Questions tagged [natural-numbers]

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

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30 views

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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we want to prove that $(n,n+1) \cap \mathbb{N}$ is empty [duplicate]

Problem: prove that for each $n \in \mathbb{N}$ : $(n,n+1) \cap \mathbb{N} = \varnothing $ attempt: Indeed, let $S(n)$ be the statement $\{ n : (n,n+1) \cap \mathbb{N} = \varnothing \}$. Clearly, $(...
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1answer
26 views

Examples of basis

Excuse me , can you see this question, For each positive integer $n$ , let $S_n=\{n,n+1,\ldots\}$ . The collection of all subsets of natural number which contain some $S_n$ is a base for a topology on ...
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Proving that some integer multiple of a real number is within $\frac{1}{k}$ of an integer.

So I'm trying to prove that for every real number $a \in \mathbb{R}$, the set $M = \{a,2a,\dots,(k-1)a\}$ contains at least one element that is within $\frac{1}{k}$ of an integer. (Note that $k \in \...
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2answers
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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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4answers
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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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1answer
49 views

How to write numbers in the language of first-order set theory. [duplicate]

I saw this Numberphile video (link at bottom), and at around 10:10 they talk about writing numbers in the language of first-order set theory. For example, to write $0$, it showed the empty set: $$\...
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1answer
59 views

Is $2^{-k}\left(\left(1+ \sqrt {k^4−k^2+1}\right)^k+\left(1−\sqrt{k^4−k^2+1}\right)^k\right)$ an integer for every non-negative integer $k$? [closed]

I want to check if this fraction $$\frac{\left(1+ \sqrt {k^4−k^2+1}\right)^k+\left(1−\sqrt{k^4−k^2+1}\right)^k}{2^k}$$ is integer for every non-negative integer $k$. I tried induction but it ...
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34 views

Prove that, for every natural number, their factorization as primes is unique

I need some feedback on this proof I wrote that: $$\forall n\in\mathbb{N} \text{ assumed the existence of a factorization of } n \text{ as } n = p_1p_2\cdots p_k, \text{ where } p_i, (1 \leq i \leq k)...
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1answer
33 views

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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3answers
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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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1answer
38 views

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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4answers
32 views

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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1answer
42 views

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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Double Induction Example

I've been looking at examples of problems using double induction and have found one that has stumped me. Here is the problem: Let $n,m\in \mathbb{N}$. Let $P(n,m)$ denote the statement $n>m \...
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Exercise 2.2.1 in Analysis by Terence Tao Induction Proof of Addition Associative Law Natural Numbers

The proposition 2.2.5 and also exercise 2.2.1 of Analysis by Terence Tao is as below: Show for any natural numbers $a, b, c$, we have $(a+b)+c = a+(b+c)$ the associative rule The proof should use ...
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2answers
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Conjecture: There are infinitely many $N \in \Bbb{N}$ such that $p$ a prime $p \leq \sqrt{N+1} \implies p \mid N$?

Conjecture. There are infinitely many $N$ such that if $p$ is a prime $\leq \sqrt{N+1}$ then $p \mid N$. Is this another hard to prove number theory conjecture, or do you have some idea of how to ...
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29 views

Sum of real numbers equal to integer number

Let $t_1, t_2, ..., t_n \in \mathbb{N}_{>0}$, $t = \sum_{i=1}^n t_i$, and $s \in \{n,n+1, ..., t\}$. Does an integer function $f: \mathbb{R}_{>0} \to \mathbb{N}_{>0}$ such that $ \sum_{i=1}^{...
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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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1answer
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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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1answer
82 views

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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1answer
25 views

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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2answers
32 views

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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1answer
44 views

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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1answer
32 views

Why are natural numbers defined with union in Zermelo–Fraenkel (ZF) set theory? [duplicate]

In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined as $0 = \emptyset$ and $n + 1 = n \cup \{n\}$ $$\begin{alignat}{2} 0 & {} = \{\} && {} = \emptyset,\\ 1 & {} ...
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1answer
54 views

Proof of the well-ordering principle

I tried to prove Well-Ordering Principle by myself, and I finally did it. However, I'm not sure if this proof is correct. Can anyone evaluate my proof? Proof: Since the set of natural numbers, $\...
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Proving Natural Predecessor existence using Well Order

I know that the Principle of Induction is equivalent to Well Order at least in Natural Numbers, but I have seen that the demonstration of Induction using Well Order uses the existence of Predecessor: ...
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1answer
38 views

Prime number $x$ and $x!$

I'm trying to show that there exists a prime $p$ between $x$ and $x!$ where $x \in \mathbb{N}, x>2$ such that, $$ x < p < x!$$ Can I say that since $x!$ and $x!-1$ are relatively prime(share ...
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1answer
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Prove trichotomy of order WITHOUT induction?

Claim. If $x,y \in \mathbb N$, then at least one of these is true: (a) $x>y$; (b) $x=y$; or (c) $x<y$. It seems that the usual proof of this claim uses induction. Is it possible to prove this ...
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Prove the following inequality regarding natural numbers

For all natural numbers $a>2$. I want to prove, $$ (a-1)! + 1 < a^{a-1} $$ How do I go about this?
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1answer
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Proof for formula for number of divisors

For any natural number $n>1$, with factorization $$ n = q_1^{\alpha_1}q_2^{\alpha_2}q_3^{\alpha_3}....q_m^{\alpha_m} $$ we can find the number of divisors by using the formula, $$ (\alpha_1+1)(\...
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1answer
42 views

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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1answer
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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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1answer
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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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1answer
37 views

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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1answer
36 views

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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How to prove formula related to $2$-adic valuation / $2$-adic absolute value and binary expansion

I would like to prove the following formula, which I have verified for every positive integer $n \ge 1$ up to $n = 10000$: $$n - \sum_{k=0}^{\lfloor \log_2{n} \rfloor}\left(\left\lfloor\frac{2n-1+2^{...
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80 views

How to prove $n = \sum_{k=0}^{\lfloor \log_2{n} \rfloor}{\left[ \left\lfloor \frac{n}{2^{k+2}} \right\rfloor + c_k \right](k+1)}$

I would like to prove that: $$n = \sum_{k=0}^{\lfloor \log_2{n} \rfloor}{\left[ \left\lfloor \frac{n}{2^{k+2}} \right\rfloor + \left(\left\lfloor \frac{n}{2^{k}} \right\rfloor \bmod 2 \right) \right](...
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1answer
19 views

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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1answer
30 views

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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41 views

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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63 views

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 ...
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1answer
23 views

Is Appert space Fréchet-Urysohn?

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 ...
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1answer
14 views

About the limit on the definition of the Appert Space

Let $X$ be the set of positive natural numbers. Define the following topology $\tau$ on $X$: $$A\in \tau \iff 1\notin A \,\, or \,\, 1\in A \,\, and \,\,\lim\limits_{n\to\infty}\dfrac{S(n,A)}{n}=1$$. ...

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