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

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5
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0answers
38 views

Heuristics of the sum of squared naturals $(1^2 + 2^2 + 3^2 \cdots + n^2)$

I'm new and this is my first question (though I've been lurking). English is not my native language. Studying on my own. I'm really interested in deriving the formula $1^{2} + 2^{2} + 3^{2} + \cdots+ ...
2
votes
3answers
93 views

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any ...
3
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1answer
52 views

Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$

It seems obvious that $$\forall n \in \Bbb Z^+, \sum_{i=1}^n{1} = n $$ However, I'm having trouble coming up with a formal proof for this. Given a concrete number like $4$, we can say that ...
1
vote
2answers
65 views

Possible not countable extension of the natural numbers?

This question comes from:Is $1234567891011121314151617181920212223......$ an integer? We define $\mathcal{A}$ as the set of infinite strings of digits $$ \bar a_i=a_0 a_1a_2a_3\cdots a_i \cdots ...
2
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1answer
99 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
2
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1answer
17 views

Number of equivalence classes of binary sequences which differ only by finitely many elements.

This question rose up when i was reading a problem the author used to argue against the axiom of choice. Consider the set of all (infinite) sequences of 0's and 1's. Q1) How many such sequences are ...
2
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3answers
109 views

Why do some accept zero as a natural number but others don't? [duplicate]

I have had many teachers who have told me that zero is a natural number but then there is those teachers who say its not. why is that ?
0
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1answer
33 views

How to derive bounds for the $n$-th term of a subsequence of $\mathbb {N} $, knowing two functions “squeezing” the number of the terms below $x$?

Let $ a_n $ be the $n $-th term of an infinite strictly increasing subsequence of $ \mathbb{N}$ and denote with $\nu(x)$ the number of terms smaller than or equal to $x$. Assume also ...
90
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11answers
6k views

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
3
votes
2answers
92 views

Is there a name for property $n+k=m+k\implies n=m$?

Monoid of natural numbers with addition have such property, that for any $n,m, k \in \mathbb{N}$ if $n+k=m+k$ then $n=m$. Does this property have some name in English?
6
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3answers
102 views

Why isn't it necessary to postulate the existence of $1$?

These are the Peano axioms, I'll focus on the second one now: If $a$ is a number, the successor of $a$ is a number. Basically, here is defined the successor function $S(n)=n+1$. My question is, ...
3
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2answers
124 views

Three digit number $ABC$ with $ABC = A + B^2 + C^3$

Is there a trick for solving this problem about number of digits? $ABC$ is a three-digit natural number, such that $ABC = A + B^2 + C^3$. According to above equation what is $ABC$ ?
3
votes
1answer
32 views

Cardinality of subsets of $\mathbb{N}$ with fixed asymptotic density

For a set $S\subset \mathbb{N}$, let $$a(S)=\lim_{n\rightarrow\infty}\frac{\#\{s\in S\>|\>s\le n\}}{n}$$ be the limiting asymptotic density of $S$ in the natural numbers if the limit exists, ...
0
votes
1answer
100 views

Is $\mathbb{N}$ a well-founded set?

I was reading about Von Neumann's construction of $\mathbb{N}$, I understood that $\mathbb{N}=\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\},...\} $. I see that, with this construction, ...
2
votes
2answers
113 views

Strictly monotonically increasing sequences of natural numbers

I have several questions with regards to these sequences: What is the cardinality of the set of all such sequences? I assume that it is equal to the cardinality of $\mathbb{R}$, is that correct? ...
1
vote
2answers
123 views

Natural numbers in set theory is {0,1,2,…}?

The set of natural numbers $\mathbb{N}$ in set theory is defined with the axiom of infinity as the smallest inductive set and then it is usually proven that $\mathbb{N}$ satisfies the Peano axioms and ...
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vote
3answers
120 views

Is natural numbers set $\mathbb N$ infinite set?

A set with uncountable number of elements is called an infinite set. Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set? As far i know $\Bbb N$ is "countably" ...
6
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2answers
53 views

Gallian: is it true that the well-ordering principle can't be proven from properties of arithmetic?

I just started reading Gallian's Abstract algebra and on page 3 it says "An important property of the integers...is the so-called Well Ordering Principle. Since this property cannot be proved from the ...
0
votes
2answers
33 views

Induction proof for $x \le y $

So these are $x \in \{1 ... i\}$ and $y \in \{ i + 1 ... n\}$, for $i, n \in \mathbb N$. I want to prove it for every $x \le y $. I know it`s easy but the solution is escaping me. I have tried with ...
0
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0answers
31 views

Solving system of inequalities, with solution in only natural numbers, with priority on variables

If I have the equations $27a+30b+33c+36c \geq x$ $a+b+c+d=4$ and want to solve them using only natural numbers (including 0) for both $x=131 $ and $x=142 $ preferably but not necessarily with ...
0
votes
1answer
23 views

Show natural numbers ordered by divisibility is a distributive lattice.

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a ...
4
votes
1answer
116 views

Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
0
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1answer
41 views

showing the natural numbers exist from axioms (help with making sense of book)

I'm now on page 40 of a set theory book and I've hit the natural numbers. I think the book has oversimplified some things. The successor of a set $x$ is defined to be $S(x)=x\cup\{x\}$ A set $I$ is ...
2
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2answers
93 views

Showing there is no natural number between two consecutive natural numbers

I want to show that: $x\subset S(x)$ where $S$ is the Successor function and $\not\exists z:x\subset z\subset S(x)$ These are obvious results, but the relation of $m<n\iff m\in n$ is given as a ...
1
vote
1answer
91 views

Conjecture: three or more decompositions into powers with a base differing by 1 means its a perfect power

If $$(i_1)^{a_1}(i_1+1)^{b_1}=n $$ $$(i_2)^{a_2}(i_2+1)^{b_2}=n $$ $$(i_3)^{a_3}(i_3+1)^{b_3}=n $$ where all the terms are positive integers and the groups ...
3
votes
2answers
69 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
votes
2answers
83 views

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

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
votes
1answer
42 views

Let $\gamma$ be the Euler-Mascheroni constant. Can there be natural numbers $a,b,c$ such that $\log a - \log b - \log \log \log c =\gamma$?

Can there be integers satisfying $$\ \log a - \log b - \log \log \log c = \gamma \ \ \ ? $$
0
votes
5answers
81 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 $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
1
vote
1answer
79 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
839 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
91 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
votes
0answers
24 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
77 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, ...
6
votes
2answers
459 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
60 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
42 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 ...
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3answers
109 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
32 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
24 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
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1answer
25 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
35 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
37 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
63 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
51 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
38 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
397 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
29 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
91 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\{ ...
0
votes
2answers
65 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 ...