# Questions tagged [natural-numbers]

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

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### Please find minimum (x+y) [closed]

If $360.x=y^3$ and $x;y$ elements N then minimum $(x+y)$ is 105. Please help explain how to calculate the answer of 105. Apologies for not asking a question correctly My workings went as follows:-...
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### well founded induction / difference to strong induction

We are given a chocolate bar with $n$ pieces (squares) and we already know by strong induction that $n-1$ are needed to break it in individual parts. https://web.stanford.edu/class/archive/cs/cs103/...
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### Find $n$ and $m$, if $n,m$ are natural numbers, such that $m^6 + 279 = 2^n$.

Suppose $n, m \in \mathbb{N}$. If $m^{6}+279=2^{n}$, find $n, m$. What are some ways to approach this question? Instinct told me to take logarithm but doesn't work to well.
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### Proof by induction that $2^{n+4}<3^n$ for large n

We have an inequality: $$2^{n+4}<3^n$$ which holds for large n. How to prove by induction that it holds for large n. I got an idea that I would make infimum of the natural numbers bigger and then ...
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### Can natural number be an ordered pair?

I’m supposed to prove that natural number $n$ cannot be an ordered pair. The definition for ordered pair is $(x,y) = \{\{x\},\{x,y\}\}$ and for the definition of natural numbers we use the definitions ...
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### How Can You Prove the Completeness of $\mathbb{N}$?

I am trying to self-study real analysis and I am finding it difficult to prove some statements even when I have intuition for it. One exercise in my book asks: Prove that $\mathbb{N}$ is complete. ...
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### Prove for all positive integers $a$ will never reach $2^b$

In the $3x+1$ problem the question is about finding out wether any initial positive integer will eventually reach $1$ after a finite iterative amount of time. Instead of reaching $1$ we could ...
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### Proving that you can pick a group of numbers from a set of 4 natural numbers that divide 4 [duplicate]

Prove that for any set of $4$ natural numbers, it is possible to pick a group of numbers (can contain $1-4$ numbers) from the set such that the sum of the group is divisible by $4$. I tried to ...
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### Bijection of the set of natural numbers onto the set of integers. [closed]

An example in Real Analysis by Sherbert and Bartle tells that the set of integers is a bijection of the set of natural numbers. How is the one to one correspondence possible for the set of integers? ...
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### Define a geometry based on a pseudometric in which the distance between distinct primes is zero?

Consider a pseudometric on $\Bbb N$ by $d(p_k,p_n)=0;$ else $d(x,y)=|x-y|.$ Here $p_k,p_n$ are distinct primes. Edit 5/14/2020: From the comment below the definition I gave does not satisfy the ...
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### Why some mathematicians axiomatized number sets?

There was sense of natural numbers before Peano. Likewise why did Peona axiomatized Natural Numbers? Or later, why some mathematicians did axiomatize the Real Numbers or Complex Numbers? And before ...
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### Given the binary operation $x*y=x^2+4xy+y^2$ show that $a*1 \in \mathbb{N}$ has infinitely many solutions, where $a$ is irrational.

Consider the binary operation: $$x * y = x^2 + 4xy + y^2$$ defined on $\mathbb{R}$. I have to show that there are infinitely many irrational numbers $a$ such that $a * 1$ is a natural number. This ...
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### Confusion in well ordering principle

Well-ordering principle states that every non-empty set of positive integers contains a least element. I have a set S which is a subset of natural numbers. Now by well-ordering principle I can ...
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### Find the maximum value $LCM$ pair in the sequence where $LCM(a, b)$ means the smallest positive integer that is divisible by both.

Problem Statement: Given a sequence $S$ of $N$ positive numbers, calculate the $\max\limits_{1 \le i < j \le n} LCM(a_i,a_j)$, where $LCM(a, b)$ is the smallest positive integer that is divisible ...
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### I have a question on Lebesgue Iterated Integrals with the Counting Measure

Here is the question: Let $X = Y = \mathbb{N}, S = \Sigma= P(\mathbb{N})$, the power set of the natural numbers. Let $\mu = \nu =$ the counting measure. Let $h: X \times Y \rightarrow \mathbb{R}$ be ...
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### How to calculate the sum of the first n natural numbers? [duplicate]

I already know one way to prove the sum of the n first natural number is equal to $\frac{n(n+1)}{2}$, but I found another way which involves calculating $(k+1)^2 - k^2, k\in \mathbb{N},\ k\geqslant 1$ ...
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### Proof that for all $m,n,l \in \mathbb{N}_0$, $m\leq n \iff m+l \leq n+l$.

I am trying to prove the claim: "for all $m,n,l \in \mathbb{N}_0$, $m\leq n \iff m+l \leq n+l$". In what follows, $s_m$ for (some natural $m$) is the unique function that exists by the recursion ...
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### Set theory: $n$ is a set in the naturals, if $x$ is in $n$, is $x$ also a natural number?

Im having trouble with a homework question from my Set Theory class. The question is, Let $n$ be a set and an element of the natural numbers. If $x$ is an element of $n$, is $x$ also an element of the ...
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### An alternative proof of set of all permutations on naturals is uncountable by Cantor's diagonalization.

I was trying to show that the set of all permutations on $\mathbb N$ is uncountable.It is quite easy to show it directly using Riemann Rearrangement Theorem of Series.Take any real $r$ and choose a ...
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### Prove that $\text{Inc}(n,m) = \text{Inc}(m-1, m) \circ \text{Inc}(n, m-1)$

Let $n, m \in \mathbb N$ and also let $n \leq m$, we define the set: $$\text{Inc}(n,m): = \{f: \{1,...,n\} \to \{1,...,m\} \mid f \text{ strictly increasing}\}.$$ I am trying to prove (for $n<m$ I ...
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### Proving Proposition 4.1.8. from Terence Tao's Analysis I

Let $a$ and $b$ be integers such that $ab = 0$. Then either $a = 0$ or $b = 0$ (or both $a=b=0$). MY ATTEMPT Let us consider that $a = m - n$ and $b = p - q$, where $m,n,p,q$ are natural nubmers. ...
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### Does $\{1\} \times B$ measurable w.r.t. product measure imply $B$ is measurable?

Let $B \subseteq X$. Let $(X, \Sigma, \mu)$ be a measure space and let $c$ be the counting measure on $\mathcal P(\mathbb{N})$. Assume that $\{1\} \times B$ is measurable with respect to the product ...
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### What's the next base-ten non-pandigital factorial number after 41!?

By pandigital number I mean a number for which each digit in a given base occurs at least once (some definitions that state each digit must occur exactly once), and since I looking for numbers that ...
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### Coprimality in a given set of consecutive natural numbers

Given the first n natural numbers, is it possible that every composite odd number is coprime with at least one even composite number and that no two odd numbers share the same even number. For example,...
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### Show that there do not exist any distinct natural numbers a,b,c,d such that

Show that there do not exist any distinct natural numbers a,b,c,d such that $a^3+b^3=c^3+d^3$ and $a+b=c+d$
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### Is there a place which has the functional (programming) definition of Numbers in various forms?

I am familiar with the Peano Natural Number definition which is what you see in intro to functional programming courses and examples. But I have also much later encountered a practical implementation ...
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### Peano arithmetic - Why is adding n to m the same as incrementing m n times?

Addition(+) is defined using the successor function(++) in Peano arithmetic as: 0 + m = m (n++) + m = (n + m)++ While these are intuitive axioms that are consistent with my previous, elementary, ...
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### Number of ways to express a natural number N as a product of K natural numbers where each number is greater than or equal to a natural number M?

I want to know the number of ways to express a natural number N as a product of K natural numbers where each number is greater than or equal to a natural number M? Or a simpler version of the problem ...
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### Prove that $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have the same cardinality iff m=n

In the following proof, $\mathbb{Z}_n= \{ 1,2,....n \}$ After $g$ is defined , they defined the composition $gof$ and then they defined $h$ by removing the last ordered pair $(n+1,m)$ from the ...
I was asked if the set of positive natural numbers $\mathbb{N}$ with the Appert topology is a Fréchet-Urysohn space, that is, for every $A\subset \mathbb{N}$ and $p\in \bar{A}$, there is a sequence in ...
Let $X$ be the set of positive natural numbers. Define the following topology $\tau$ on $X$: $$A\in \tau \iff 1\notin A \,\, or \,\, 1\in A \,\, and \,\,\lim\limits_{n\to\infty}\dfrac{S(n,A)}{n}=1$$. ...