with no ...
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How Many Rational Slopes?

Given an $N$ by $M$ grid with integer coordinates (e.g. like pixels in an image), how many slopes are defined by the set of lines passing through the each grid point pair? Note that because the ...
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Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
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Is $\mathbb{Z}[\sqrt{2}]$ a lattice?

Is $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ a discrete subgroup of $\mathbb{R}$? How to prove that?
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Solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$

How to solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$? The origin of this problem is the following question: Show that rank-2 free $\mathbb Z$ module $\mathbb Z^2$ has $p+1$ ...
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Dual lattice and extreme value

Suppose I have a periodic function which has minimums only at lattice sites. Say the lattice is a honeycomb lattice, do I have my maximums at sites of a triangular lattice (which is the dual lattice ...
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Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
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Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
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Determining if $\mathbb{Z}[a]$ is a discrete subring of $\mathbb{C}$.

Let $a \in \mathbb{C}$ and consider the ring $\mathbb{Z}[a]$. Is there some nice criterion which will tell me whether $\mathbb{Z}[a]$ is discrete in the sense that there is some $\delta >0$ such ...
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What conditions are necessary for this property?

Let L be an ideal lattice. choose four elements a,b,c and d in L Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ 2 by 2 matrix. I think that $A=BCB^{-1}$ for some invetible matrix B ...
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Given a random unit cell's defining vectors in a random basis, what kind of lattice am I looking at?

So, as stated in the title, I have a lattice basis $B \in \mathbb{R}^{3\times 3}$ (given as a non-singular Matrix containing its unit cell vectors) which is arbitrarily rotated, scaled and reflected. ...
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Possible areas within an integer grid

Given a 1x1 grid with 4 lattice points $[(0,0),(0,1),(1,0),(1,1)]$ (equivalent to a $2 \times 2$ grid of vertices), there are 2 shapes and areas that can be formed: a triangle and a square. There are ...
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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
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Which planar angles on an integer lattice are possible?

As shown in this question, you can construct an angle $A$ on 3 integer points on a plane only if $\tan A$ is rational. A natural generalization is to ask which values can planar angles based on 3 ...
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Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
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Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
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Is it true that for a lattice $L$, $\mathbb{R}L = \mathbb{R}^{n}$?

I have as a definition A lattice $L \subseteq \mathbb{R}^{n}$ is a subgroup that is free of rank $n$ such that $\mathbb{R}L = \mathbb{R}^{n}$. I don't know if I am misinterpreting the statement, ...