Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
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$H(n)=\lfloor\dfrac{b}{n}\rfloor- \lfloor \dfrac{a}{n} \rfloor=$ (roughly) # odd pairs $o, o+2 \in [a,b]$ such that $n \mid o$ or $n \mid o+2$
I came up with the following formula and deleted that question so that I don't have two questions on the same formula.
Conjecture. Let $a, b, n \in 2\Bbb{N} + 1$ be odd natural numbers. Then the ...
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How to prove formula related to $2$-adic valuation / $2$-adic absolute value and binary expansion
I would like to prove the following formula, which I have verified for every positive integer $n \ge 1$ up to $n = 10000$:
$$n - \sum_{k=0}^{\lfloor \log_2{n} \rfloor}\left(\left\lfloor\frac{2n-1+2^{...
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Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
I would like to show that
$$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$$
I'm interested in more ways of proofing this.
My method :
suppose that $\sqrt{\frac{n}{n+1}}\in ...
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Why don't we use Presburger's arithmetic instead of Peano's arithmetic?
I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it:
It is decidable, complete and consistent.
It omits multiplication ...
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Is aleph-$0$ a natural number?
Would I be right in saying that $\aleph_0 \in \mathbb N$?
Or would it be a wrong thing to do?
user108343
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Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I)
In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms:
Axiom 3.4 If $a$ is an object, then there exists a set
$...
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Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?
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What's the next base-ten non-pandigital factorial number after 41!?
By pandigital number I mean a number for which each digit in a given base occurs at least once (some definitions that state each digit must occur exactly once), and since I looking for numbers that ...
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A matrix w/integer eigenvalues and trigonometric identity
Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated:
Let $n$ be a natural number.
(a) Consider the following Toeplitz/circulant symmetric matrix:
$...
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Is it correct to say that the natural numbers are a proper subset of the integers?
Is it correct to say that the natural numbers are a proper subset of the integers? $\mathbb{N} \subset \mathbb{Z}$.
Just want to be absolutely sure.
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How can I expand mathematical induction to rational numbers?
I know mathematical induction can be used to prove that a statements is true for all natural numbers (or those belonging to a certain subset of N). However, it is pretty obvious, unless I'm terribly ...
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Define a model for $\mathbb N$ without set theory
I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.
It led me to do a bit of elementary ...
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How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.
While answering a question here I noticed that:
$$n=1+\sum_{k=1}^{n}{\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor}$$
for every natural number $n$.
I tried to demonstrate it using Legendre's ...
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Explicit bijection between $\mathbb N$ and $\mathbb N \times \mathbb N$ [duplicate]
We can consider the quadratic scheme above for a possible explicit bijection between $\mathbb N$ and $\mathbb N \times \mathbb N$.
The part $\mathbb N \times \mathbb N \to \mathbb N$ is easy via $(m,...
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Proof that binomial coefficient is a natural number [duplicate]
Possible Duplicate:
Proof that a Combination is an integer
What is the proof that the binomial coefficient is a natural number?
$$k\ge0,n\ge k \implies {n \choose k} \in N,$$
I guess it's a ...
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