Questions tagged [parity]

This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

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76 views

If $\frac{p^2}{q^2} + \frac{r^2}{s^2} = 1$, then $q,s$ are odd and one of $p,r$ is even

Suppose $\frac{p}{q}$ and and $\frac{r}{s}$ are rationals in lowest terms (so $\gcd (p,q) = \gcd(r,s) = 1$) and $\frac{p^2}{q^2} + \frac{r^2}{s^2} = 1$; i.e. $p^2s^2+r^2q^2=q^2s^2$. Then exactly one ...
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Proving that if Generator Matrix is in standard form then parity check matrix is a generator matrix for not C and so a parity check matrix

I have to prove that: Is $G=\left(I_{k} P\right)$ a generator matrix for a $[n, k]$ -Code $C$ then $H=\left(-P^{\top} I_{n-k}\right)$ is a generator matrix for $C^{\perp}$ and so it is a a parity ...
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1answer
44 views

Floor function parity problem

Prove that for every natural k this expression is always odd $⌊(5+\sqrt{19})^k⌋=A^k$ Progress that I' ve done is: I noticed $9^k<A^k<(9.5)^k$ Also I tried an induction approach, I used Binomial ...
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2answers
54 views

Determining the parity of the coefficients of a quadratic given that there is a rational solution

Que. If the quadratic $ax^2 + bx + c$ has a rational root, and $a$, $b,$ and $c$ are integers, then A) at least one of $a, b, c$ is even B) all of $a,b,c$ are even C) at most one of $a,b,c$ is odd D)...
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5 views

periodic aperiodic signal where Od is the odd part of the signal

Given the next signal $x(t)=Od(|sin(\frac{\pi}4t|u(t))$ where Od( ) is the odd part of the signal and has the next properties A.non periodic B.periodic,$T_0=4$ C.periodic,$T_0=2$ D.periodic,$T_0=\frac{...
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31 views

Finding the most likely correct words from a coded message which contains more than one error per string

I am studying a module called Linear and Discrete mathematics and have been given an assignment which contains the following question: Using the parity matrix H, as well as encoded alphabet ...
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2answers
60 views

What is a generator for an ideal such that $I=\{a+bi|a+b \text{ is even}\}$?

I had this problem where i had the application $\varphi: \mathbb Z[i] \Rightarrow \mathbb Z/(2)$ where $\varphi(a+bi)=\bar{a}+\bar{b}$. I had to find the kernel and prove that is a factor ideal. I ...
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43 views

let $a, b \in \mathbb{Z}$. Prove that if $a-b$ is odd, then $a$ and $b$ have opposite parity.

Just wondering if this is the correct way to write this proof. Thank you! Assume $a$ and $b$ have opposite parity. We’ll consider two cases: $a$ is even, $b$ is odd or $a$ is odd, and $b$ is even. ...
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2answers
69 views

Is $3^{\aleph_0}$ odd? [closed]

I'm actually interested in the range of answers. Is an odd number to the infinite power still odd? Must $3^{\aleph_0}$ be odd? I can't tell if the answer is yes or no. It seems like it should be ...
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1answer
30 views

Is there a way to test parity of fractional part (only period) of irredecible rational number without calculation?

I search in the web to get any way to test parity of fractional part of irredicible rational number by means to know if that fraction (period) even or odd but i didn't get , for example the fraction ...
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37 views

What is $\sum\limits_{i=1}^{2012} b_{4i}$?

Let $a_k$ be the number of ordered 10-tuples $(x_1, x_2,\ldots,x_{10})$ of nonnegative integers such that $x_1^2 + x_2^2 +\cdots+ x _{10}^2 = k.$ Let $$ b_k= \begin{cases} 0 & \text{if}\,\, a_k\,...
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1answer
42 views

Determining the parity (even or odd) of pi notation

I am trying to disprove a conjecture, and I have gotten it such that the conjecture is only true if $$\prod_{i=1}^{g}{(\frac{j_i^{L_i+1}-1}{j_i-1})}$$ is singly even (of form $2m$ where $m$ is odd). ...
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1answer
50 views

Leibniz formula for determinants and the sign function

I'm trying to build up an intuition for the Leibniz formula for matrix determinants. I understand that it relies on the sign or signature function of permutations, as described here. What I don't ...
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3answers
144 views

If $\sigma$ is an odd permutation, explain why $\sigma^2$ is even but $\sigma^{-1}$ is odd.

If $\sigma$ is an odd permutation, why is $\sigma^2$ even and $\sigma^{-1}$ odd? Edit: If $\sigma$ is even, then ${\rm sign}(\sigma)= 1$, and if $\sigma$ is odd, then ${\rm sign}(\sigma) = -1$. Based ...
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18 views

Does parity definite linear ODE have nonzero solution for homogeneous Dirichlet boundary condition?

Let's say we have a 2nd-order linear ODE $$\mathcal{L}\,y(x)=0,$$ whose differential operator $\mathcal{L}$ is parity even. We want to solve it for homogeneous Dirichlet boundary condition (b.c.) $y(-...
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1answer
274 views

Conjecture : an odd perfect square $n>1$ raised to the third power is never divisible by the sum of $n$'s divisors

This is a conjucture that I created : Let $\,\,n = (2k+1)^2 \,\, $with $k\in \mathbb{N}$ and so $n>1$, and let $$\,\,A = \sum_{d \in \mathbb{N}; \ d|n} d.$$ Then $n^3$ is never divisible by ...
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1answer
55 views

How is the parity condition determined for modular forms corresponding to $L(s, \chi_1) L(s, \chi_2)$?

This answer to a related question on Math StackExchange indicates the following: "This leads you to the conclusion that products like $\zeta(s)^2$ or $L(s, \chi_1) L(s, \chi_2)$ should correspond to ...
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1answer
35 views

How many ways are there to permute the integers from 1 to 1000 under the condition that two consecutive numbers must have different parity?

How many ways are there to permute the integers from 1 to 1000 under the condition that two consecutive numbers must have different parity? I know that there are two possible cases: either the ...
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2answers
137 views

Parity of a boolean function

I was doing a course on Quantum Computing. There was a question in problem set. Which is given below (Question No.5) Here the parity function is defined as(As per the instructor): $$y_i = 1-2x_i$$ $$...
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59 views

Sign of permutation with first all odd numbers and then even numbers

Let $\sigma \in S_{2n}$ be the permutation defined by $$\sigma := \begin{pmatrix} 1 & 2 & 3 & \ldots & n & n+1 & n+2 & \ldots & 2n \\ 1 & 3 & 5 & \ldots &...
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47 views

Problem with Functions and relations

Here's the question: Let $A$ be a nonempty finite set and let $f:A\rightarrow A$ be a function. Suppose that $f\circ f = Id_A$. Let $R$ be a relation on $A$ defined by $$ aRb \iff a=b \lor f(a)=b $$ ...
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27 views

$\epsilon_{ijk}x_j p_k$ under orthogonal transformation including those involving parity

The easiest way to see that angular momentum $\vec{L}=\vec{r}\times\vec{p}$ is an axial vector is to note that $\vec{r}\to -\vec{r}$ and $\vec{p}\to-\vec{p}$ under parity. But this should mean that ...
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1answer
23 views

How many of the following permutations are even?

This is from a problem from a past exam. How many of the following permutations on the set $\{1,2,3,4,5,6,7,8\}$ are even? $$(a) \quad \left(\begin{array}{llllllll}{1} & {2} & {3} & {4} &...
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46 views

Why should the sum of three numbers be even?

It says in "104 Number Theory Problems" by Titu Andreescu, Dorin Andrica, Zuming Feng (October 25, 2006) in the solution to this problem Example 1.6. Find all positive integers n for which 3n − 4, ...
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1answer
81 views

Parity function definition and intuition, characteristic function of a set.

I have a question of two (and a half) parts relating to parity functions. I) Pertaining to the definition of a parity function. II) Pertaining to the intuition behind checking some specific functions' ...
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2answers
107 views

Number of ordered triples $(a,b,c)$ such that $abc=n$

I am trying to investigate the situation above, or more weakly I am wondering about the parity of this number depending on $n$. This came about because I know that the number of ordered pairs $(a,b)$ ...
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2answers
52 views

Divisibility of a statement without the induction principle

I want to prove that the following statement is divisible by $4$ with a direct proof. $1 + (-1)^n ( 2 n -1 )$, $n$ natural number. My solution : Because $n$ is a natural number, we can look at two ...
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1answer
65 views

Finding Parity of Exponent in Congruence

Prove, z is odd if both x and y are odd and $3^x + 2^y ≡ 15^z \pmod {20}$ where $y > 1$. Taking modulo 4 gives $ - 1 + 0 ≡ 3^z \pmod {4}$, then what? There are several such problems, what is the ...
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123 views

What is meaning of even (odd) operator?

Let $ V^{\bullet}=\bigoplus_{i\in \mathbb{Z}} V^{i} $ is graded vector space. What is the meaning of even(odd) operator on this graded vector space? It related to shifts the grading by even(odd) ...
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1answer
24 views

Given a time-dependent function, complete it and find $c_0$

Complete graphically and analytically the function $f(t)$ so that the coefficients of the exponential Fourier series are pure imaginary: $$f(t)=\begin{cases}2t+1&\text{if $0\leq t\leq2$},\\\...
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1answer
343 views

Is $\pi$ even or odd? [duplicate]

This question is a question in my textbook, and I cannot stop thinking about it. The question: Is $\pi$ even or odd? I don't know if even or odd is defined for decimals or for irrational numbers, or ...
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3answers
55 views

Parity of two expressions of the same variable

For example: Given a variable $n$, and two expressions $n(n-1)/2$ and $n^2 - 1$. If they have the same parity or under what condition, they are all odd, or all even? I want a general solution ...
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2answers
52 views

Proof: sum of even and odd integer is odd

Statement: Sum of even and odd integer is odd $$ \forall(a,b) \in \mathbb{Z} : a \text{ mod } 2 = 0 \wedge b \text{ mod } 2 \neq 0 \implies a + b \text{ mod } 2 \neq 0 $$ Proof: $$ a \text{ mod } 2 ...
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2answers
39 views

Is it always true that $\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$?

My question is pretty basic. Here it goes: Is it always true that $$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$$ where the $s_j$'s are positive integers, and may be odd or even? We can ...
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1answer
56 views

How many digits do I need to determine if the product of a whole number an irrational number is odd or even? [closed]

So, say you have a really huge whole number like $5^{2000}$ and an irrational number like $\sqrt(5)$. If you were two multiply the two would you get an even or odd number after rounding to the ...
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1answer
51 views

Find the remainder Of an equation

Let, $S = 1^1 + 2^2 + 3^3 + 4^4 + \ldots + 2016^{2016}$. What is the remainder when $S$ is divided by $2$? Please give us an explanations for newbies like us.
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67 views

Parity of some binomial coefficients

Consider for fixed integer $n>1$ the binomial coefficients $\binom{n-1+2^j}{n+1-2^j}$ for $j>0.$ It seems that precisely one of these numbers is odd. Is there a simple proof of this fact?
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Prove that $n^2$ is even if and only if $n$ is even [closed]

I am practising exam questions and have come across the following. Prove that $n^2$ is even if and only if $n$ is even A contrapositive proof comes to mind, since it must be the case that if $n$ ...
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1answer
116 views

Parity of an infinite exponential function (What shape is $y=x^\infty$)

When you have functions of the form $x^c$, the shape of the graph is symmetric with even integers for c (└┘-shape) and for odd integers it is non-symmetric (┌┘-shape) When dealing with limits as c ...
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2answers
65 views

What are linear codes having minimum distance 2 used for?

Consider the following parity check matrix $$H = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \...
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1answer
31 views

Is the parity of error function enough to show :$\int_{-l}^{l} \exp ({\operatorname{-x^2erf(x)})dx=\int_{-l}^{l} \exp({\operatorname{x^2erf}}(x)})dx$?

I have tried to show the below identity using the parity of both error function and exp function but I didn't succeed, then my question here is there any analytical way to show this identity or Is ...
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1answer
84 views

Maximal likelihood Error/Syndrome table for $[16, 11]$ hamming code

I think I have to start with a parity check matrix for $[16,11]$ Hamming code. $$H = \left( \begin{array}{cccccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 &...
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71 views

What is $\{0,1\}^8$?

Recall the parity-bit error detecting method, which uses the function f : B8 → B9 , where B = {0, 1}, defined by f(b1, b2, . . . , b8) = (b1, b2, . . . , b8, b9) where b9 is the sum of the previous 8 ...
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57 views

Continuous function that outputs even and odd numbers from natural inputs in a non-constant, non-alternating order?

For example, plugging 1, 2, 3, 4 into this function would produce results which are even, odd, odd, even, and that pattern would repeat. Is this possible for a continuous function? Can you make any ...
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0answers
109 views

Equation $\frac{1}{a_1}+\ldots +\frac{1}{a_{2018}} = 1$

Let $A_{n}=\{(a_1,a_2,\ldots,a_{n}): a_i\in\mathbb{Z_{>0}}|\ \ \frac{1}{a_1}+\ldots \frac{1}{a_{n}} = 1\}$. My question. What is $|A_n|\operatorname{mod}2$, for $n=2018$? That is what is the ...
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1answer
135 views

Understanding the received vector in syndrome decoding

I have an exercise, which I do have solutions to but cannot understand the problem text. As far as I understood, syndromes are computed by checking the received vector against the parity check matrix. ...
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1answer
127 views

Parity of the result of modular multiplication and modular multiplicative inverse

I want to know the parity of the result of applying modular multiplication and modular multiplicative inverse with a prime modulo knowing only the operands, i.e. without performing the actual ...
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1answer
48 views

Coloring proofs and conditional statements

My question isn't specifically about the problem below, but about coloring proofs in general. From what I've seen about coloring proof problems thus far, I've noticed that we form a conditional and ...
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1answer
66 views

Differential Equation with y(-x)

Please how can I solve $$y''(x)+y'(-x)=e^x$$ I tried everything I could I can't even find the complementary solution Any help would be gladly appreciated Thanks In Advance
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89 views

Dealing with Turing Machines

I have an easy problem with Turing Machines: Let $f$ be a function in $\#P.$ Then there is some polynomial-time TM $M$ such that for every input $x, f (x)$ is the number $\#M(x)$ of strings $u ∈ \{0,1\...