Questions tagged [natural-numbers]

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

23
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736 views

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 ...
6
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0answers
99 views

Can you make all natural number from 3?

Can you make all natural number from 3 with only four function that $$x!,\; \sqrt{x},\;\lceil x\rceil,\;\lfloor x \rfloor $$ ? ex) $1=\lfloor \sqrt3 \rfloor$ $\;\;\;\;\;2= \lceil \sqrt3 \rceil$ $\...
4
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0answers
452 views

Crititism of the set-theoretic definition of natural numbers

A while ago I read in a book (or a paper?) that a very well-known mathematician (Saunders Maclane?) in his lectures used to mock the classical set-theoretical definition of natural numbers: 0 = {}, 1 ...
3
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53 views

Defining Addition of Natural Numbers as the Algebra of 'Push-Along' Functions

Let $N$ be a set containing an element $1$ and $\sigma: N \to N$ an injective function satisfying the following two properties: $\tag 1 1 \notin \sigma(N)$ $\tag 2 (\forall M \subset N) \;\text{If } ...
3
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0answers
110 views

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 ...
3
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0answers
98 views

Do there exist any cycles for these number sequences?

We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots *p_m^{...
3
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0answers
36 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum $$\...
2
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287 views

Sequences of Natural Numbers without Arithmetic Subsequences

Let's call a sequence $k^+$-free if it is contains no arithmetic subsequence of length $k$. Define the $\bf{density}$ of a sequence of natural numbers $s_n$ as $$\lim_{n\to \infty} \frac{n}{s_n}$$ ...
1
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56 views

Non trivial bijective maps from $\mathbb{N}^{3}$ to $\mathbb{N}^3$

The main reason I ask this question is because I want to implement an information scrambling or permutation algorithm on a $3$-D cube. The input to my bijective map would be the coordinates $(x,y,z)$ ...
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0answers
10 views

Is there a specialized formula for Lagrangian interpolation on equispaced points?

If we know $f(0),f(1),f(2),\cdots f(n)$, is there a specialized version of the Lagrangian interpolation formula and a shortcut to compute the coefficients ? (Stability is not a concern.)
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53 views

Is the NNO in Cat isomorphic to $\mathbb {N}$ viewed as a discrete, skeletal category?

Suppose N is any category which satisfies following axioms: There exists a distinguished object, z. There exists a distinguished functor, $\sigma': N\rightarrow N$. To any category $X$ with a ...
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36 views

Properties of this notion of density in $\Bbb{N}$?

For a given $S \subseteq \Bbb{N}$, the asymptotic density of $S$ is defined as $$d_\text{asy}(S) := \lim_{n \to \infty} {\#(k \in S : k \le n) \over n}$$ If the limit exists. Wikipedia says this is ...
1
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0answers
28 views

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$ for all natural numbers $X,Y$ where $X>Y$. Further, there exist integers $...
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0answers
140 views

Proving that addition in $\mathbb{N}$ is associative and commutative

I would like to prove the associative and commutative property of the natural numbers by using Peanos axioms only. Did I justify every step in my proofs correctly? Definition. $\forall n,m \in \...
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0answers
27 views

$3n = 8x+4$ control flag inside of a function

I want to be able to evaluate wether the result of a function satisfies an expression. Is this possible to do? Example I have an expression, a linear polynomial: $$8x+4$$ for $x\in\mathbb{N}$ If I ...
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0answers
40 views

Is possible to represent natural number X without 'splitting' it as additive-finitely repeated element set property?

I have a $Set$ of elements $n$ (a multiset). Each element of this set $n$ is the same, is a multiset because one element $n$ is replicable using addition operation, in other words we can have finitely ...
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0answers
42 views

Writing down consecutive natural numbers until a certain number of digits $k$ is reached.

A person starts writing consecutive natural numbers from $5$ until $k$ digits are reached. For some values of $k$, this will be impossible, for example $6$ or $8$ are impossible as then after writing ...
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27 views

Natural Number Object and multiplicity of its points

As defined, the definition of natural number object seems odd to me, for it does not ensure the multiplicity of its elements. For example, category with exactly one object which has only 1 map ...
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0answers
77 views

Weird natural numbers function problem

So I came across with the following problem, of a Brazilian elementary number theory book, which did not came with the resolutions: "Let $f:\Bbb N^{*} \to \Bbb N^{*}$ a function such that $f(a)=f(1995)...
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0answers
25 views

How to create an injective function to generate pseudo-random numbers with seed

Let's call A the set of all the n-digit natural numbers (base 10). So with n=3, they would be 000, 001, 002, ... 999 Basic question: I need to create a mathematic function with this features: it ...
1
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0answers
36 views

A combinator-ish way to construct a set $\mathbb N_0$

We can define $0$ to be the number of elements of an empty set. Then we can define successor of $0$ as the number of all empty sets and we can denote it as our familiar $1$, since there is only one ...
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0answers
31 views

I'm interested in the solution set satisfying the equation $\log_{10} p\times\log_{10} q=\log_{10} r$

The equation interested in is $\log_{10} p\times\log_{10} q=\log_{10} r$ where $p,q,r\in\mathbb N$ are natural numbers. Here, I want not to consider some trivial solutions that make any one of ...
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0answers
28 views

Is there a difference in the rate of decrease between $f(x)$ and $g(x)$ for increasing $x$?

I have the following two functions of $x$: $ f(x) = \frac{c}{c + (N-1)o + Nd + xl}$ $g(x) = ae + (1-a)\frac{1}{x+2N}$ with $0 \leq a, e, c, o, d, l \leq 1$ and $N, x \in \mathbb{N}^+$. For both ...
1
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0answers
83 views

Does there exist a positive integer which is a power of two, such that by rearranging it's digits we can get another power of two?

Does there exist a positive integer which is a power of two, such that by rearranging it's digits we can get another power of two? I know a quite good solution, that involves working with sum of ...
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0answers
39 views

Let$\ H$ be a hyperinteger. If$\ f(n)=g(n)$ is true for all$\ n \in \mathbb{N}$, will it be so for all$\ H$?

I do know that all true first order statements in$\ \mathbb{N}$ are also valid in$\ \mathbb{N^*}$, so for example$\ \sin(H \pi) =0$. As a consequence, my question is equivalent to: is$\ f(n)=g(n)$ a ...
1
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0answers
18 views

Random Algebra Problem 2

Prove that if a, b, c are integers and x, y, z are non-integer real numbers and $\alpha$ is a real number, for every given set of x, y, z, the number $\alpha$ obtained from the following equation: $$\...
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0answers
766 views

Natural Numbers Equation

I am trying to find the $(k_1,k_2,...,k_N)$ tuples solutions to an all natural numbers equation in the following form : Given $n\in\mathbb{N}^{*}$, $N\in\mathbb{N}^{*}$ and $n_i\in\mathbb{N}^{*}\leq ...
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0answers
71 views

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? ...
1
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0answers
50 views

Is it correct? $\lim_{n\rightarrow \infty} \frac{c^{n}}{n!^{\frac{1}{k}}}$

This is what we have $$\lim_{n\rightarrow\infty} \frac{c^{n}}{n!^{\frac{1}{k}}},$$ $$n \in N, k>0, c>0$$ If n->inf $$\frac{{x}_{n+1}}{{x}_{n}}=\frac{{c}^{n+1}}{{(n+1)!}^{\frac{1}{k}}}*\frac{{n!}^...
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0answers
361 views

Peano system vs natural numbers

What exactly is the difference between natural numbers and an arbitrary peano system? In particular there is a proof in my book for recursion on natural numbers, as well as an erroneous proof of ...
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0answers
103 views

Operations between temporal discrete intervals

I am going to present you my problem and ask you for solutions' references. I have a discrete temporal series, which is the sequence of Natural numbers $T = \lbrace 1, 2, \dots, N \rbrace$. An Event ...
0
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0answers
24 views

Foundations of analysis - Natural number addition operator proof/definition

In Landau's "Grundlagen der Analysis" the author states the following proposition which at the same time is a definition. $$\text{Proposition 4/at the same time Definition 1. There is precisely one ...
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0answers
43 views

Hierarchy of subsets of $\mathbb{N}$

I was wondering if there is an interesting way to build an "hierarchy" of natural numbers subsets in a transfinite sequence: $$(U_\alpha)_{\alpha < \lambda} \quad \text{with } U_\alpha \subset\...
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0answers
12 views

convergence velocity

I've been given some succession to work on, in particular I've been asked the variation of the velocity of the convergence of the following succession, which obviously depends on $\alpha$: $$a_n=\Big(...
0
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0answers
22 views

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: $\left(x + m \dfrac{b}{\text{gcd}(a,b)}\,,\,y-m \dfrac{a}{\text{gcd}(a,b)}\right)$ ...
0
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0answers
32 views

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 ...
0
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0answers
54 views

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 ...
0
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0answers
25 views

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 ...
0
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0answers
30 views

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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0answers
31 views

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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0answers
53 views

Boolean algebras/ Shelah/ Unclear step in the proof

Here on the page $10$, what does the displayed formula in the $6$th line $$\text {lg}(\eta_\ell)<\omega\Rightarrow \bigcup\{\text{Rang}(\nu_\ell(k):k<\text{lg}(\nu_\ell))\}\cap\bigcup_{k<\...
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0answers
14 views

Are the natural numbers definable in the (2nd-order) theory of complete ordered fields?

I read that the natural numbers are not definable in the theory of real closed fields (RCF), which captures the 1st-order properties of the real numbers. That's why RCF doesn't contradict Gödel's ...
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0answers
8 views

Existence part of Iteration theorem

Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → ...
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0answers
41 views

Why does $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$ exhibit a bijection with the pairs (x,y)?

Consider $$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$ I discovers it essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So intuitively, given any P(x,y) we follow ...
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0answers
23 views

Proving Euclidean 'Division' in (N,+) Using Sigma Representation?

Is the following argument valid? We are working in the natural numbers, $0 \in (\mathbb N,+)$. Let $b \gt 0$. Lemma 1: If $a \lt b + b$ then $a \lt b$ or $a = b + r$ with $r \lt b$. Proof Assuming ...
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0answers
36 views

Reference request on Number theory

Which books are good for beginners on this topic? I don't know much about this, just did a quick search on Wikipedia on Peano's axioms. I got very exited about the topic. And could you tell me ...
0
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0answers
83 views

“Ordering” n-tuples of Natural Numbers

I'm interested in a value $r>0$ defined like this: $$r^2=l^2+n^2+m^2,\qquad (l,m,n)\in \mathbb{N}^3$$ This can be thought of as a relation $\mathcal{R}$ between $\mathbb{R}$ and $\mathbb{N}^3$: $$\...
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0answers
25 views

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 $n=6$...
0
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0answers
32 views

A sequence of numbers easily computable with no apparent order and with defined inverse function

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 ...
0
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0answers
60 views

Does the closure of natural numbers under addition need to be proven or is it an axiom?

From the way content is written online I am having trouble discerning which properties of numbers require proof and which are taken to be axioms because so few take the time to explain anything from a ...