# Tagged Questions

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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### 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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### 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 ...
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### Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
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### Is zero odd or even?

Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real ...
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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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### Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
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### 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?
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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 ...
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### 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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### 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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### 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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### Generalizations of the number theory concepts of “even” and “odd”?

One of the very first number theory concepts introduced to students -- even before primeness, divisibility, etc. -- is the idea that a natural number can either be "even" (that is, evenly divisible by ...
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### 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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### 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 and ...
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### Do different notations imply different properties of a number?

I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit. So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### $\left[\frac n1\right]+ \left[\frac n2\right] + \cdots+\left[\frac nn\right]+\left[\sqrt n\right]$ 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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### $(2^a -1)(2^b -1)=2^{2^c}+1$ 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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### 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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### 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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### 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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### 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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### 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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### 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 \left(\left(-1\right)^{k-1}\binom{n}{-\left|...
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 \$n≥...