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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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 ...
5
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3answers
54 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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2answers
31 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
81 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.
2
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1answer
77 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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3answers
29 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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2answers
354 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
38 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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0answers
30 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
185 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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0answers
28 views

Integral parity

Is there any rule for a complex function that integral of an odd function should be even? For real functions it is evident, but what about complex functions?
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1answer
47 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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1answer
57 views

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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1answer
124 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 ...
2
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1answer
57 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 ...
2
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1answer
85 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 ...
2
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2answers
95 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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1answer
223 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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1answer
120 views

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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4answers
200 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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0answers
88 views

Finding a parity check matrix of a binary code

I'm supposed to find a parity check matrix of a binary [6,3,3] code. Given a generator matrix G I can find a parity check matrix by row reducing until I get the identity matrix, then take $-A^{\top} ...
2
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0answers
50 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): $ ...
3
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2answers
216 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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0answers
48 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 ...
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2answers
601 views

Frog Jump Problem

Lotus leaves are arranged around a circle. A Frog starts jumping from one leaf in the manner described below. In the first jump it skips one leaf,next jump it skips two,three the next jump ...
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3answers
38 views

Series representation of $\sin(nu)$ when $n$ is an odd integer?

So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer: $$\sin(nu) = \sum_{k=1}^\frac n2 ...
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2answers
45 views

$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions

$(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers. Any solutions using parity Approach: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
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1answer
391 views

Proof by induction involving fibonacci numbers

The Fibonacci numbers are defined as follows: $f_0=0$, $f_1=1$, and $f_n=f_{n−1}+f_{n−2}$ for $n≥2$. Prove each of the following three claims: (i) For each $n≥0$, $f_{3n}$ is even. (ii) For each ...
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2answers
38 views

If $f$ is of given parity, what can be said of its derivative and primitive?

If $f$ is of given parity, what can be said of it's derivative and primitive? Clearly for power functions or simple trigonometric functions it seems that the parity of the derivative and ...
2
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3answers
72 views

Prove that if $n^{2} - \left(n - 2\right)^{2}$ is not divisible by $8$ then $n$ is even

Let $n$ be an integer. Prove that if $n^{2} -\left(n - 2\right)^{2}$ is not divisible by $8$ then $n$ is even. Can anyone help me step by step to understand this?
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3answers
68 views

Is it right to say that: if $2a+1=2b$ we have a contradiction?

I am trying to prove by contradiction and I have reached the conclusion that $2a+1=2b$. Now I am tempted to say it's a contradiction and call it a night. Is it a contradiction? because one is even and ...
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2answers
227 views

Hamming Code Error Detection

I am learning few things about hamming code and error detection so my question may sound stupid. So i know that lets i ahve (7,4) hamming code and i made transpose of parity check matrix H(t). Now say ...
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0answers
95 views

parity of powers of prime factors

lets consider the prime factorisation of a number N let the powers of the primes in this factorisation be a,b,c ....and so on. Is there a way to determine whether the number of powers that are even ...
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3answers
376 views

Knight on a chessboard moving from a1 to h8

I was given a puzzle to solve which goes as:- Can a knight start at square a1 of a chessboard, and go to square h8, visiting each of the remaining squares once on the way ? I reasoned that this won't ...
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1answer
105 views

Infinite Parity Function

I was looking at this problem, and I have a solution for a finite board with $2^n$ squares, that I want to extend to a countably infinite board. Label the squares from $0$ to $2^n-1$. Consider the ...
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4answers
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$\sqrt 2$ is even?

Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ ...
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1answer
130 views

An intutive explanation of natural density (asymptotic density)

I was wondering if someone can provide an intuitive explanation to natural density. I understand the concept very basically (pretty much the definition) but I can't seem to understand what natural ...
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2answers
533 views

Odd or even permutation with matrices

I know that the number of transpositions would determine the parity of a permutation like: A = (1,2,3,4,5) = (1,5),(1,4),(1,3),(1,2) = even But how would that apply to a matrix? Example: 1 2 ...
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1answer
60 views

Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?

In a section on fourier transforms, my textbook contains these steps for an example: $$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$ $$= ...
6
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3answers
146 views

Is odd continuous function differentiable at $x=0$?

Suppose that $f(x)$ is continuous and odd: $f(-x) = - f(x)$. Does it have a derivative at $x=0$? Here is what I got so far: First we calculate $f(0)$ using $f(-0) = -f(0)$, from which $f(0) = 0$. ...
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1answer
1k views

Obtaining a generator polynomial from a parity check matrix for a binary cyclic code

In general, what is the strategy for obtaining a generator polynomial (and a check polynomial) given a parity check matrix $H$ for a binary cyclic code? Things I know: Each codeword $c(x)$ ...
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0answers
154 views

How should I interpret Johnson's “Note on the '15' puzzle”?

Following the suggestion of Gerry Myerson who commented below, I went ahead and read Johnson's "Note on the '15' Puzzle". Bearing in mind that I am not well versed in the English of 19th century ...
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2answers
159 views

Similarity between integer and logical operations through parity

Lets observe the parity property of integers while adding them or multiplying. It's simple to notice that when we add two numbers, the parity of the result depends on parity of summands: ...
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766 views

Proving a statement regarding a Diophantine equation

FINAL EDIT : Prove that if $p^z|n^2-1$ $$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$. Here $p>3$ is an odd prime , $x=2y+z, \ ...
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$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?

The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$ Where: n,k are natural numbers and $k\le n$. t is taken over all ...
5
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1answer
246 views

Why is this function odd?

Suppose a complex valued function $f$ is entire, maps $\mathbb{R}$ to $\mathbb{R}$, and maps the imaginary axis into the imaginary axis. I see that $f(x)=\overline{f(\bar{x})}$ on the whole real ...
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8answers
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Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may ...
25
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5answers
16k views

Can decimal numbers be considered “even” or “odd”?

Is the concept of even/odd numbers is applicable to decimal numbers? For e.g. - 4.222 is a even number?
195
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4answers
13k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
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2answers
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Parity and Inverse of Permutations (Odd and Even)

I want an explanation on knowing how to know whether a permutation is odd or even. For example, I have a few permutations of [9] that I need explained for parity, inverse, and number of inversions if ...