Questions tagged [natural-numbers]

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

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Range of an integer $t$ for which the following inequality holds

Consider the following inequality given by $$ -3at(2t-7)+ 3ta^2 +\binom{2t-8-2a}{3}-\binom{3t-8-2a}{3} \leq \binom{2t-8}{3}-\binom{3t-8}{3}$$ Is there a nicer way (without simplifying the expression) ...
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How to prove that $\forall a \forall b \exists x(a+x=b \lor b+x=a)$

The universe is the set of natural numbers including 0, defined by the Peano Axioms. I tried and failed to prove this by induction on b.
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How do I prove that $\forall a \forall b(\neg a<b\iff b\leq a)$ [duplicate]

The universe is the set of natural numbers including 0, which are defined in accordance with the Peano Axioms. We define the inequalities as: $a\leq b \iff \exists x(a+x=b)$ $a<b \iff \exists x(a+x=...
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Contradiction of axioms of real numbers

I am just starting out in real analysis, so please bare with me. My questions concerns three specific properties of the real numbers, at least as far as i understand them. Those are: The natural ...
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Convex games and a convex function $f:\mathbb{N}\to\mathbb{R}$ [closed]

I'm stocked with this exercise in Game Theory... A function $f:\mathbb{N}\to\mathbb{R}$ is called convex if $\forall i,j,k\in\mathbb{N}$ such that $i\le j\le k$ and $j<k$, $f$ satisfies $\left(k-i\...
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If I define addition in the following way, how can I prove that it's commutative?

$a+b=a$, if $b=0$ $a+b=S(a)+S^{-1}(b)$, if $b\not=0$ Here a and b are natural numbers defined according to the Peano axioms, while S represents the successor function. Basically, I am trying to prove ...
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Does the infinite limit probability of additive closure for pairs of elements from $(0, 1, 2 \cdots, n)$ not exist?

Suppose that we have a sequence of non-negative integers $S_n = (0, 1, 2 \cdots, n)$. At a given value of $n$ we can uniformly sample two elements of $S$ with replacement. We can check if adding those ...
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What exactly is zero? [duplicate]

I'm learning set theory and the construction of set of natural numbers by Peano's axioms, which says "$0$ is a natural number", and everything starts at $0$. Then what is $0$? In set theory, ...
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Numbers preceded by perfect cube and followed by perfect square

I read a puzzle about finding a natural number that is preceded by a perfect square and followed by a perfect cube. The answer was $26$, which is preceded by $25=5^2$ and followed by $27=3^3$. Are ...
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Natural Density For Subsets of $\mathbb{N}^2$?

Suppose that we have an set $S\subset\mathbb{N}^2$. Taking the natural order on $\mathbb{N}^2$ induced by $f((n,m))=$ max$(\{n,m\})$, what can we say about the "natural density" of S in $\...
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Numerical semigroup gerated by two elements

I have a question about numerical semigroups. It is known that if $a, b\in \mathbb{N}$ and $\gcd(a, b)=1$, then the numerical semigroup $\langle a, b \rangle$ has genus $\frac{(a-1)(b-1)}{2}$. My ...
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Prove that the following recursive functions (defining addition) are equivalent for all natural numbers

$S$ represents the successor function. $S(0)=1, S(1)=2, S^{-1}(3)=2$ and so on. First definition: $$a+b=sum1(a,b)\\ sum1(a,b)=a, \rm{if\ } b=0\\ sum1(a,b)=S(sum1(a,S^{-1}(b))), \rm{if } \neg b=0$$ ...
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How do I prove formally that for all natural numbers $a\cdot S(c)=b\cdot S(c)\implies a\cdot c=b\cdot c$

Natural numbers, addition, multiplication, and the successor function S, are defined in the wikipedia article regarding Peano axioms. https://en.m.wikipedia.org/wiki/Peano_axioms Originally I was ...
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Find the least natural number such that its cube is less than its factorial.

Basically, find a natural number $k$ such that $k^3 > k!$ but $(k + 1)^3 < (k + 1)!$. Now I know that the answer is 5. The issue is in the deriving this through algebra. Here's what I did: $(k + ...
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Prove that for all natural numbers $\neg\big(S(a)+b=a\big)$

The natural numbers are defined as per the Peano axioms. Addition is defined recursively as follows: $$ \begin{cases} a + 0 &= a,\\ a + S(b) &= S(a+b). \end{cases} $$ Prove that $\forall a\...
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Prove that the set of reciprocals of natural numbers has no positive infimum.

Prove that for each $x>0$, there is an $ n \in \mathbb N $ such that $\frac{1}{n}$ $<$ $x$, WITHOUT using the Archimedean theorem. Its simple enough with the theorem, but without the theorem I ...
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How do I prove that, for any positive integer $n > 2$, $n^{n/2} < n!$ [closed]

I tried using Induction, but I couldn't prove the inequality. Any proof would work. Rewriting the question for clarity, here is its statement: For any positive integer $n > 2$, prove that $n^{n/2} &...
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Does an open Interval of natural numbers have a minimum and maximum? (unlike one of rational numbers)

Like for example (1, 9) The maximum would be 8, because we are talking about natural numbers, so the problem of an undefined maximum (the number right before 9) doesn't exist? Am I right?
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Adequately defining the fundamental theorem of arithmetic.

Adequately defining the fundamental theorem of arithmetic. So after sifting through the internet, I realized that there are a few ways the fundamental theorem of arithmetic is defined. Paraphrasing, ...
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Analytic number theory-Are these sums equal?

I am studying analytic number theory by myself and I came across with two posts that seem to confuse me . Let $S(x)=\sum_{p\le x,\; q\le x,\; pq\gt x}\frac{1}{pq}$, where p and q are primes. Find the ...
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Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?

As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form: $z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$ in which $\omega$ is a formal ...
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Where does this proof that $P(\mathbb{N})$ is countable fail?

Let $A_k$ be the set of all subsets of $P(\mathbb{N})$ with size $k$. $A_k\in\mathbb{N}^k$ is countable for $k\in\mathbb{N}$. Since $$P(\mathbb{N})=\bigcup_{k=1}^\infty A_k$$ which is a countable ...
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Roots of polynomials with non-negative integer coefficients

Is there a way to describe/characterize all complex numbers $z$ with $|z| = 1$ such that $z$ is the root of some polynomial with non-negative integer coefficients? For instance, I've found that such $...
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Do rearrangements of $\mathbb{N}$ necessarily have fixed points with respect to placement distance?

Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection. Does there exist $\ n_1,n_2\in\mathbb{N},\ n_2>n_1, $ such that $\vert f(n_2)-f(n_1) \vert = n_2 - n_1\ ?$ I have tried coming up with counter-...
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Monoid actions - is this action inelegant?

I’m recently delving into abstract algebra, and I’ve attempted to devise a monoid action on the natural numbers. I think I must be missing something here—is there a better way to represent these same ...
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Does a distribution over $\mathbb{R}^n$ s.t. $\prod_{j}^n X_j \in \mathbb{N}_0$ exist?

This question was motivated by the fact that $\mathbb{E}[U] = \sum_{k=0}^{\infty} Pr[U > k]$ if $U \in \mathbb{N}_0$. The easiest use of this fact is to simply use a single random variable whose ...
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Difficulty with diophantine approximation: building sets with arbitrary lower and upper natural densities

$\newcommand{\d}{\mathrm{d}}\newcommand{\du}{\overline{\mathrm{d}}}\newcommand{\dl}{\underline{\mathrm{d}}}\newcommand{\card}{\operatorname{card}}$This was left as an exercise: Show that for all $0\...
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Finding Integer Approximations

The Saros is the time period for the draconic month ($T_d$ = 27.212220815 days), synodic month ($T_s$ = 29.530588861 days) and anomalistic month ($T_a$ = 27.554549886 days) to approximately match. ...
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bijection between naturals

in https://www.coursehero.com/file/69596247/hw0pdf/ problem 3 asks to find a bijection between $\mathbb{N}$ and $\mathbb{N} \times \mathbb{N}=\mathbb{N}^2$. Recalling Cantor diagonal proof it is easy ...
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What are the solutions for $y = \frac{a^n}{b^m}x$, with $a, b \in \mathbb{R}, x, y, n, m \in \mathbb{N}$?, how can they be approximated?

Came to this equation while trying to find if any natural number $x$ can be transformed into another natural number $y$ by multiplying $x$ by a real number $a$ a number of times $n$, and dividing it ...
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Is ZFC arithmetically sound?

I apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "real&...
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Why does my topology textbook (Munkres) define positive integers as the intersection of all inductive subsets of the reals?

This is how the topology textbook I'm reading (Munkres) defines integers: A subset of the real numbers is "inductive" if it contains 1 and $1+x$ for all $x$ in the subset. The intersection ...
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The $a+nd$ topology for $\mathbb{N}$

For each $a, d \in \Bbb N$, let $B(a,d) = \{a+nd \mid n \in \mathbb{N}\cup\{0\} \, \}$. I seek to show that $\mathscr{B} = \{B(a,d)\}$ is a basis for a topology on $\mathbb{N}$. I have already shown ...
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Order preserving map from interval $(0,1)$ to the ultrafilter

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$. Does there ...
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4 votes
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Question about the product of some consecutive integers being factorials. [duplicate]

A nice little curio. 1 = 1! 2(3) = 6 = 3! 4(5)(6) = 120 = 5! 7(8)(9)(10) = 5040 = 7! Are there any other examples of products of 'some' consecutive integers equalling factorials? Is there a proof ...
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1 answer
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Induction axiom and its logical properties

Is the induction axiom (postulated in the Peano Axioms or in Dedekinds version using only FOL) a logical truth or a contigent truth (so that it could turn out false)? IMO the induction axiom would be ...
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Addition and Multiplication Independent of Permutations

Note: In this discussion, $0$ is included in $\mathbb{N}$. In Chapter 1, Section 5.13 (a) from Analysis I by Amann and Escher, there is the following statement: ... If addition $+$ and multiplication ...
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Does the side of the successor function in the second postulate of the definition of addition matter?

In Peano arithmetic addition is usually defined with the following two postulates: $(1a):p + 0 = p$ $(2a):p + S(q) = S(p+q)$ Lets say I put the successor term of the second postulate on the left? ...
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How do I define the proper subset $\bigcup\limits_{i=1}^\infty[n^2,n^2+1]$ of a set $\bigcup\limits_{i=1}^\infty[n,n+1]$

I have to determine if the set $T=\bigcup\limits_{i=1}^\infty[n^2,n^2+1]$ is bounded and find the supremum and infimum if they exist. Clearly, $T=\{[1,2], [4,5], [9,10]...\}$ it is clearly bounded ...
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4 votes
3 answers
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What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct?

$1+2+3+\dots+7+8+9+10+12+13+\dots+96+97+98+102+103+104+\dots+985+986+987+1023+1024+1025+\dots+9874+9875+9876+10234+10235+10236+\dots+98763+98764+98765=$ The only thing I can do is to evaluate a (bad) ...
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3 votes
0 answers
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How many elements of the inverse of an $n\times n$ matrix of natural numbers can be the same as itself?

What is the maximum number of elements of the inverse of an $n\times n$ matrix consisting of natural numbers $(\geq1)$ that are identical to itself? This question arose from my previous question, How ...
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Question regarding natural numbers in Tao’s Analysis 1.

This doubt might be a mistake on my part, but its confused me for a while now so I decided to ask here. In Terrence Tao’s “Analysis 1”, at the beginning of the chapter about Natural Numbers (where he ...
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How to solve this problem on alligations?

Question: "The diluted wine contains only 8 litres of wine and the rest is water. A new mixture whose initial concentration is 30 %, is to be formed by replacing wine. How many litres of mixture ...
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Question about inductive definition of natural number [closed]

in "An Intro to Complex Analysis and Geometry, John P. D'Angelo', It has the following: Definition 3.4            A subset S of R is called inductive if whenever x∈S, then x+1 ∈ S. Definition 3....
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A proof of recursion theorem in "Elements of Set Theory"

In "Elements of Set Theory" by Enderton 74~75p, Recursion Theorem on $\omega\;\;$ Let $A$ be a set, $a\in A$, and $F:A\to A$. Then there exists a unique function $h: \omega\to A$ such that $...
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Probability of second random number is larger than the first one?

Imagine n1 is the first random number and n2 is the second one, both between A and B and both are natural. What is the probability that n2 is greater than n1? What if there is another n3 random number ...
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Does there exist a subset of N satisfying a certain density property

If $A\subset \mathbb{N}$ we say that the set $A$ has a natural density if the limit $$\lim_{n \to \infty}\frac{|A\cap [1,n]|}{n}$$ exists and we denote it by $d(A)$. Also for every $k\in \mathbb{N}$ ...
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Prove that intersection of a non-empty set of natural numbers is itself a natural number using ZF set theory and Peano axioms

Update I was going to use this question to solve another question using intersection approach. I could find an answer for that question using another approach and posted an answer there but I still ...
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2 votes
2 answers
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Halmos's set theory If $E$ is a non-empty set of natural numbers, exists some $k$ in $E$ such that $k \leqq m$ for all $m\in E$. Prove by intersection

Update I saw a similar question which provided a proof by using axiom of induction and I wrote an answer by adapting to one of the answers of that question. But I hope someone can give another proof ...
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1 answer
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Describing $\mathbb{N}$ with multiples of $4\mathbb{N}$

Suppose we index subsets of the natural numbers in the following way. $$\begin{matrix} X_2 = 4 \mathbb{N} &Y_2 = 8\mathbb{N} \\ X_3 = 16\mathbb{N} & Y_3 = 32\mathbb{N} \\ X_4 = 64\mathbb{N} &...
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