Questions tagged [natural-numbers]

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

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What would the 3D graph of GCD(x, y) and LCM(x, y) look like?

To find the answer out, I was going to use a graphing calculator but I couldn't find any one that supports the two operations. I would try to draw by hand but since it'll a 3D graph, I would have to ...
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1answer
48 views

If $x,y \in \mathbb{N}$ then $x+y=0 \iff x=y=0$

Let $x,y \in \mathbb{N}$. The operation $(+)$ is defined by: $$x+0=x$$ $$ x+(y+1)=(x+y)+1$$ Then prove that $x+y=0 \iff x=y=0$. The second implication $x=y=0 \implies x+y=0 $ is simple and ...
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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 $n=6$...
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Prove set of primes is equal to set of natural numbers

I was studying for an upcoming test in college and was looking at an old test. I'm struggling to understand how to prove this problem and was hoping someone could help me out. Prove that |P| = |N| ...
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Show on $\mathbb{N}$ there are $2^{\aleph_0}$ nonisomorphic linear orders.

How can I attack this problem? My idea is for $X \subset \mathbb{N}$ set up a linear order $O_{X}$ such that if $X \not =Y$ then $O_X \not = O_Y$.
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1answer
30 views

Injective map $\phi _k :P_k (\mathbb{N})\rightarrow \mathbb{N}^k$

Let $\mathbb{P}_k (\mathbb{N})$ be the set of subsets of $\mathbb{N}$ with $k\geq 1$ elements. Find an injective map $\phi _k :\mathbb{P}_k (\mathbb{N})\rightarrow \mathbb{N}^k$ (Beware, the sets $\{...
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Prove that the product of three consecutive natural numbers is divisible by 6

My math teacher asked me this question, and I told him that in every three consecutive natural numbers we have one multiple of 3, and at least two multiples of 2. And for a number to be divisible by 6,...
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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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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 ...
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1answer
51 views

Can $\mathbb{N}$ be an element of $\mathbb{R}$

Is $\mathbb{N}$ a element of $\mathbb{R}$ I understand that $\mathbb{N}$ ⊆ $\mathbb{R}$, but does that also imply that $\mathbb{N}$ ∈ $\mathbb{R}$?
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Numbers that are different repeated digits in different bases

Q. What are all the numbers $n$, such that each is represented as repeated digits—different digits—in two different bases $b_1$ and $b_2$. So in base $b_1$, $$ n_{b_1} = c c c \cdots c \...
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How does the well-orderedness of the set of natural numbers follow assuming the inductive-set definition of natural numbers

Assume that $\mathbb R$ is an ordered field (i.e. $\mathbb R$ is a model of real numbers). We define the set of natural numbers $\mathbb N$ as the smallest inductive set containing $1_\mathbb R$ (...
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1answer
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Prove that if $\\gcd(a,b)=1$ then $a\mathbb{N}\cap b\mathbb{N}=(ab)\mathbb{N}.$

Prove that if $\\gcd(a,b)=1$ then $a\mathbb{N}\cap b\mathbb{N}=(ab)\mathbb{N}.$ We have $1=\gcd(a,b)\implies au+bv=1$ for some $u,v\in \mathbb{Z}.$ Is it the correct way to prove the above result? ...
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1answer
39 views

Find the zeros in integer multiplication

How many zeros at the end of multiplication of natural numbers from 10 to 75? What i did: $$\left[\frac{75}{5^1}\right]+\left[\frac{75}{5^2}\right]=18=\text{Zeros at the end of 75!}$$ $$\left[\frac{...
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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 ...
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Set of natural and rational numbers

Just a quick question: Is it correct to say that the set of rational numbers cannot be a subset of the set of natural numbers? Certainly, we know these two sets have the same cardinality and there ...
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1answer
64 views

An infinite division and balls problem

Suppose you have a bottle with infinite volume and infinite number of balls. Now it's 11 o'clock and an hour left till 12. So you put 10 balls in and take one out 30 minutes later. You repeat this ...
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1answer
60 views

Bijection between two Peano triples.

Let $(P_1, \sigma_1, s_1)$ and $(P_2, \sigma_2, s_2)$ be two Peano triples. Can we show that there exists a bijection $g:P_1 \to P_2$ such that $g(\sigma_1) = \sigma_2$ and $g \circ s_1 = s_2 \circ g$...
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Being $f_n$ a Fibonacci-like sequence, prove that if $f_n$ and $f_{n-1}$ are coprimes, then $f_n$ and $f_{n+1}$ are coprimes.

$f_{n}=f_{n-1}+f_{n-2}$ I think I proved this by combining mathematical induction and modus tollens, but not sure about the correctness of the proof. Maybe we can get a fix or a totally different ...
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The HCF of two numbers is $13$ and their LCM is $4095$. If one of the numbers is $819$, find the other numbers

Stuck on this question. My Workings: $$4095 = 3^2 * 5 * 7 * 13$$ $$819 = 3^2 *7 * 13$$ and I'm lost after this part. Help would be appreciated. Thank You
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3answers
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Find the smallest four digit number which is divisible by $15,25,40$ and $75$

I'm stuck on this question. My working: \begin{align*} 15 & = 3 \cdot 5\\ 25 & = 5^2\\ 40 & = 2^3 \cdot 5\\ 75 & = 3 \cdot 5^2 \end{align*} LCM $= 600$ And I'm not sure what to do ...
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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
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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 ...
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1answer
65 views

Is there a general mathematical method that determines whether any sequence of natural numbers is generated by a particular mathematical law?

My question is: In mathematics is there a general method that determines whether any sequence of natural numbers is generated by a particular mathematical law/function/closed-form expression/...
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2answers
469 views

How many prime numbers in a given interval?

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ? Example: [2, 10] --> 4 prime ...
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1answer
328 views

How to Guess This Game's Number Mathematically?

So, I and my friends have a game named "Strike Ball". It's basically number guessing game. There are minimum 2 player. Both of them think a n digit number where every digit is different (ex. 1210 isn'...
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28 views

Proof of a practical method of a natural number's equation

Example Question : $3m + 4n = 70$, $m,n$ are natural numbers. How many values can $m$ have? I learned a method to solve this kind of problem, but I've never thought about that before. for n=1 =...
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3answers
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Linear ordering $\leq$ on $\mathbb{Z}$ in ZFC

Given a set of natural numbers $\mathbb{N}$ in ZFC, we define $\mathbb{Z}$ by first defining an equivalence relation $\simeq$ on $\mathbb{N}\times\mathbb{N}$: $(n,m) \simeq (n',m') \Longleftrightarrow ...
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2answers
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An endofunction on natural numbers defined by induction on $n$.

A function $f \colon \mathbb{N} \to \mathbb{N}$ fulfils the following conditions, for all $n \in \mathbb{N}$: $f(4n) = f(2n) + f(n)$; $f(4n + 2) = f(4n) + 1$; $f(2n + 1) = f(2n) + 1$. Question 1: ...
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Writing numbers with fewer symbols using expressions with powers

For example, it takes 7 symbols to write the natural number $n=9999999$ but we can also write it with 5 symbols as $n=10^7-1$. (Of course, with even larger exponents we can save even more symbols.) ...
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1answer
81 views

Proving the commutative law for 1-based $\Bbb{N}$

We will work with $1$-based natural numbers, that is $0\notin\Bbb{N}$. Let $s:\Bbb{N}\rightarrow\Bbb{N}$ be the function axiomatically given in peano axioms, we define addition and multiplication as ...
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1answer
61 views

$n$-tuples as nested ordered pairs: a formal definition using recursion on $\mathbb{N}$

Using conventional set theory as a foundation, there are two most popular definitions of an $n$-tuple of elements of $X$: A function $n\to X$. For any $x_1,...,x_n \in X$, $(x_1,...,x_n, x_{n+1}) = ((...
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Properties of floors, ceilings and modulus

I'm trying to reduce one calculation in an iterative Successive Over-Relaxation procedure for a program I'm writing. The code that works does this calculation: $$ s = \left\lfloor\frac{b + \left\...
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2answers
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Generating function for sequence of natural numbers?

How do I find the generating function for the sequence of natural numbers 1,2,3... by formulating the closed form of the recurrence relation $a_{n+1}=a_n+1, n \geq 0$
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1answer
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Distance between point p and origin [closed]

How to prove that the distance between the origin and the point $P$ is a natural number, where $P=(n, n+1, n(n+1))$.
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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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If the order in a set doesn’t matter, can we change order of, say, $\Bbb{N}$?

I’m given to understand that the order of the elements of a set doesn’t matter. So can I change the order of the set of natural numbers or any set of numbers ( $\mathbb{W,Z,Q,R}$ for that matter) as ...
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6answers
283 views

Can we prove that $\lim_{n\to\infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}}$ is finite for any $n_0\in\Bbb N$ without a direct computation?

Can we prove without direct calculation that this limit is finite for any natural number $n_0 \in \mathbb{N}$? $$ \lim_{n \to \infty} \frac{1^{n_0}+2^{n_0}+\cdots+n^{n_0}}{n^{n_0+1}} $$
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Is this a “good” definition of addition over the set $\mathbb N_0$?

I was thinking of defining addition and the concept of successor with the help of addition by defining addition like this: $$+(0,1)=1$$ $$+(m,1)=+(+(m-1,1),1)=s(m)$$ and, more generally: $$+(m,n)=+...
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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 ...
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Can we turn $\mathbb N_0$ or $\mathbb Z$ into a field? [duplicate]

I think that these two sets cannot be turned into fields by re-defining addition or multiplication (or both) but I am not sure how to prove this only from axioms of the field and (if needed) some ...
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Divisor Function of Sums in Fractions

I have a question that I've been working on for a while now. It says, "Let $A=\{0,1,2,\dots,2018\}$. Prove that $\forall n\in\mathbb{N},\exists\{a_0,a_1,a_2,\dots,a_{2018}\}\subseteq\mathbb{N}$, $$(...
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4answers
61 views

Show that the function $f: \mathbb {N} \to \mathbb {N}$ given by $f(n) = n + 2$ is not onto

Show that the function $f: \mathbb{N} \to \mathbb{N}$ given by $f(n) = n + 2$ isn't onto (surjective) Any advice on what to do here would be much appreciated! This has been taken from a past ...
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4answers
359 views

Find the sum of the First $50$ Natural Numbers starting from $11$. Is it from $11-50$ or $11-60$?

This a simple question yet confusing for me, I found the answer as 1220 by taking sum from $11$ to $50$, by inferring the question as first 50 natural numbers {$1,2,3,4,5,6,7,8,9,10,11,...49,50$} ...
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1answer
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A question on Goldrei Theorem 3.13 : prove that if $n>m$ and $a>0$ then $a\cdot n>a\cdot m$

This question is about the proof of Theorem 3.13 b) in Goldreis' "Classic Set Theory": For all natural numbers $n,m,a$, if $a>0$ and $n>m$ then $a\cdot n>a\cdot m \quad(1.)$ It has ...
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1answer
53 views

Function problem with Sigma summation

As a programmer learning how to write math-notation, I want to be able to exit when some function inside the Sigma summation has reached a particular value, but keep the value that the Sigma has ...
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6answers
169 views

What is a number in math? [closed]

Before I begin, let me give you so background. I previously asked a question on "How to prove that −x is not equal to x just because they yield the same result when in $x^2$". This got me thinking. ...
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1answer
104 views

Terry Tao's strong induction formulation

So I began to read Terry Tao's "Analysis 1" and I got confused by his strong induction formulation. The way he puts it is: "Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an ...
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2answers
203 views

Making the sum of 5th power of integers, a perfect square.

Yesterday this question was posed in a contest. It contains pretty easy questions like asking range of $ab+bc+ca$ when $a^2+b^2+c^2=1$, etc. But this question is something else. I haven't been able ...
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1answer
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How to prove that $m < n \Longleftrightarrow m + 1 < n + 1$ when defining natural numbers from scratch in ZFC?

An important results for natural numbers and their ordering by $<$ (that is, $\in$, $m < n$ means $m \in n$) is that for any natural numbers $m,n$ and $k$, we have $m < n \Longleftrightarrow ...