The parity tag has no wiki summary.
153
votes
4answers
8k 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 ...
65
votes
8answers
6k views
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 ...
32
votes
8answers
2k views
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 ...
24
votes
7answers
2k views
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!
10
votes
0answers
247 views
$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - 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 ...
7
votes
3answers
840 views
$C(n,p)$: even or odd?
Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value?
($p\lt n$, $p$ and $n$ are positive integers)
7
votes
3answers
633 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, \ ...
6
votes
3answers
104 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 $f(0) = 0$.
...
5
votes
2answers
570 views
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 ...
4
votes
1answer
154 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 ...
2
votes
2answers
603 views
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 ...
2
votes
3answers
160 views
Parity confusion
I am confused. I have to show that $f(x)$ that satisfies the ODE below "has definite parity or can be chosen to have def parity".
$$\frac{d^2}{dx^2}f(x) = (h(x)+c)f(x)$$ where $h(x)$ has even parity ...
2
votes
2answers
86 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:
...
1
vote
2answers
53 views
Linear independence of linear combination of functions
This question is inspired by this question. Is it always true that if $\{f_n(x)\}_n$ are linearly independent then so is $\{f_n(x)+f_n(-x)\}$ given that each $f_n(x)$ do not have definite parity?
1
vote
1answer
33 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 ...
1
vote
2answers
26 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 ...
1
vote
1answer
34 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:
$$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$
$$= ...
1
vote
1answer
310 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 $c(x)$ ...
1
vote
0answers
94 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 ...
0
votes
1answer
64 views
Parity, Set of functions
Given linearly independent $\{f_n\}_{n=1}^N$ how can we form $n$ linearly independent functions $\{F_n\}_{n=1}^N$ such that each $F_n$ is either an even or odd function? Thanks.
-2
votes
2answers
335 views
Is $0$ an even number? [duplicate]
Possible Duplicate:
Is zero odd or even?
Is $0$ an odd number?
$1$ is an odd number. $10$ is an even number.
NOTE: There's nothing in $0$ which can be divided to check for evenness or ...
