Questions tagged [natural-numbers]

For question about natural numbers $\Bbb N$, their properties and applications

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of and , respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to , such as the distribution of , are studied in . Problems concerning counting and ordering, such as partition enumeration, are studied in .

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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?
jon jones
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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 fast computation of Bernoulli numbers. Here we have…
Luckymaster
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Examples of polynomial injections $f:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$

I've seen that there are polynomial bijections $f:\mathbb{N}\times \mathbb{N}\to \mathbb{N},$ for example $f(m,n)=\frac{1}{2}(n+m)(n+m-1)+m.$ I'm looking for more examples of injective polynomials from $\mathbb{N}\times \mathbb{N}\to \mathbb{N}.$…
subrosar
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Finding all functions $f: \mathbb{Z^{+}}\to \mathbb{Z^{+}}$ such that $f(a)| (f(b)+a-b)$ for all $a,b\in \mathbb{Z^{+}}$

I managed to find the two trivial solutions - that is $f(n)=n$ and $f(n)=1$. I am not sure if there are any more. How to go about finding the full set of solutions and proving it is complete? Thank you for your help
nowepas
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Consistency of Peano axioms (Hilbert's second problem)?

(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary mathematicians believe that Peano's axioms are consistent,…
miku
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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 entirely. It is weaker than Peano's…
Red Banana
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$\mathbb{Z}^{+}$ includes zero or not?

Does $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer.
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The number of ways to represent a natural number as the sum of three different natural numbers

Prove that the number of ways to represent a natural number $n$ as the sum of three different natural numbers is equal to $$\left[\frac{n^2-6n+12}{12}\right].$$ It was in our meeting a year ago, but I forgot, how I proved it. Let the needed number…
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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 are not pandigital in base ten at least one of the…
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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 head. So my question is pretty straight-forward, why…
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Prove that $4$ is the only solution to $2+2$.

This question was featured on Saturday Morning Breakfast Cereal and I haven't been able to find a proof. Can anyone help?
user119137
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The history of set-theoretic definitions of $\mathbb N$

What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages? I read about Frege's definition not long ago, which involves iterating over all other members of the…
Charles
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Extending the primes

I had an idea and I'd like to find out whether it has a name or has been studied before. Imagine the natural numbers and the operations of addition and multiplication, but with the following restriction: multiplication can only be carried out $d$…
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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: $$ \Lambda_{kl} = \begin{cases}\frac{2n(n+1)}{3}, &…
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