Mathematics Stack Exchange
Mathematics Stack Exchange

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.

In mathematics, parity is a term we use to express if a given integer is even or odd.

  • The parity of a number depends only on its remainder after dividing by $2$.
  • An even number has parity $0$ because the remainder after dividing by $2$ is $0$, while an odd number has parity $1$ because the remainder after dividing by $2$ is $1$.
  • Parity is often useful for verifying whether an equality is true or false by using the parity rules of arithmetic to see whether both sides have the same parity.
  • In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code.
  • Parity is an important idea in quantum mechanics because the wavefunctions which represent particles can behave in different ways upon transformation of the coordinate system which describes them.

References:

https://en.wikipedia.org/wiki/Parity_(mathematics)

http://mathworld.wolfram.com/Parity.html

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Parity of an odd integer

I'm not sure if I'm just tired and I'm missing something obvious, but how come I'm obtaining the following: $$(2m+1)^n=\sum_{k=0}^n \binom{n}{ k} (2m)^k = 2 \sum_{k=0}^n \binom{n}{ k} 2^{k-1}m^k $$ This seems to imply any power of an odd integer is…
Andrew Tawfeek
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Can $n$ be an odd natural number?

Consider $n$$(>1)$ spots created around a circle. A man jumps from one spot to another in the following manner. He starts from some selected spot. From there, he skips exactly one spot in the clockwise direction and jumps to the next one. Then skips…
Coherent Sheaf
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Question on parity from Mathematics Circles book

Forty-five points are chosen along line $AB$, all lying outside of segment $AB$. Prove that the sum of the distances from these points to point $A$ is not equal to the sum of the distances of these points to point $B$. The answer given behind the…
Rishabh Shrivastava
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Prove: $\text{$n$ is even} \iff n^n\equiv 1\mod{(n+1)}$

Prove: $\text{$n$ is even} \iff n^n\equiv 1\mod{(n+1)}$ where $n\in\mathbb{N}$. First, to prove $n^n\equiv 1\mod{(n+1)}\implies\text{$n$ is even}$, I supposed $n^n\equiv 1\mod{(n+1)}$ is true. It goes like this: The supposed proposition could be…
Garmekain
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Is a storage-space-efficient RAID with 3 arbitrary disks failing without data loss possible?

Beforehand: A RAID 1 or RAID 1+0 is not a solution because they're not storage-space-efficient as they don't share the property of RAID 5 and RAID 6 that the number of additional disks needed is constant as the total number of disks increases. In…
UTF-8
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"Mathematical Circles (Russian Experience)", Chapter 1, P5

Three hockey pucks, A, B and C, lie on a playing field. A hockey player hits one of them in such a way that it passes between the other two. He does this 25 times. Can he return to three pucks to their starting position? This question from the book,…
Coherent Sheaf
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If $f(x) = 4x^2 - 4ax + b$ and $a^2-b$ is a perfect square of a rational number then which one is the right statement

If : $f(x) = 4x^2 - 4ax + b$ and $a^2-b$ is a perfect square of a rational number then among the following statements below which one is the right statement : (a)If $a$ is odd , b is even , roots of $f(x) = 0$ are not integer (b)If $a$ is even , b…
RDT
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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. WLOG, suppose $a$ is even, and $b$ is odd. By…
Wng427
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Parity of two numbers if given parity of one of them and their binomal coefficient

If we are given that $n$ is odd and for some $1 \leq k \leq n-1$ we have $n \choose k$ is even. Can we conclude anything about the parity of $k$?
TheNotMe
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Parity of Permutations

Here's a question I'm working on: Show that if $\rho$ and $\sigma$ are any two permutations, then $(a)$ $\sigma\rho$ and $\rho\sigma$ have the same parity and $(b)$ $\sigma$ and $\rho\sigma\rho^{-1}$ have the same parities For the first part of…
John W. Smith
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Prove that the following statements are equivalent. $n^3$ is odd, $n^2$ is odd, $1-n$ is even, $n^2+1$ is even

So I started with if $n^3$ is odd, then $n^2$ is odd and I used a direct proof. $n^3 = n^2 \cdot n$ and the product of 2 odd integers is odd so $n^2$ is odd. Is this an okay proof? Then I have to prove that if $n^2$ is odd then $1-n$ is even,…
Yanni Papaioannou
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Property of Positive/Negative

Is there a word for whether something has the property of being positive or negative such as "parity" for being odd or even?
Alexander Charters
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A question on parity from book "Mathematics Circles"

Three hockey pucks, A,B, and C, lie on a playing field. A hockey player hits one of them in such a way that it passes between the other too.He does this 25 times. Can he return the three pucks to their starting positions.(THIS IS A PART OF QUESTION…
Rishabh Shrivastava
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