# Questions tagged [natural-numbers]

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

580 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
### 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 ...