Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
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Why does induction only allow numbers connected to $0$ to be natural?
When learning Peano axioms normally we use induction to suggest that if a property $P(0)$ is true, and if $P(k)$ being true implies $P(k + 1)$ being true (for natural number $k$), then if $P$ is the ...
3
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Solve in $\mathbb N^{2}$ the following equation : $5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
Question :
Solve for natural number the equation :
$5^{2x}-3\cdot2^{2y}+5^{x}2^{y-1}-2^{y-1}-2\cdot5^{x}+1=0$
My try :
Let : $X=5^{x}$ and $Y=2^{y}$ so above equation
equivalent :
$2X^{2}+(...
- 2,323
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Show natural numbers ordered by divisibility is a distributive lattice.
I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with $\operatorname{gcd}$ as "$\wedge$" and $\operatorname{lcm}$ as "$\vee$".
I ...
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Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers
I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).
I've already constructed two non-isomorphic total ...
- 31
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Addition of natural numbers in Edmund Landau's Foundation of Analysis
I am reading the proof of addition of numbers.
In the proof author first shows uniqueness of $x+y$ and then the existence of plus operation with the above listed properties.
The second proof is as ...
- 273
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2
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Proof of the cancellation law for natural numbers without the axiom of the succesor
We are to show $\forall m,n,p\in \mathbb{N_{0}}:m+p=n+p\implies m=n$.
Proof. Let $m,n\in \mathbb{N_{0}}$.
Base case: If $p=0$, then $m+p=m$ and $n+p=n$ which clearly establishes $p=0\implies (m+p=n+p\...
user923938
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The best symbol for non-negative integers?
I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable?
$\mathbb{N}_0$
$\mathbb{N}\cup\{0\}$
$\mathbb{Z}...
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4
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Sum of sum of $k$th power of first $n$ natural numbers.
I was working on a problem which involves computation of $k$-th power of first $n$ natural numbers.
Say $f(n) = 1^k+2^k+3^k+\cdots+n^k$ we can compute $f(n)$ by using Faulhaber's Triangle also by ...
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What is the meaning of set-theoretic notation {}=0 and {{}}=1?
I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
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How are the elementary arithmetics defined?
In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that "minus"...
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Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
I try to prove the following formula:
$$n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$$
I noticed that
$\upsilon_{p}(\operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)) = s$ iff $\...
- 119
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Proving $n \leq 3^{n/3}$ for $n \geq 0$ via the Well-Ordering Principle
I'm attempting to prove:
$$n \leq 3^{n/3} \quad \text{for }n \geq 0$$
I'm having a little trouble continuing. This is what I have so far:
Suppose for a contradiction there is a subset of nonnegative ...
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Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following:
...
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Proof that binomial coefficient is a natural number [duplicate]
Possible Duplicate:
Proof that a Combination is an integer
What is the proof that the binomial coefficient is a natural number?
$$k\ge0,n\ge k \implies {n \choose k} \in N,$$
I guess it's a ...
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1
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Determine the divisibility of a given number without performing full division
My question is slightly more complicated than what's implied on the title, so I will start with an example. Given any number $N$ on base $10$, we can easily determine whether or not $N$ is divisible ...
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