# 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 ...
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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 ...
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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 ... ...
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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 ...
1 vote
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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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### 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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### 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 $2n+1 \in A$." ...
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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 ...
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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, ...
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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 ...
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### 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 $X \hookrightarrow \{1, \ldots, n\}$ ...
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### 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 ...
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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 ...
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### 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 ...
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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 $x+k$. ...
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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 ...
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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?...
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