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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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What is the solution of this algebraic problem?

Let $a, b, n, X$ and $Y \in \mathbb{N}$. Find $X$ and $Y$ such that $(a+b)^{n} = X + Yb$.
Alezigl
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Are there any integers that are considered uncomputable based on their extremely large size alone?

This is difficult to frame, but my question is if there are any finite integers that are so large that they are uncomputable in principle. Meaning that they cannot be handled by mathematics even in theory. So is there some kind of limit in…
Juel Herbranson
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Find all natural (+0) solutions of a system of linear equations with natural (+0) coefficients

Consider the following equation: $$ \begin{pmatrix} n_1 \\ n_2 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 0 & 0 & 1 & 0 \\ 2 & 1 & 2 & 3 & 0 & 1 \end{pmatrix} \begin{pmatrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \end{pmatrix} $$ where $n_i \in…
Miguel
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Why does an intersection of all inductive sets comprise natural numbers?

I suppose we cannot say "Let natural numbers be $1, 1+1, 1+1+1, \dots$", but I fail to put into words why we can't. What's the reason to define natural numbers as an intersection of inductive sets? If we could define it as I wrote above, we wouldn't…
Alexander Vronsky
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How to approach proving that walking this tree infinitely will product all natural numbers?

A binary tree with a root of "empty" always has a left node of 0 and a right node of 1. If we follow some path along this tree we can produce binary numbers with the least significant bit as the first step. For example travelling L->R->L->R would…
allumbra
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Total numbers formation by permutation method

How many numbers greater than $3400$ can be formed using first five natural numbers? Without and with repetition.
SandyM
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Are the natural numbers closed under exponentiation?

Is it true that $\forall n \in \mathbb Z^+ \land \forall i \in \mathbb Z \land i \geqslant 0: n^i \in \mathbb Z^+$? The reason I ask is because this would allow you to create a countably infinite set $$\forall k \in \mathbb Z \land k \geqslant 0,…
keluj
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How to recursively define a sum of any finite sequence

I'm stuck at exercise 4.9 of chapter 3 in "Introduction to set theory" of Karel Hrbacek and Thomas Jech (Third revised edition) The exercise is as follows: For each finite sequence $\langle k_i: 0 \leq i < n \rangle$ of natural numbers define $\sum…
Eparoh
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The less period of "1/b"

Which is the less "b" natural number, for which the less period of "1/b" is 7? I have already tried this way: I have the following formula: 10^i ≡ 1 (mod b), where "i" is the less period of "1/b". So I replaced this "i" with 7 (b | 10^7 - 1), and…
mrsbean
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Official name of "sized integers", a kind of number where $00$ is not equal to $0$?

Hierarchical identifiers, labels and indexes... All can use digits as character-strings, differenciating $0$ and $00$, $1$ and $001$, but preserving all other numeric interpretations, like order ($002>001$) and freedom for represantation (some…
Peter Krauss
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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 empty set. Now, we can define successor of $1$,…
Right
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Is there a listing of natural numbers with their properties?

I am looking for a category of natural numbers (about <1000 is enough) with its properties. Here's some examples : 2 - It is the first prime number. 1729 - It is the smallest number expressible as the sum of two cubes in two different ways. Here's…
Analysis
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Biggest 8 digit number following two specific rules

Which is the biggest 8 digit number of the form "abcdefgh" which is made up only of 1, 2, 3 and 4 and which follows the rule: the digit 1 is one digit away from another 1, the digit 2 is two digits away from another 2, the digit 3 is three digits…
Noifma
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Counting how many natural numbers satisfy a given condition.

I've defined a sequence of sequences $\{x^n\}$ as follows $x^1=(1^2,2^2,3^2,4^2,5^2,...)$ $x^2=(1,2^2,3^2,4^2,5^2,....)$ $x^3=(1,2,3^2,4^2,5^2,...)$ . . . and for each $n$ fixed, I am trying to determine $|\{j: x^n_{j} \leq k\}|$. In other words,…
user100106
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Function of two integers that has not repeated values

In order to implement a software I need a function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ such that: $$(x' \neq x \lor y' \neq y) \Rightarrow f(x,y) \neq f(x',y') $$ It's very easy to implement an algorithm that implements such a…
Aslan986
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