Parity is a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is 'evenly divisible' by two and odd if it is not even. (Def: http://en.m.wikipedia.org/wiki/Parity_(mathematics))

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If $\sigma(N)$ is odd, $N = 2y^2$, and $y$ is not a power of two, does it follow that $\gcd(2,y) = 1$?

Let $\sigma(N)$ denote the sum of the divisors of the number $N$. It is well-known that $$\sigma(N) \equiv 1 \pmod 2 \iff \left\{\{N = x^2\} \lor \{N = 2y^2\}\right\}.$$ Here is my question: If ...
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5answers
77 views

Proof that if $(n+1)^2 -1$ is even then $n$ is even?

The forward implication, if $n$ is even then $(n+1)^2 -1$ is even, was simple. I can't figure out the other implication: if $(n+1)^2 -1$ is even then $n$ is even. What type of proof do I want to ...
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32 views

General results on the change of the parity of a number by repeatedly dividing by 2 [closed]

I know this question may seem open, but I'm a bit interested in figuring out, getting some ideas, or at least getting some sources on how the parity of a number is affected by repeatedly diving it by ...
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17 views

Can I find number of “even” subarrays in less than $O(n^2)$?

You have an array called $A$ of length $n$ (indexed from $0$), filled with numbers from interval $[1,n]$, not necessarily different. You need to know how many subarrays this array has in which every ...
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2answers
29 views

Question on Parity

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 ...
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19 views

How to find the base and the dimension of the space? [on hold]

I understand this task as inserting for n is equal and n is odd. Then, probably write down related polynoms.t Am I right ? Natural number is >3 and we have a differnet parity and inital "record": ...
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1answer
46 views

is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
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1answer
33 views

Why is $\int_{-∞}^∞ x^k*e^x/(e^x +1)^2 dx = 0$ for odd k?

Here's what I've considered: $e^x$ and $(e^x + 1)^2$ are of no parity (neither odd nor even functions). $e^k$ for odd k is an odd function. I've always thought one could only multiply functions of ...
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42 views

Regular grammar with parity

Give a regular grammar that generates the set of strings over {a, b, c} with an odd number of occurrences of the substring bc. How can you limit the number of recursions for a regular grammar to be a ...
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3answers
75 views

Prove that an odd integer is always of the form $2m+1$

I have that by definition an integer $n$ is even if $n = 2m$ for some integer $m$. By definition an integer is odd if it is not even. I would like to prove that if $n$ is odd, then $n = 2m + 1$ for ...
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63 views

Even-odd multiplicative algebraic structure with idempotency? [closed]

What is the algebraic structure for the multiplications of even elements and odd elements? Please notice that $o*o=o$, $e*e=e$ (idempotency) and $o\not = e$. 1st structure is such that even times ...
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1answer
28 views

Determine the parity and the inverse for each of the following permutation of {1, 2, . . . , 9}:

So i was given this question. Determine the parity and the inverse for each of the following permutation of {1, 2, . . . , 9}: (a) (987654321) (b) (135792468) I don't understand how to go about ...
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1answer
39 views

Find all permutations such that $\sigma=a\tau a^{-1}$

For (b) and (c) we note that $\sigma$ and $\tau$ have different parity so there cannot be any $a\in S_4$ that will fix that parity mismatch. For (a) we have the cycle $a^{-1}=(3 2 4)$ and it is ...
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22 views

decoding a block code in non systematic form

I've a Generator matrix in Gf(2) in non systematic form( no seperate, data bits, and no separate parity). It is a full rank matrix. Now to find H[7x20] from this matrix I used the property ...
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2answers
28 views

8-puzzle which has the numbers in order but has gap in between

If the numbers of the 8-puzzle are all in order, but the blank tile is somewhere in between, is this puzzle solvable? (for example, $\begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 6 & 7 ...
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1answer
91 views

Induction to prove parity

Let x1,…,xn be binary variables, i.e. they can be either 0 or 1. Prove by induction that parity(x1,…,xn) = x1 ⊕⋅⋅⋅⊕ xn, where ⊕ is exclusive or. The parity function returns 1 when the number of 1s in ...
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31 views

Deriving explicit formula from recursive function

I have a recursive function R that is defined as follows: R(0) = 1 R(1) = 1 R(2) = 2 R(2n) = R(n) + R(n + 1) + n (for n > 1) R(2n + 1) = R(n - 1) + R(n) + 1 (for n >= 1) Is it possible to define a ...
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5answers
521 views

Prove that sum $(\sqrt2+1)^n+(\sqrt2-1)^n$ is rational for even numbers

Let $n \in N$ Prove that $(\sqrt2+1)^n+(\sqrt2-1)^n$ is rational iff $n$ is even I have tried to do it in induction but got stuck... Any ideas for how to solve this? Thanks
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7answers
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How to prove $\frac xy + \frac yx \ge 2$

I am practicing some homework and I'm stumped. The question asks you to prove that $x \in Z^+, y \in Z^+$ $\frac xy + \frac yx \ge 2$ So I started by proving that this is true when x and y have ...
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5answers
109 views

Prove: If $n^2$ is odd, then $n$ is odd. [duplicate]

$n$ is a natural number. I want to prove that, if the square of $n$ is odd, then $n$ itself is odd. Any hints welcome and preferred. Thank you!
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7answers
94 views

$n^2 + 7n + 1$ is odd

Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd. I have $n=2k+1$ for some $k\in Z$ I really do not how to do this problem. any help in understanding would be greatly appreciated.
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170 views

Prove that it is not possible to completely cover a 6 × 6 chessboard by tiles which have dimensions 1 × 4.

I think I have some sort of understanding of how to solve this but I'm not sure. I would colour the board with 4 colours such that every 1x4 rectangle would cover one of each colour. Then cover the ...
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79 views

Maximal $k$ such that ${{n+k+1}\choose{n+1}}$ is odd

How to compute the maximal $k$ such that $ 1\leq k\leq n $ and $$ {{n+k+1}\choose{n+1}} $$ is odd? Given $n$, suppose this maximal number is $f(n)$. I want to obtain the explicit expression of ...
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Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
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1answer
86 views

$\left[\frac n1\right]+ \left[\frac n2\right] + \cdots+\left[\frac nn\right]+\left[\sqrt n\right]$ is even [duplicate]

Let $n$ be any natural number. Prove that $\left[\dfrac n1\right]+ \left[\dfrac n2\right] + \left[\dfrac n3\right]+\cdots+\left[\dfrac nn\right]+\left[\sqrt n\right]$ is even. I tried this by ...
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Proving that all integers are even or odd [duplicate]

I know that $\mathbb{Z}$ is a group under addition with a multiplication defined. I have just the definition of even and odd integers: $n$ is even if $n = 2k$ for some integer $k$ and $n$ is odd if $n ...
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1answer
64 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$, for a fixed $p$. I will consider only prime fields, not $GF(p^n)$. Represent the $p$ elements of the field as integers $\{0,1,\ldots ...
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Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
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107 views

Odd and Even Parity in Proofs

The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points. For example, ...
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153 views

How to determine the parity of a function about a specific point.

A student asked me to demonstrate why a specific function $f(x)$ was odd about a point $a$. I realized that I never had actually formalized, let alone proven such a claim, but had rather rode the ...
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1answer
102 views

How to convince people that 0 is even [duplicate]

Some people say that 0 is neither even nor odd. I say that 0 is even. Is there a simple way to convince people that 0 is even and the statement that "0 is neither even nor odd" is false.
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1answer
408 views

Prove by induction that numbers of Fibonacci of the form $F_{3n}$ are even

I am not quite asking to solve this problem, but I am asking what they are asking me. This is the problem: The Fibonacci numbers $F_n$ for $n \in \mathbb{N}$ are defined by $F_0 = 0$, $F_1 = 1$, ...
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32 views

What function would do this

I need a function $f(n)$ that if $n$ is odd it returns $-1$, but if it is even it returns $1$. Is there a function like this? If so what is it? Also I would appreciate if it is not a trig function ...
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370 views

Sum of series with binary parity in the numerator

I'm now stuck with this question, and I don't even know where to start: Find sum of series$$\sum_1^\infty \frac{f(n)}{n(n+1)}$$, where f(n) - number of ones in binary representation of n. I wish I ...
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1answer
48 views

The parity of the function $f(x)=x^{2^{x^2}}$?

Someone claims that $f(x)=x^{2^{x^2}}$ is an even function which I highly doubt. I claim that $f(-0.5)=(-0.5)^{\sqrt[4]{2}}$ which is an imaginary number and $f(0.5)$ is a real number. Can I have ...
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67 views

The parity of the sum of a multiset (of integers) and of the cardinality of the odd submultiset are equal

This is a fairly simple concept, yet surprisingly difficult [for me] to state simply. Given a multiset of integers, $M$, of finite (but arbitrary) cardinality. And, $S_m$ is the sum over the ...
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3answers
11k views

Is zero an even number? [duplicate]

A quick google returns the answer on the parity of zero: Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. The simplest way to prove that zero is ...
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4answers
184 views

Is this attempt of showing the square root of two is irrational valid?

Any odd integer squared is always odd, and likewise, any even integer squared is always even. Therefore, the square root of an odd number must be odd and the square root of an even number must be ...
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55 views

Equal number of black and white neighbors in circle

There are $n\geq 12$ points on a circle, each colored black or white. When we consider the $5$ neighbors to the left of any point and $5$ to the right, then among the $10$ points, exactly $5$ are ...
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Erasing numbers from circle and writing down sum

There are $50$ copies of the number $1$, and $50$ copies of the number $-1$, written alternately in a circle. In each step, we pick an arbitrary number, write down the sum of the number and its two ...
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758 views

Generator matrix of a binary cyclic code

I need to find the Generator and Parity check matrix of a binary cyclic [9,2] code. If I calculated right, the Generator polynomial is x^7 + x^6 + x^4 + x^3 + x + 1 and the check polynomial is x^2 - x ...
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1answer
69 views

Parity Bit Detecting Odd Bit Errors

I'm going over a past paper which has a true or false question with the following statement. A single parity bit computed over 128 data bits can detect an error when bit-flips occur in exactly 93 ...
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1answer
262 views

Undetected Errors in 2 Dimensional Parity

Given a two dimensional parity with a data block of two rows and two columns what is the probability that a four bit error goes undetected? The naive method would be to look at all ways in which an ...
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182 views

Prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous

How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin. Pertinent definitions: ...
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879 views

Palindromic binary words with even or odd amounts of letters

Say you have one bag of apples and one bag of oranges. Each bag contains at least one fruit, and the number of fruit in each bag is an odd number. I have found that it is impossible to line up these ...
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Coset Leaders and Syndromes

This is my Parity check matrix For Coset Leader 100010 Syndrome is 101 Could any one help me with the procedure, since I figured ...
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976 views

Can decimals/fractions be odd or even? [duplicate]

At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
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61 views

Number of divisiors of $n$ less than $m$

I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ ...
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558 views

Why is the cube-root of $x$ 'odd'?

I am trying to understand why $\sqrt[3]{x}$ is an odd function; can anyone explain how I could come to this conclusion?
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56 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...