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

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3
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2answers
50 views

Are these two definitions of the natural numbers equivalent?

If we consider two definitions of the natural numbers: Definition 1 $N$ is the set that satisfies all of: There is an element $0$ in $N$. For each element $n$ in $N$, there is the successor of ...
0
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2answers
52 views

An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
0
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1answer
37 views
2
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1answer
39 views

Is this version of Lagrange's four-square theorem true?

Lagrange's four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1\times a_1 + a_2\times a_2 + a_3\times a_3 + a_4\times a_4$ ...
0
votes
5answers
75 views

What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
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0answers
33 views
1
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1answer
76 views

What sets does $\mathbb{N}$ include?

My text states that the set $\{1, 2, 3...\}$, and the set $\{101, 102, 103, 104...\}$ are elements of $\mathbb{N}$. Doesn't this imply that $\mathbb{N}=\{1, 2, 3... 101, 102, 103, 104...\{1, 2, 3 ...
20
votes
3answers
784 views

List of powers of Natural Numbers

Greatings,   Some time ago a friend of mine showed me this astonishing algorithm and recently i tried to find some information about it on the internet but couldn't find anything... Please help. ...
6
votes
2answers
85 views

Count with only certain digits allowed - And feel a fractal

I have a friend ~200 years old mathematician who has forgotten some digits and now he counts things in very strange manner: when he is going to count the $n$-th thing and $n$ contains a digit he ...
3
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0answers
22 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
6
votes
1answer
75 views

An interesting problem with natural numbers

While, preparing for competition, found an interesting problem, but absolutely don't know how to start. $600$ natural numbers from $1$ to $600$ are written in a string(each once) in a certain order, ...
2
votes
3answers
142 views

How are the elementary arithmetics defined?

In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that ...
0
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1answer
50 views

Construction of natural numbers in ZF/ZFC without class predicates.

I'm told by some websites that it's possible to formally construct the naturals in ZF without recourse to either predicates involving classes ("all inductive sets", yuck) or some sort of "intuitive ...
0
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1answer
35 views

Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers

I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title). I've already constructed two non-isomorphic total ...
1
vote
3answers
99 views

Proof of the Principle of mathematical Induction [duplicate]

We always use the Principe of Mathematical induction and we have two versions of it. I myself have been using it for many years. But it just came to my mind that I have never seen a proof of the ...
0
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4answers
30 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
1
vote
0answers
22 views

Is there a difference in the rate of decrease between $f(x)$ and $g(x)$ for increasing $x$?

I have the following two functions of $x$: $ f(x) = \frac{c}{c + (N-1)o + Nd + xl}$ $g(x) = ae + (1-a)\frac{1}{x+2N}$ with $0 \leq a, e, c, o, d, l \leq 1$ and $N, x \in \mathbb{N}^+$. For both ...
3
votes
1answer
24 views

Submagmas of natural numbers

What is known about submagmas of natural numbers under addition/multiplication? For example, all subgroups of integers under addition are of the form $~n \mathbb{Z}~$. Are there similar results for ...
2
votes
4answers
32 views

Use Induction to prove: $(1+2x)^n \geq 1+2nx$

Show by induction that: for all $x>0$ that $(1+2x)^n \geq 1+2nx$ So far I have: for $n=1 \rightarrow (1+2x)^1 \geq 1+2x$. True! for $n=k+1 \rightarrow (1+2x)^{k+1} \geq 1+2(k+1)x$ ...
1
vote
0answers
33 views

Does there exist a positive integer which is a power of two, such that by rearranging it's digits we can get another power of two?

Does there exist a positive integer which is a power of two, such that by rearranging it's digits we can get another power of two? I know a quite good solution, that involves working with sum of ...
0
votes
2answers
50 views

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$. Interested if there is a nice quick way other than mine.
1
vote
2answers
36 views

Open/closeness of subsets of natural numbers

So I've just started reading about neighbourhood and Hausdorff space. It makes me wonder if $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ is Hausdorff and why, and are sets in $\mathbb{N}$ open or closed or? ...
0
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1answer
35 views

Calulus on the set of natural numbers

Can we think about Calculus on $\mathbf{N}$? How would work the notion of neighborhood, limit, continuity and differentiability and analyticity? I need these to understand discrete dynamics over ...
4
votes
3answers
393 views

Difference of inverse squares

Given that the positive number $a$ is the difference of inverse squares: $$a = \frac{1}{n^2} - \frac{1}{m^2}, m, n \in \mathbb{N},$$ could it well be that the $pa$ is also a difference of inverse ...
0
votes
1answer
24 views

Finite binary representation of float number

How can I prove that float number $x$ has the finite binary representation if and only if it is written like that: $ x = m / 2^n $, where $m, n \in \mathbb N$. Should I consider something like 3 ...
1
vote
1answer
87 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
1
vote
2answers
57 views

How to prove that if m and n are natural numbers than m+n is also a natural number?

Problem sounds easy enough - prove that if $m$ is in set of all natural numbers (let's call it $\mathbb N$) and so is $n$ than $m+n$ also must be there. Probably it should be done using induction. But ...
0
votes
0answers
48 views

Sum/Product of two natural numbers is a natura number

I wanted to prove that the sum and the product of two natural numbers is a natural number. Intuitively it's clear to my why that is true, however I couldn't prove it. So our lecturer first defined ...
1
vote
2answers
46 views

What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

I've been working on a particular set theory problem for a while and essentially I've hit a roadblock because of this question. I just need to know what this ordinal is, because I have a sneaking ...
1
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0answers
28 views

Let$\ H$ be a hyperinteger. If$\ f(n)=g(n)$ is true for all$\ n \in \mathbb{N}$, will it be so for all$\ H$?

I do know that all true first order statements in$\ \mathbb{N}$ are also valid in$\ \mathbb{N^*}$, so for example$\ \sin(H \pi) =0$. As a consequence, my question is equivalent to: is$\ f(n)=g(n)$ a ...
2
votes
3answers
75 views

Sum of squares of two integers divisible by five [closed]

Supposing $x,y$ are natural numbers, what is the probability that the sum of their squares are divisible by 5? I am getting $1/3$ as squares can only end with $0,1,4,5,6,9$. So $36$ pairs are ...
1
vote
2answers
58 views

Natural numbers object via initial morphism

I assume that a natural number object (or see nLab) can be defined as an initial morphisms. (edit: as in the title, I ment initial morphism, not objects) $\hspace{1cm}$ Thoughts: Probably $X:=1$, ...
3
votes
3answers
60 views

Unique element m in N

Let $0 < x$. Show that there is a unique $m \in \mathbb{N}$ such that $m-1 \leq x < m$. Hint: Consider the set $\{ n \in \mathbb{N} : x < n\}$ and use the well-ordering of $\mathbb{N}$. The ...
25
votes
2answers
5k views

Demonstration that 0 = 1 [duplicate]

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e ...
0
votes
1answer
72 views

Prove with combinatorial arguments this equation [duplicate]

Prove with combinatorial arguments, that, $\forall n \in \mathbb{N}$. $$\sum_{k=0}^n (-1)^k {n \choose k} =0$$
1
vote
2answers
48 views

Metric on natural numbers united with infinity

can anyone give me an example for the following metric $d$? Let $\Omega = \mathbb{N}_+ \cup \{ \infty \}$ and $d$ be a metric such that all points $n \in \mathbb{N}_+$ are isolated w.r.t. $d$ and ...
1
vote
0answers
34 views

Why is multiplication commutative - intuitive explanation [duplicate]

While I know that both addition and multiplication are commutative operations, I can easily visualize that, e.g. 3 + 4 = 4 + 3 = 7 by thinking of seven objects in a row and separating them into two ...
11
votes
5answers
1k views

What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
4
votes
2answers
62 views

Encyclopedia of integers

Many years ago I read something that mentioned a book I would like to find. Apparently this book is sort of an encyclopedia for integers; each entry lists interesting mathematical facts about that ...
3
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1answer
59 views

Determine the divisibility of a given number without performing full division

My question is slightly more complicated than what's implied on the title, so I will start with an example. Given any number $N$ on base $10$, we can easily determine whether or not $N$ is divisible ...
0
votes
2answers
124 views

Reciprocal of 81 being the sequence of all natural numbers?

According to this document: http://www.answering-christianity.com/fakir60/81.htm describing the theory of scientist Peter Plichta, the reciprocal of 81 is: the ...
3
votes
2answers
59 views

Find the 6 Digit Number

$N$ is a 6 digit Natural number such that its sum of the digits is $43$. Find $N$ if Exactly One of the statements below is False: $(1)$ $N$ is a Perfect Square $(2)$ $N$ is a Perfect Cube $(3)$ N ...
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vote
0answers
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How solve the equation $a^x+\left(2a+1\right)^y=\left(a+1\right)^z$ for $a\in N - \{1\}$ and $x,y,z\in N\cup\{0\}$?

How solve the equation in natural numbers $a^x+\left(2a+1\right)^y=\left(a+1\right)^z$ for $a\in N - \{1\}$ and $x,y,z\in N\cup\{0\}$?
2
votes
2answers
114 views

coin problem with two coins, inductive proof

Adjustment This proof is flawed. I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime ...
1
vote
1answer
74 views

Natural and Real sets of numbers, which one is bigger than another?

From the years ago, it has always been this question in my mind which a teacher of high school talked about in a class but I never found it's correct answer. We have set of natural numbers ...
-1
votes
1answer
22 views

How can we generate a $2$-digit number $XY$ on base $B$, such that $BX+Y=Y^X$?

For example, $25$ on base $10$ is equal to $5^2$. This should be pretty easy to solve using fairly simple arithmetic. But I'm finding it hard to generate any other solutions besides the one ...
0
votes
1answer
43 views

Is it possible to represent subsets of natural numbers as groups with prime generators?

I'm learning group theory and I'm trying to consider the "symmetry" of a certain group of natural numbers: Here's the idea, all natural numbers are comprised of multiples of primes. So a subset would ...
3
votes
2answers
68 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
7
votes
3answers
407 views

How to find natural solutions of an equation?

When I'm solving problems, I'm often confronted to solving equations, and when I'm solving equations, I'm often confronted to find the natural solutions of these equations. My actual personal ...
2
votes
2answers
70 views

Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?

This question is very similar to Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set? Consider the following set-theoretic definition of natural ...