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Questions tagged [natural-numbers]

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

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how to find explicit formula for A008303?

I tried to find formula for $\Omega_n(a,m)$ for natural $n,a,m$ with $n\ge a , a\ne 0 , n\equiv a \text{ mod2}$ where $$ \Omega_n(a,m)=\int_0^\infty \frac{\sin^n\left(\tan_mx \right)}{x^a} dx$$ where ...
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What is equivalent of Peano axioms for Real numbers?

I know Peano arithmeic defines natural numbers and their behaviour. I am interested in whether there exists equivalent for real numbers, or even for rational numbers. And whether there is a consensus ...
Tereza Tizkova's user avatar
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How to show that the increment operation for natural numbers is well defined?

Let $a,b$ be natural numbers and $++$ be an increment operation. Based on Peano axioms alone, how to show that if $a=b$, then $a++ = b++$? Do note that the book I am reading (Real Analysis by Terrence ...
Stokolos Ilya's user avatar
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3 answers
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How to phrase the proof of $m \lt n$ if and only if $m \le n-1$

I have been reading Knuth's "The Art of Computer Programming" and in the mathematical preliminaries chapter of volume 1 there is on page 476 the answer to an exercise where he states ... ...
branco's user avatar
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How to prove $a ≤ b$ OR $b ≤ a$ for all $a, b$ in $\mathbb{N}$?

I'm currently reading Terence Tao's "Analysis I" and I'm at the beginning where he defines the natural numbers and proves some of their basic properties using the Peano axioms. I've nearly ...
TheBanker22's user avatar
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1 answer
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Pairs of infinite subsets of natural numbers that intersect on an infinite set give the same property to triples?

Let $A \subseteq P(\mathbb{N})$ such that $A$ is infinite and for each pair $B,C \in A$ then $B \cap C$ is infinite, this implies that for any triple $B,C,D \in A$, $B\cap C \cap D \neq \varnothing$? ...
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2 answers
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Why does $\forall n \in \mathbb{N} \vdash P(n)$ not imply $\forall n \in \mathbb{N}P(n)$?

I am trying to understand why $\forall n \in \mathbb{N} \vdash P(n)$ doesn't imply $\forall n \in \mathbb{N}P(n)$ by studying the concepts in this answer, and had 2 questions about the difference ...
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Find all $(a,d)\in\{1, 2, ..., 9\}^2$ such that $ad$ is a square [duplicate]

Find all six-digit numbers of the form $abcbcd$ for which $a \cdot b^2 \cdot c^2 \cdot d$ is a perfect square of a natural number (note: 0 is not a natural number, so obviously $a, b, c, d \neq 0$, ...
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Consecutive multiplication of natural numbers problem [duplicate]

Prove that the product of any three consecutive natural numbers is not a perfect square. If there were four numbers, I know how to solve the problem, as I would somehow mention the number $ n(n+1)(n+2)...
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2 answers
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Find all pairs of natural numbers $m$ and $n$ that satisfy the following conditions: [closed]

Find all pairs of natural numbers $m$ and $n$ that satisfy the following conditions: $m$ and $n$ are two-digit numbers, $m - n = 16$, the last digit of the numbers ...
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2 answers
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Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$

This is a trivial task: Given $a \in \mathbb{N}$ and $$a=4$$ Show $$a > 1$$ Part of the challenge for newcomers like me is that "easy" tasks actually make it harder to think about the ...
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14 votes
8 answers
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Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I)

In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms: Axiom 3.4 If $a$ is an object, then there exists a set $...
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$f((x,y))=(x*gcd(x,y),y) $is injective and surjective?

The function $f:\mathbb{N^+}\times\mathbb{N^+} \rightarrow \mathbb{N^+}\times\mathbb{N^+} $ definide as $f((x,y))=(x*gcd(x,y),y) $is injective and surjective? Let $(a,b)\in\mathbb{N^+}\times\mathbb{N^+...
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Characterizing cofinite submonoids of $\langle \mathbb{N}, + \rangle$?

A set $S \subseteq \mathbb{N}$ is a submonoid of $\langle \mathbb{N}, + \rangle$ when $0 \in S$ and $S$ is closed under addition (that is, $m+n \in S$ whenever $m$ and $n$ are). For example, $\mathbb{...
templatetypedef's user avatar
5 votes
1 answer
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how to find expression of $a\text{ mod } n$ using polynomial of $e^{\frac{2a\pi i}{n}}$?

I tried to find a polynomial for $a\text{ mod } n$ (for $a,n\in N$) by using powers of $\exp\left(\frac{2a\pi i}{n}\right)$ which mean find the coefficients $f(n,k)$ in $$ a\text{ mod } n=\sum_{k=0}^{...
Faoler's user avatar
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1 answer
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Do we have complete understanding of $\mathbb N$?

We have some understanding of natural numbers. We have PA theory and we believe that $\mathbb N$ is one of the PA models. But PA can't prove some statements about $\mathbb N$ even though they are true ...
user341's user avatar
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1 answer
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These constrained alternating series always satisfy an inequality.

Let $x_0 \in \Bbb{N}$ and suppose that $x_1 \lt \frac{x_0}{2}$, while $0 = x_n \leq \dots \leq x_3 \leq x_2\leq x_1$. Then is it possible that: $$ \sum_{i = 0}^n (-1)^i x_i \gt 1 $$ no matter what ...
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2 answers
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proving the set of natural numbers is infinite (Tao Ex 2.6.3)

Tao's Analysis I 4th ed has the following exercise 3.6.3: Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$

Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational numbers. $f$ is strictly increasing in both arguments. Can $f$ be one-to-one? This question is related to many ...
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2 votes
2 answers
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Proof that each natural number has a unique successor

I've proven that every positive natural number has a unique predecessor using Peano's axioms. But now, I was wondering how I could prove that every natural number has a unique successor using the same ...
Aryaan's user avatar
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2 answers
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Prove that the only natural value of $g$ that makes $8g+4p^2-4p+1$ ($p\in\mathbb{N}$) a perfect square is $g=p$

This problem came up while I was working on a larger problem and I've looked at a few different ways of solving it, the main approach being mathematical induction; however, I've been unable to prove ...
b_rop's user avatar
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1 answer
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Proof sum of first n natural numbers is natural

Im sure most of you know about the proof of the sum of $n$ consecutive numbers $\sum_1^n = \frac{n(n+1)}{2}$. The proof to this identity is quite common and I had no problems finding it. However I'm ...
st30's user avatar
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2 answers
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Why does the principle of mathematical induction work for integers?

I took a course on the foundations of mathematics a while ago and we went through the construction of the natural numbers and then the integers. We did prove the principle of mathematical induction (...
nazorated's user avatar
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2 answers
283 views

Can we modify the Peano axioms like this? [closed]

I am wondering if the following modifications of the Peano axioms result in a set of axioms equivalent to the Peano axioms, in the sense that any set of numbers satisfies these modified axioms if and ...
Princess Mia's user avatar
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Picking random natural number

It is not possible to pick a random natural number (out of all natural numbers), such that each number has the same probability. But is it possible to define a probability distribution over the ...
tmlen's user avatar
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2 answers
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What is the definition of an infinite sequence?

Usually when an infinite sequence is described in simple words, someone writes something like this: $(1,2,3,...)$. So it's clear to everyone that the first element is $1$, the second is $2$, the third ...
user3635700's user avatar
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1 answer
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Factorial for Natural Number Object

It is Awodey Exercise 17 in Chapter 9. It asks to define factorial as an arrow $N \to N$ for a natural number object. Awodey page 246 and 247 defines how to add a natural number by recursion. That is, ...
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Reason for property of convex decreasing sequences of reciprocals of positive integers; property defined with greedy algorithm

Let $a_0 < a_1$ be positive integers. Define recursively the sequence $(a_n)$ so that $a_{n+1}$ is the greatest positive integer, if one exists, such that $\left( \frac{1}{a_{n-1}}, \frac{1}{a_n}, \...
Adam Rubinson's user avatar
-1 votes
1 answer
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What is the intuitive meaning of natural numbers as constructed in ZF set theory?

My understanding is that in elementary set theory, the natural numbers are defined so that $0 = \emptyset$ and $n+1 = n \cup \{ n \}$. I understand that this gives us some very pleasant properties ...
Kevin's user avatar
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1 answer
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Proving that the set of non-negative half-integers satisfies Peano's axioms

I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define ...
Aryaan's user avatar
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2 votes
5 answers
443 views

Clarification about Cantor's Diagonal argument compared to Natural Numbers

I'm not a mathematician but I am a software engineering student. From what I've understood so far, the Cantor diagonal argument proves that the real numbers are infinite and uncountable. My biggest ...
peachyoana's user avatar
1 vote
1 answer
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how to deal with prime number and mod operation?

I had list of equivalent relations that for prime numbers in mod operation. for example its easy to prove that for prime $p$ $$ p\equiv 1 \mod2 \space \space \space \space\space \space\space \...
Faoler's user avatar
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1 answer
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Show that $A = \mathbb N$.

I have a question regarding the following exercise: "Let $A \subseteq \mathbb N$ with the following properties: $1 \in A$ and $\forall n\in A$ it ist true that $2n \in A$ and $2n+1 \in A$." ...
klonedrekt's user avatar
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How to prove that the product of two numbers, each of them being a sum of four perfect squares, is also a sum of four perfect squares? [duplicate]

This is problem # 165 from my beloved handbook of ancient math problems written by Chistyakov. One must prove that if we have two sums and each of these two sums represents a sum of four perfect ...
Alexander's user avatar
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1 vote
1 answer
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How to find three numbers such that the sum of any two of them as well as the total sum is a perfect square?

Chistyakov, problem # 51. One must find three numbers such that the sum of any two of them and also the total sum would be a perfect square. The author does not specify that numbers must be natural, ...
Alexander's user avatar
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-1 votes
1 answer
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Integer solutions of $a^3+2=3b^3$ [closed]

I checked this one with my computer and it seemed that there is no solution except $a=b=1$ , but I don't know if it's true or how to prove it. Here is the full question: a, b are natural numbers. Are ...
Fort Nite's user avatar
2 votes
1 answer
33 views

Equivalent characterizations of finite sets

How can we show that the following notions of finiteness for a nonempty set $X$ are equivalent? There exists $n \in \mathbb{N}$ such that there is an injection $X \hookrightarrow \{1, \ldots, n\}$ ...
Smiley1000's user avatar
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0 votes
2 answers
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Confusion about the validity of the proof of Trichotomy of order for natural numbers in Tao's Analysis

It's well-known that in Tao's Analysis I P28, he provides a provement of Trichotomy of order for natural numbers as follows. Denote the number of correct propositions among the three (i.e. $a<b,\ ...
Richard Mahler's user avatar
3 votes
2 answers
51 views

Determine all nonzero natural numbers $x$, for which, $2023\times(\frac{1}{3}+...+\frac{1}{4x^2-1})\in \mathbb N$

the question Determine all nonzero natural numbers $x$, for which, $$2023\times(\frac{1}{3}+...+\frac{1}{4x^2-1})\in\mathbb N$$ my idea Lets note $2023\times(\frac{1}{3}+...+\frac{1}{4x^2-1})=N$ , ...
IONELA BUCIU's user avatar
2 votes
3 answers
91 views

Find the natural numbers $x,y$ if $\sqrt{x-1}+\sqrt{x+2023}=y$

Question Find the natural numbers $x,y$ if $\sqrt{x-1}+\sqrt{x+2023}=y$ my idea $\sqrt{A}+\sqrt{B}=N$ where we noted $A=x-1$ and $B=x+2023$ and $N=y$ $\sqrt{A}=N-\sqrt{B}|^2$ $A=N^2+B-2N\sqrt{B}$ $\...
IONELA BUCIU's user avatar
3 votes
1 answer
29 views

Which is the smallest integer $c>0$ making $a\cdot (b+c)$ a pairing function for positive integer $a,b$ with $a \le a_{max}$ and $b \le b_{max}$?

The goal is to generate $a_{max} \cdot b_{max}$ different numbers with $$a\cdot (b+c)$$ using an integer offset $c>0$ as small as possible. We know $a \in [1,a_{max}]$, and $b \in[1,b_{max}]$ I'm ...
J. Doe's user avatar
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2 votes
0 answers
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Is there a name for this set-theoretical definition of natural numbers, or has it been invented?

I'll call it the binary encoding with sets. I think it's nice and trivial, should have been discovered by many genius brains, but i can't find it by searching with efforts. Prior arts are Zermelo's ...
Farter Yang's user avatar
1 vote
1 answer
123 views

Definition of natural numbers in category theory

It is said that category theory serves as an alternative foundation of mathematics, as such it must define natural numbers in terms of categories as it is done in set theory, when we consider set ...
RataMágica's user avatar
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0 answers
23 views

Prove $\forall{m}\in \mathbb{N} - \{ 1 \}. \exists B_m \in \mathbb{N}. \forall{n} \in \mathbb{N}. n \geq B_m \implies n! > m^n$

Im trying to prove that $\forall{m}\in \mathbb{N} - \{ 1 \}. \exists B_m \in \mathbb{N}. \forall{n} \in \mathbb{N}. n \geq B_m \implies n! > m^{B_m}$ There seems to be a pattern. For $n=2 \...
xxtensionxx's user avatar
0 votes
1 answer
76 views

Does the following conditions ensure the infinite set B is enumerable?

Let $B$ be an infinite set. Suppose we take input set as $B$ and output set as $\mathbb{N}$ (1) $∀\ b∈ B, ∃!\ n∈ \mathbb{N}, \text{output $(b)$ is $n$}$ This ensures $f:B→N$ exist. (2) $∀\ n∈ \mathbb{...
lorilori's user avatar
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1 vote
1 answer
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Is addition by a specific nonzero natural number a term function in this structure?

Consider the structure $(\mathbb{N};+,\times,0)$. I know that every nonzero natural number $k$ is definable by a first-order formula in that structure, and hence, so is the unary function $x+k$. ...
user107952's user avatar
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Is addition a term-function in this structure?

This is a follow-up to my previous model theory question, here: Is addition definable from successor and multiplication?. I asked whether addition is definable by a first-order formula in the ...
user107952's user avatar
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1 vote
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Is addition definable from successor and multiplication? [duplicate]

Consider the structure $(\mathbb{N};\times,S,0,1)$, where $\times$ represents multiplication and $S$ represents successor. Is the addition function definable by a first-order formula in that structure?...
user107952's user avatar
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Strong finite induction

If I have a proposition $P(n), n \in \{ 1,2,\ldots, m\}$ and I want to prove it by induction, can I proceed like this: Show that $P(1)$ is true Suppose that $P(n)$ is true for all $n \in \{1,2,\ldots,...
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1 answer
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A case of induction! $ n^3 − n^2 + 1 $ is an odd number for all n ∈ N.

So I am learning about induction and I am trying to prove that $(n^3 - n^2 + 1)$ is an odd number for all n ∈ N (natural number) I want to see if I am on the right track to understanding this and if ...
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