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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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 2) or "odd" (all other numbers). For all…
Justin L.
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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 could post some moves, that I've tried but I don't…
DoctorMoisha
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$\lfloor (2+\sqrt{3})^n \rfloor $ is odd

Let $n$ be a nonnegative integer. Show that $\lfloor (2+\sqrt{3})^n \rfloor $ is odd and that $2^{n+1}$ divides $\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $. My attempt: $$ u_{n}=(2+\sqrt{3})^n+(2-\sqrt{3})^n=\sum_{k=0}^n{n \choose…
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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 , $x=2y+z, \ \{\{x,y,z\}>0\} \in \mathbb{Z}$ . There $n$ is an even number. If…
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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 so on. If the frog can reach all the leaves, show…
user96172
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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 form the subsets $S_1 = \{100 , 64 , 25 , 4\}$ …
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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 $A$ for every $m \in \mathbb{N}$ . I found a proof for…
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Why are an even number of flips required to get back to the original list?

Consider the list of numbers $[1, \cdots, n]$ for some positive integer $n$. Two distinct elements $i$ and $j$ of the list can be switched in a so-called flip. For example, let $f$ be a flip that switches $2$ and $4$. Then $f([1,2,3,4]) =…
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Odd Binomial Coefficients?

By Newton's Formula: $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}\,b^k $$ Prove that$\dbinom{n}{k}$ is odd for every $k=0,\dots, n$ if and only if $n=2^r-1$. I have already shown that if $n$ is of the form $2^r-1$, having used the…
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Using Group Theory to Solve this IMO problem

A few weeks ago, I found a fascinating solution to a USAMO combinatorics problem that used group theory. Look at the 2nd solution on this link to view it. I think there might be a way to use group theory to solve the following IMO problem from…
Jackson
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Differential Equation with y(-x)

Please how can I solve $$y''(x)+y'(-x)=e^x$$ I tried everything I could I can't even find the complementary solution Any help would be gladly appreciated Thanks In Advance
arsene stein
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Do more numbers from 1 to 10000 inclusive have an even or odd sum of their digits?

I have tried using modulo arithmetic with $\bmod9$. However, I have found that it doesn't always help. It seems that for most numbers the parity of their digit sum is the same as the parity of the number $\bmod9$. But this is not always true. For…
John Smith
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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 the same parity, but I'm not sure how to proceed when x…
pickle
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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 axis, and thus the identity theorem implies that…
Ben Nevis
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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 for OEIS sequence A169950: [T]riangle read by rows, in…
Peter Kagey
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