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 $2^{n}+1$ is divisible by $n=3^k$ for $k≥1$

prove that : $2^n+1$ is divisible by all number from : $n=3^k$ for $k≥1$ I find this problems in book and I need ideas to approach it Problems :
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Is it true that $n \leq 2^{n-1}$ for all natural numbers $n > 0$?

Seems to be true but I want to make sure: $n \leq 2^{n-1}, n > 0$
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Prove using induction that $n^6 < 3^n$,for all $n > 18$

Prove using induction (or using any other elementary precalculus techniques) that $$n^6 < 3^n, \forall n \geq 19.$$ I have no idea how to do this. Writing the induction step, I get that I need to prove that $$3n^6 > (n+1)^6,$$ and I don't know how…
user606835
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Distance between point p and origin

How to prove that the distance between the origin and the point $P$ is a natural number, where $P=(n, n+1, n(n+1))$.
jofernando
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Countable set example

Another crazy question. Is this a countable set; $$A = \{-4, -2, -1, 0, 1, 2, 3, 4\}$$ I think it is. But my teacher says it isn’t due to the concept , he says, is of one to one mapping with a some subset of natural number. Since some elements…
Pyr James
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How do I prove that, for any positive integer $n > 2$, $n^{n/2} < n!$

I tried using Induction, but I couldn't prove the inequality. Any proof would work. Rewriting the question for clarity, here is its statement: For any positive integer $n > 2$, prove that $n^{n/2} < n!$
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Can '0.999...' be used instead of "1"?

As we know 0.999... = 1 I wonder in which contexts is it acceptable to use the recurring decimal "0.999..." as it is a representation of the number 'one' can it be used as an alternative to the number 'one', do we draw a distinction between one as…
user1007028
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How can I prove this with induction?

Being $p,\,m\in\mathbb{N}$ such that $p>m\ge1$ and $p$ is not a multiple of $m$. How can I show that there are $q,\,r\in\mathbb{N}$ with $r
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How can I prove 0<1? ,What axioms do I use?

I need to prove 0<1, I'm not able to give any of what I tried because I'm not sure how to prove it, maybe starting by some axioms I would be able to prove it.
Steve12
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Can I write e(mathematical constant) in fraction?

Proof of e can be in fraction e is 2.718281828..... so as you can see, there is a pattern. So, I break it down into 2.7 + 0.01828 + 0.0000001828 ..... and then I use sum of infinity to get the fraction Well, in mathematics theory , e cannot be in…
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Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ that satisfy $2f(m+n) = f(m)f(n)+1$ for all $m,n \in \mathbb{N}$

Problem: Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ that satisfy $2f(m+n) = f(m)f(n)+1$ for all $m,n \in \mathbb{N}$
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Why natural set is an infinite set with each element a finite number?

I can not well understand that the natural set $\mathbb{N}$ is an infinite set (which contains infinite many elements) while each natural number is finite? I already find the same question in Why set of natural numbers is infinite, while each…
X Leo
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