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1
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1answer
58 views

Why does an odd number plus one, not necessarily entail it being even?

Why does an odd number plus one, not necessarily entail it being even? For example, sqrt(5) + 1 is not even.
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3answers
30 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
41 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: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
0
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2answers
22 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 ...
0
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3answers
62 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 ...
0
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2answers
112 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 ...
0
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0answers
69 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 ...
4
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3answers
208 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 ...
2
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1answer
69 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 ...
18
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4answers
1k views

$\sqrt 2$ 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 $\sqrt [n]{2}$ ...
1
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1answer
76 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
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2answers
272 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
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1answer
53 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
134 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 $f(0) = 0$. ...
1
vote
1answer
565 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
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0answers
125 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 ...
2
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2answers
121 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
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2answers
386 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 ...
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3answers
729 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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0answers
396 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 ...
4
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1answer
190 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 ...
77
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8answers
8k 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 ...
172
votes
4answers
10k 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 ...
2
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2answers
1k 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 ...
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2answers
59 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?
0
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1answer
73 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
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3answers
208 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 ...
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7answers
3k 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!
5
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2answers
934 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 ...
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10answers
5k 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 ...
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3answers
1k 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)