# Questions tagged [natural-numbers]

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

92 questions
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### Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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### Similarities of CA Rule 150 and Odd Collatz-function outputs

I made a "discovery" a couple of weeks ago in regards to the first iterates of (odd) numbers of the form $2^n-1$ where $n\in\mathbb{N}$. First iterates is a bit loose term here; what I mean is all of ...
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### natural number reorder problem

Suppose the original natural numbers are sorted as 1, 2, ..., N. The distances of two neighbors are 1. Is there any method to reorder the natural number list to maximize the distance of ALL neighbors? ...
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### General Solution of Linear Diophantine Equation with 2 and more than 2 variable

We know the theorem of linear diophantine equation from Bezout's identity that the solution is in ordered pair form: $\left(x + m \dfrac{b}{\text{gcd}(a,b)}\,,\,y-m \dfrac{a}{\text{gcd}(a,b)}\right)$ ...
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### Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto.

Question: Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto. Then show that if $i\in m$ and $j \in n$, then $p(i,j) \in p(m,n)$. The book offers a ...
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### Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
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### Do two pairs of distinct natural numbers exist such that AGM(A,B) equal to AGM(C,D)?

Here AGM is arithmetic-geometric mean. Are there natural numbers A,B,C,D such that $1\leq A<C<D<B$ and arithmetic-geometric mean AGM(A,B)=AGM(C,D) ? In other words, is AGM a homomorphism ...
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### Can arithmetic-geometric mean of two distinct natural numbers be a natural number?

Are there solutions to equation $AGM(A,B)=C$ such that $1\leq A<B$ and $A,B,C\in\mathbb{N}$?
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### Find the natural number which is divisibled by another number

Let $n= \overline{a_1a_2a_3}$ ($a_1 \neq a_2$, $a_2 \neq a_3$, $a_3 \neq a_1$). 1/ How many possible values does $n$ have which is divisibled by $7$? 2/ How many possible values does $n$ have which ...
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### Determine the elements of the set $\{ n \in \mathbb{N} :( \exists x,y \in \mathbb{N} )(n = 2x+3y) \}.$

I let $n=1-9$ and solved for $x$ and $y$ to find the elements of the set. When $n=1-4$, you can't find an $x$ or $y$ to make the equality true. When $n=5$, it is true that $x=y=1$. But then, when $n=6$...
I am looking for a function that for any natural number n returns a natural number m. Inverse function should exist for any $m=F(n), n \in \mathbb{N}$. Sequence should be simple enough for a person to ...