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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60 views

A question regarding cyclic codes [closed]

Let $C$ be the smallest cyclic code over $F_3$, which contains the word $11020201$. A.What is the generating polynomial $g(x)$ of the code $C$ ? B. Write the generating matrix and the parity check ...
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49 views

Is it possible to determine if $\frac{xy^2}{2}$ is an even number?

The problem Given $x, y \in \mathbb{Z}$, is it possible to determine if $\frac{xy^2}{2}$ is an even number? $x$ and $y$ are consecutive numbers and $x$ is even. My attempt Assuming $n$ is an integer, ...
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1answer
33 views

Function that maps any given positive even number and its subsequent (odd) integer to the same value? [closed]

I’m looking for a function $f$ such that $f(2n)$ is uniquely equal to $f(2n+1), n \in \mathbb{Z}^+$. The only one I came up with is the floor function, i.e. $f(x)= \lfloor{\frac{x}{2}}\rfloor$, but ...
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108 views

A proof of the theorem $1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ with $k$ odd and $m$ even

I tried to prove the following theorem : If $n>1$ then $\displaystyle1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ where $k$ is odd and $m$ is even. and I'd like to know if there is any flaw in ...
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1answer
40 views

1000 divisors, alternating parity, many digits

Prove that any positive integer with exactly $1000$ divisors, which when arranged in increasing order have alternating parity (so the first divisor is odd, the second is even, the third is odd, etc. ...
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1answer
47 views

Using parity properties to evaluate the inner product of spherical harmonics

I would like to know how to argue if the inner product of two spherical harmonics is zero using symmetry arguments. If the inner product is given by the following integral, $$\left\langle Y_{\ell}^{m},...
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1answer
132 views

Rearranging a $10 \times 10$ matrix of naturals $1\le n\le 100$ s.t. the sum of every two neighbouring numbers is composite in max. 35 steps

We are given a $10 \times 10$ matrix which contains every natural number between $1$ and $100$ in arbitrary order. We are to prove that it is always possible to rearrange the matrix by swapping any ...
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1answer
85 views

The Sierpiński triangle and the number of $(0,1)$-polynomials $p(x)$ where $p(x)^2$ has largest coefficient $k$.

My Twitter bot @oeisTriangles randomly selects an OEIS "table"-style sequence and draws an image, where even terms are light-colored and odd terms are dark-colored. Today it tweeted an image ...
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2answers
62 views

Direct Proof of “If $5x^2+8$ is odd then $x$ is odd” [closed]

From my basic understanding of proofs, the best way to tackle this proof is by contrapositive which I have done but I need it to be using direct proof. Also, there is a hint in the question stating &...
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(notation question) what does $k//2$ means as in $\tilde{x}=\sum(-1)^{k // 2} x_{k}$?

I found a notation this notation in this result: (The reverse involution) If $x=\sum x_{k}\left(x_{k} \in \wedge^{k} E \right)$ is a multivector, we define $\tilde{x}=\sum(-1)^{k // 2} x_{k}(k / / 2=\...
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2answers
72 views

A rectangle P is divided into smaller rectangles by segments parallel to its sides. We call a point a t-point if its a vertex of two small rectangles. [closed]

A rectangle P is divided by segments parallel to its sides into smaller rectangles. We call a point a t-point if it is a vertex of exactly two such small rectangles. Prove that the number of t-points ...
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3answers
98 views

Function to Check the Parity of an Integer

Is there any function, f(x), that outputs whether a given integer is odd or even? For example, if x were 1237, the function would output 1; and if it were 80, it would output 2. I know this is quite ...
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3answers
87 views

Could $(2n-1)\left( \dfrac{1}{a}- \dfrac{1}{b} \right) $ be an integer?

Let $ a,b \in \mathbb{R}_{>0} \backslash \left \{ \tfrac{2N-1}{2M-1} : N,M \in \mathbb{N} \right \} $ and $n \in \mathbb{N}$. I want to prove that the quantity \begin{equation}\label{key} (2n-1)\...
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0answers
47 views

Parity of (unique) number of prime factors

Analysis on the parity of the number of prime factors of an integer $n$ was performed. Here I displayed the parity as an random walk till $n=100.000.000$: Parity of the total number of prime factors (...
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3answers
107 views

How would I prove that if 11n-5 is odd, then n is an even number using only a direct proof?

I've been able to prove this statement through contraposition and contradiction but I'm struggling to prove it through a direct proof. It seems I always get it in the form where 11n=2(k+3).
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2answers
76 views

Trouble calculating a integral

I've difficulties calculating the following integral $$\int_0^\infty y^k\frac{1}{\sqrt{2\pi}y}e^{\frac{-(\ln y)^2}{2}}[\sin (2\pi \ln y)]\,dy$$ This integral is a part of a density distribution of a ...
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4answers
108 views

Prove that every natural number is either even or odd using induction

It is a pretty basic question at first sight. It seems to be intuitively correct but I can't figure out a way to prove it. I think we have to use strong induction but even after sitting on this ...
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4answers
101 views

Improper even integral

I have this improper integral: $$I = \int_{-\infty}^{\infty} x^4 e^{\frac{-x^2}{2t}} \ dx$$ I used a substitution to get ($u=\frac{x^2}{2t}$): $$I = \int_{-\infty}^{\infty} u^{\frac{3}{2}} e^{-u} \ du$...
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373 views

Confirming a easy proof: the product of two consecutive numbers is always even.

Can someone confirm if my prove is right? Theorem. The product of two consecutive integers is always even. Proof. Define a number $n$ such that $n:=2k$ where $k\in\mathbb{Z}$, this ensures that $n$ is ...
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A question regarding parity involving spherical coordinate

Any quantity that changes sign under parity is called odd under parity and if it does not then it is even under parity. The basis vectors for spherical polar co-ordinate are {$e_r, e_θ, e_\phi$}. ...
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Review of parity of integers acquaintances proof from “Permutations and the 15-Puzzle” by Peter Trapa (2004)

I am reading the paper Permutations and the 15-Puzzle by Peter Trapa (2004). Section 2. Parity of integers says the following: Let's start by considering the integers $\mathbb{Z}$. One of the ...
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1answer
76 views

Parity relation for Legendre polynomials [closed]

Can anybody please tell me the parity relation for $(-1)^{-n} P_n(x)$. Note that $P_n(x)$ is the $n$th Legendre polynomial.
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1answer
191 views

Finding cosets and coset leaders of [6,3] code

I was given a generator matrix: $$ G =\begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{bmatrix} ...
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1answer
135 views

On the existence of solutions to two integer quadratic equations of the same “parity”

Let $a,b,c,d\in\mathbb{Z}$ such that $b\mid a^2+1$, $d\mid c^2+1$, $a\equiv c\,(2), b\equiv d\,(2)$. I have the equations $$b^2 x^2-2abxy+y^2(a^2+1)=b,\quad d^2 x^2-2cdxy+y^2(c^2+1)=d$$ and I am ...
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2answers
90 views

Proving by contradiction check if correct

Consider the statement: For all x, y, z ∈ Z. At least one of x− y, x− z and y − z is even I want to check if my proof is correct via contradiction: Suppose(for a contradiction) that all of x-y, x-z, ...
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5answers
143 views

Proving by contradiction

Consider the statement: For all $x, y, z ∈ Z$. At least one of $x−y$, $x−z$ and $y−z$ is even. Prove this statement by contradiction. So the contradictory statement would be that neither one of $x−...
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1answer
49 views

Is there a way to prove that the results of quadratic equation are always odd?

Given the following quadratic equation: $$f(x) = x^2-3x-1$$ For $x, 1\leq{x}\leq{100}, x \equiv 1\mod 2$ Is this true for all $x \in{\mathbb{Z}}$? If so, is there a way to prove it?
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72 views

What is the proof of graded Jacobi identity?

$\newcommand{\parcir}[2]{\frac{\partial^R #1}{\partial #2}}$ $\newcommand{\parcil}[2]{\frac{\partial^L #1}{\partial #2}}$ $\newcommand{\vprcir}[2]{\frac{\overleftarrow{\partial} #1}{\partial #2}}$ $\...
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1answer
307 views

Parity of spherical harmonics

I would like to proof $Y_{\ell m}(-\mathbf{r}) = (-1)^\ell\, Y_{\ell m}(\mathbf{r})$. In this formula, $Y_{\ell m}$ are the spherical harmonics given by \begin{equation} Y_{\ell m}(\theta, \varphi) = \...
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93 views

Symbolic definition for $i$?

Please help! I am trying to find a definition for $i$ that doesn't work for $-i$. let $j$ be either $i$ or $-i$. saying $j^2=-1$ doesn't help since $(i)^2=-1$ and $(-i)^2 = -1$ saying $j=\sqrt{-1}$ ...
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50 views

prove that if $m$ and $n$ are odd integers, then $mn+2$ is odd.

I need to prove that if both $ m$ and $n$ are odd integers, then $mn+2$ is odd. I found several similar answers for a problem that asks if $m$ and $n$ are odd the prove that $mn$ is odd as well. This ...
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30 views

Which of partitions of 5 correspond exclusively to even permutations?

I am ultimately want to prove that $A_{5}$ is simple and the first step in doing so is to: $(a)$ Write out all partitions of $5.$ Which of these correspond exclusively to even permutations? I was able ...
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1answer
46 views

Is it possible to construct an odd function and an even function so that their sum is a constant?

Is it possible to construct an odd function and an even function so that their sum is a constant (let's say in interval $(-1, 1)$)? Personally, I feel it is impossible.
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1answer
143 views

Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. Can you make all labels divisible by 3?

Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. You’re allowed to change the labels by picking an edge of the cube, and adding 1 to the labels of both of its endpoints. ...
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Determinant game: a generalization of Putnam 2008, A2 [duplicate]

The initial question is: Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real ...
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224 views

For $n\ge 6$, can we partition the set $\{1 , 4 , 9 , …,n^2\}$ into two subsets whose sums are equal or differ by one?

For $n\ge 6$, can we partition the set $\{1 , 4 , 9 , ...,n^2\}$ into two subsets such that the sums of the elements in the two subsets are equal or differ by one? For example : for $n = 10$, we can ...
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2answers
158 views

Prove that $x^2 + 6x + 9$ is even iff $x$ is odd

Prove. $x^2 + 6x + 9$ is even iff $x$ is odd. I had first proved this using a contrapositive statement. However, I was told I was not making a distinction between an if and only if statement, and a ...
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29 views

A closed path is made up of 11 line segments. Can one line… [duplicate]

A closed path is made up of 11 line segment. Can one line, not containing a vertex of the path, intersect each of its segments? Can someone please explain the answer to this parity question to me with ...
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36 views

If $S=(s_1,s_2,\ldots)$ is defined by $s_ i=\max\{j\in\mathbb{Z}_{\ge0}|2^j\text{ divides }i\}$, then no subsequence of $S$ can appear twice in a row.

Let $S = (s_1, s_2, s_3, \ldots)$ be an infinite sequence where $$s_ i = \max\big\{j \in \mathbb{N} \cup \{0\} \big|2^j\text{ divides }i\big\}\,.$$ I need to prove (i'm quite sure it's true) that a ...
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3answers
98 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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1answer
68 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
110 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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2answers
76 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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2answers
608 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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1answer
48 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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40 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
62 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
128 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
563 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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1answer
374 views

Conjecture : an odd perfect square $n>1$ raised to the $m$-th 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 $n^m$ is never divisible by $A$ ...

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