0
votes
1answer
22 views

How many ordered bases can be found for $\mathbb{Z}_p^n$ over filed $\mathbb{Z}_p$?

Take $\mathbb{Z}_p^n$ as a linear space over $\mathbb{Z}_p$. Now you can imagine multy bases for this space. (please leave a comment or have an edit if question is not clear enough.)
0
votes
0answers
18 views

Is it a case of a mixed-integer problem?

I have been faced with a problem for quiet some time and don't know how to go further. The problem is as follows and would appreciate any help on formatting this problem in terms of mixed-integer ...
3
votes
0answers
50 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
6
votes
1answer
65 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
0
votes
0answers
16 views

Finding discriminant of this quadratic form

My question is what are the discriminants of $X^2+Y^2 $ and $X^2-Y^2$ over $\mathbb{R}$ and $\mathbb{C}$ and why? It should be $1$ and $-1$ respectively over $\mathbb{R}$. But shouldn't they be same ...
1
vote
1answer
32 views

Discriminant of a Quadratic form

Let $V$ be a vector space over field $K$ and $Q$ is the quadratic form on it, and $A$ be the matrix w.r.t. $e_1,e_2,...e_n$ of $V$. Now $discr(Q)$ is defined as $det(A)$ mod ${K^{*}}^{2}$. Now my ...
1
vote
4answers
47 views

Proof by Induction - Algebra Problem (Steps included but not understood)

I do not quite understand this proof, if anyone could explain the steps for me it would be greatly appreciated. It's probably something glaringly obvious I'm not seeing, thanks in advance. Prove that ...
1
vote
1answer
27 views

Discriminant of the characteristic equation

I want to know, what is the relationship between a matrix A and the discriminant of its characteristic polynomial. Thanks!
6
votes
4answers
163 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
12
votes
1answer
356 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
0
votes
1answer
27 views

Linear Constraints Solution Existence

how can one decide if $$A*t\ge b$$ $A$ is a Matrix with integer Entries and $t$ is a Vector with integer Entries, $b$ is a fixed Vector with integer Entries exists?
0
votes
0answers
19 views

Integer Solution To System of Linear Function

Hello i reposted the question, because the equation title was a bit misleading. [and i could not edit the question] $c,f$ are not known, so it is not a Diophantine equation, it is more like a system ...
2
votes
0answers
58 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
1
vote
1answer
72 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
0
votes
3answers
37 views

How many are possibilities to build count $n$ summing $k$ other counts?

I have got an integer $n$. I have to build it by summing $k$, not necessary, different integers. Is there any overall formula to count how many are possibilities to build count $n$ summing $k$ other ...
0
votes
1answer
35 views

How construct simulations using the Markov chain in R or Matlab

help please I need to simulate a set of random numbers which are belongs to the polulation of unknown distribution. still I'm not sure whether I should use a transition matrix to do the simulation? ...
1
vote
0answers
49 views

How do you evaluate $a^b$ where b is irrational using only basic operators.

How would you evaluate $a^b$ where b is irrational and you can only use +,-, multiplication, division, and rational powers. For example $2^\pi.$ We know $2^2$ = $2\times2$ etc... but when the power ...
1
vote
2answers
61 views

definition of a function 1

Kindly asking for any hints about the following questions: Define a function $ S : \{0,1\}^4 \to \{0,1\}^4$ with this conditions: 1- For any $\ x$, $S(x) \neq x $ 2- for any $x \neq y$, $\ S(x) ...
1
vote
0answers
63 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
0
votes
1answer
17 views

Find subset of rows whose entries sum to an even number in each column

I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows. How does one go about finding such a linearly ...
3
votes
0answers
61 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
1
vote
2answers
26 views

Why are the congruences $p^2-1 \equiv 0(\mod 8)$ and $p^e \equiv 1 + e(p-1) (\mod 4)$ for odd prime $p$ and $e \ge 1$ true?

Why are the congruences $p^2-1 \equiv 0(\mod 8)$ and $p^e \equiv 1 + e(p-1) (\mod 4)$ for odd prime $p$ and $e \ge 1$ true ? Suppose $p$ is an odd prime. I see easily that $p-1 \equiv 0 (\mod ...
1
vote
1answer
37 views

How to convert linear recurrence to a tiling question

If I have some linear recurrence of form $$f(n) = a_1f(n-1) + a_2f(n-2) + a_3f(n-3) + \cdots + a_kf(n-k)$$ How does this translate to tilings? For example the Fibonacci sequence is the same as ...
0
votes
2answers
107 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
0
votes
1answer
56 views

Functions determined by characters are linearly independent?

Let $X$ be a set with an action on $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi \pmod N$ we set $$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in ...
2
votes
0answers
61 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
1
vote
1answer
114 views

Linear Diophantine Equations

I was asked to find i) all integer solutions, and ii) all non-negative integer solutions to the equations below. I know (a) has no answers, but have no idea how to go about proving the rest. ...
4
votes
1answer
57 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
0
votes
1answer
69 views

non-division quaternion algebra is isomorphic to $2\times 2$ matrices

Let $k$ be a field of characteristic $\neq2$. Let $a,b\in k$ be nonzero elements. Let $A:=\left(\frac{a,b}{k}\right)$ be the quaternion algebra over $k$ with parameters $a,b$. Suppose $A$ is not a ...
4
votes
1answer
272 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
2
votes
2answers
65 views

Linear Diophantine equation in two variables with additional constraints

Given, $$aX + bY = c$$ where, $$c > b > a > 0;\quad X, Y > 0;\quad b\nmid c, a\nmid c$$ I want to find out if a solution exists as efficiently as possible (I'm not interested ...
2
votes
1answer
80 views

How to find $A,B,C$ such that $M=A^2+B^2+C^2$

Let $M=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in M_{2}(\mathbb Z)$, prove that there exist $A,B,C \in M_{2}(\mathbb Z)$ such that $M=A^2+B^2+C^2$. My idea: if we can find ...
0
votes
0answers
46 views

Producing integer combinations of irrational numbers in sequence?

Let $\mathbf{w}=\{w_0,w_1,\cdots,w_n\}$, $\mathbf{k}_i=\{k_0^i,k_1^i,\cdots,k_n^i\}$ and $\mathbf{m}_i=\{m_0^i,m_1^i,\cdots,m_n^i\}$, where $w_j\in\mathbb{R}$, $k_j^i\in\mathbb{Z}$ and ...
0
votes
0answers
50 views

Given a set of inequalities, use what algorithm to minimize one variable?

I have the following set of inequalities: $$C_1 \le y-T_1x_1 \lt C_1+D_1$$ $$C_2 \le y-T_2x_2 \lt C_2+D_2$$ $$C_3 \le y-T_3x_3 \lt C_3+D_3$$ where $x_1, x_2, x_3$ and $y$ are non-negative integer ...
1
vote
1answer
38 views

Is it possible to extract any encoded $x, y \in \mathbb{N^*}$ from $z=ax + by$

Is there any specific $a, b \in \mathbb{R}$, $\forall x,y \in \mathbb{N^*}$, take $z=a\cdot{}x+b\cdot{}y$ (then $z\in\mathbb{R}$), we can always extract $a,b$ from $z$. Here below are some trials I ...
1
vote
3answers
108 views

Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime

I am looking for ways to prove that $\mathbb{Z}_m=\{0,1,2,\dots,m-1\}$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime. I tried it this way: If $m$ is a prime $p$,it is ...
4
votes
3answers
173 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
11
votes
2answers
349 views

eigenvalues of integral matrices

Is it possible that a $3$-by-$3$ matrix with integer values and determinant 1 has a real eigenvalue with algebraic multiplicity 2, that is not equal to $\pm 1$? Doing some elementary computations ...
3
votes
1answer
102 views

Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
0
votes
1answer
48 views

Iterating over every permutation of factors of 800 OR possible isomorphism types of abelian groups of orders 74, 147, 666, 800 and 1221

this is self learning This may smell of homework but I am doing this http://homepages.warwick.ac.uk/~masdf/alg1/p4.pdf worksheet I found, you can see this question is in the "practice" section and ...
2
votes
2answers
196 views

Solving two simultaneous equations

Suppose that $x$, $y$ and $z$ are three integers (positive,negative or zero) such that we get the following relationships simultaneously $x + y = 1 - z$ and $x^3 + y^3= 1 - z^2$ Find all such ...
1
vote
1answer
255 views

Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
7
votes
1answer
114 views

When does this matrix have an integral square root?

Let $d_1$, $d_2$, ..., $d_n$ be positive integers. Let $B$ be the $n \times n$ matrix $$\begin{pmatrix} d_1 & 1 & 1 & \cdots & 1 \\ 1 & d_2 & 1 & \cdots & 1 \\ 1 & ...
2
votes
1answer
48 views

On invertibility of a special matrix - Hilbert matrix [duplicate]

I want to know how to prove that the below matrix is invertible \begin{pmatrix} 1 & \frac { 1 }{ 2 } & ... & \frac { 1 }{ n } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 3 } & ... ...
4
votes
3answers
114 views

How to prove that there does not exist a natural number '$n$' whose product of digits is $n^3-25n^2+151n$.

How to prove that there does not exist a natural number '$n$' whose product of digits is $n^3-25n^2+151n$. I don't know where to start. NOTE: I do not want the answer a hint should do it. Any help ...
3
votes
1answer
353 views

Why n! equals sum of some expression?

Why n! equals sum of some expression? Especially I need to know why this expression is true? $$ n!= \left(\frac{n+1}{2}\right)^{p(n)} \; \prod_{j=0}^{q(n)}\sum_{i=0}^j(n-2i), $$ Where \begin{gather*} ...
-1
votes
1answer
90 views

Simple Question about Induction?

let x be a natural number i want to prove that f(x)=$x^2$. suppose that f(x)=$x^2$, f(0)=0 holds we'll prove that f(x)= $(x+1)^2$, in the functional equation we have f(x-y)+f(x+y)=2f(x)+ stuff, ...
9
votes
3answers
287 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
2
votes
4answers
123 views

Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions.

How can i solve this, Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions. I thought let $\frac ...
1
vote
2answers
87 views

n is+ve integer, how many solutions $(x,y)$ exist for $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ with $x$, $y$ being positive integers and $(x \neq y)$ [duplicate]

I wanted to know, how can i solve this. For a given positive integer n, how many solutions $(x,y)$ exist for $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ with $x$ and $y$ being positive integers and $(x ...