Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [natural-numbers]

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

4
votes
1answer
85 views

How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

While answering a question here I noticed that: $$n=1+\sum_{k=1}^{n}{\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor}$$ for every natural number $n$. I tried to demonstrate it using Legendre's ...
0
votes
0answers
15 views

Can we re-read natural numbers as a “binary operation” of Integers Numbers? [on hold]

I start from Binary Operator definition I see an affinity between this definition and the 'difference' between Integers and Natural numbers. I ask if I can value in this way +1 -1 as $x, y$ |1| as $...
1
vote
0answers
30 views

Writing down consecutive natural numbers until a certain number of digits $k$ is reached.

A person starts writing consecutive natural numbers from $5$ until $k$ digits are reached. For some values of $k$, this will be impossible, for example $6$ or $8$ are impossible as then after writing ...
2
votes
1answer
22 views

Determine all $a$ and $b$ natural numbers such that $\frac {a^2+2b} {b^2-2a}$ and $\frac {b^2+2a} {a^2-2b}$ are whole numbers.

I proceeded in the following way: It is clear that $a \ne 0$ and $b \ne 0$. Let $\frac {a^2+2b} {b^2-2a} = k, k \in \mathbb{Z} \tag 1$ and $\frac {b^2+2a} {a^2-2b} = m, m \in \mathbb{Z} \tag 2$ ...
0
votes
1answer
75 views

How is induction (without hypothesis) to be used in this proof that $a\ne{b}\implies{a<b\lor{b<a}}$ for $\mathbb{N}$?

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. It's part ...
0
votes
0answers
17 views

Do two pairs of distinct natural numbers exist such that AGM(A,B) equal to AGM(C,D)?

Here AGM is arithmetic-geometric mean. Are there natural numbers A,B,C,D such that $1\leq A<C<D<B$ and arithmetic-geometric mean AGM(A,B)=AGM(C,D) ? In other words, is AGM a homomorphism ...
0
votes
0answers
21 views

Can arithmetic-geometric mean of two distinct natural numbers be a natural number?

Are there solutions to equation $AGM(A,B)=C$ such that $1\leq A<B$ and $A,B,C\in\mathbb{N}$?
0
votes
2answers
56 views

How should we parse this proof that every infinite set has a subset which is eqivalent to the set of natural numbers?

This is again from the chapter Construction of the System of Real Numbers in The Fundamentals of Mathematics, Volume 1. It may be the case that the original wording in the German language would be ...
1
vote
1answer
45 views

Is this a logically satisfacotry proof of the associativity of natural number addition?

This is again related to my previous question How should this proof of the associativity of natural number addition be understood? Here the natural numbers are defined as starting with one. Peano's ...
0
votes
1answer
38 views

Is this sufficient to prove the associativity of natural number addition?

This question is regarding the same proposition as is discussed in How should this proof of the associativity of natural number addition be understood? Here I wish to ask if a much simpler proof than ...
0
votes
0answers
27 views

Find the natural number which is divisibled by another number

Let $n= \overline{a_1a_2a_3}$ ($a_1 \neq a_2$, $a_2 \neq a_3$, $a_3 \neq a_1$). 1/ How many possible values does $n$ have which is divisibled by $7$? 2/ How many possible values does $n$ have which ...
-1
votes
0answers
35 views

What's the algorithm to determine whether a string is magic or not?

A string is said to be magic string if It consists of only digits Its digit form 8 unique numbers between 1 and 80 All digits must be used Digits must be used in order No two numbers should not use ...
3
votes
1answer
90 views

What class of natural numbers known to be infinitely large occurs least frequently? [closed]

By "class" I mean sets of natural numbers with a given specific property (i.e. prime numbers or perfect numbers). Obviously all infinitely large sets have the same "size", but for example solitary (as ...
0
votes
1answer
52 views

How many $10-$digit numbers are divided by $11.111$ and all the digits are different?

The Problem: How many $10-$digit numbers are divided by $11.111$ and all the digits are different? A) $3250$ B) $3456$ C) $3624$ D) $3842$ E) $4020$ The Problematic point ...
4
votes
2answers
95 views

If a prime $p$ divides $n^2$ then it also divides $n$ - Is this proof correct?

I'm learning Real Analysis by myself and I wanted to prove that if a prime $p$ divides $n^2$ where $n$ is an integer, then $p$ divides $n$ itself. I saw that proving this is the same as saying that ...
3
votes
0answers
48 views

Define a model for $\mathbb N$ without set theory

I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops. It led me to do a bit of elementary ...
-1
votes
1answer
44 views

$\omega^\omega$ correspondence with $\mathbb R$-irrationality

Here in the second comment I do not understand why $\omega^\omega$ corresponds to irrational numbers? : In my experience one typically identifies $ω^ω$ with the irrational elements of R; and then we ...
0
votes
1answer
27 views

How to solve for the sides of a rectangle whose sides are natural numbers given its area is a known natural number?

This may not be the best way to formulate the question but I am looking for a method to solve the following equation $d \cdot n'=n$ where $d, n', n \in \mathbb{N}$ and $n \neq1$ is known. How should I ...
3
votes
1answer
48 views

Does there exist a higher-dimensional 5-sided “tetrahedron + 1”?

The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. ...
0
votes
1answer
19 views

Conditions for solutions to $p^a = q^b$

When (for what conditions on $p,q$) can we solve this equation for integers. I know that $p = q^b$ can only be solved when $p$ is an integer to the power of $b$. By analysing the prime factorisation ...
2
votes
4answers
54 views

Prove that $a_n \in [0,2)$

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence, with $a_0=0$, $a_{n+1}=\frac{6+a_n}{6-a_n}$. Prove that $a_n \in [0,2)$ $\forall n \in \mathbb{N}$ Here's what I did: I tried to prove this by ...
3
votes
2answers
50 views

Bijection between $\{0,1\}^*$ and the natural numbers.

So the tasks is to show that $\{0,1\}^*$ is countable. So the idea that i am having is that each number can be mapped to it's own in decimal. $f(1001)= 9$ $f(101)=5$ But what happens with all the ...
4
votes
5answers
338 views

How to solve this Diophantine equation?

Can anyone say how one can find solutions to the Diophantine equation $$x^3+y^4=z^2$$ in General? Only a few triples of numbers have been found, and most likely this equation has infinitely many ...
1
vote
1answer
53 views

$\omega^\omega$ correspondence with $\mathbb R$

How does the natural continuous bijection between $\omega^\omega$ and $\mathbb R$ look like? I.e. why elements of $\omega^\omega$ are called reals?
0
votes
2answers
79 views

proving that f:N^2->N is bijective

I'v got a function $f:\omega^2\to\omega$ defined $$ f(n,k)=\frac{\left(n+k+1\right)\left(n+k\right)}{2}+n $$ This function suppose to be a bijection between $\omega$ and $\omega^2$, but I can't find ...
7
votes
1answer
66 views

Build a wall with three types of bricks. What is the max length of wall less than 1000 cm you won't be able to lay?

You need to build a wall of length no longer than $1000$ cm. You can use bricks of three sizes: $23$ cm, $27$ cm or $36$ cm, and you are not allowed to cut bricks. What is the maximum length ...
3
votes
1answer
40 views

What is a minimal set of rules that determine the usual order on $\Bbb{N}$ given that $1 \lt p_1 \lt p_2 \lt \dots$?

Let $p_i$ always mean the $i$th prime number. Given two numbers $a, b \in \Bbb{N}$ in the form $a = \{(p_i, e_i) : p_i^{e_i} \mid a, e_i \text{ maximal}\}$ ie. essentially their unique ...
2
votes
1answer
80 views

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
3
votes
0answers
48 views

Defining Addition of Natural Numbers as the Algebra of 'Push-Along' Functions

Let $N$ be a set containing an element $1$ and $\sigma: N \to N$ an injective function satisfying the following two properties: $\tag 1 1 \notin \sigma(N)$ $\tag 2 (\forall M \subset N) \;\text{If } ...
4
votes
1answer
69 views

On randomly permuting digits in a large random number

Suppose you're given a natural number with $N$ digits, randomly chosen except that none of the digits are $0$. Now shuffle its digits to obtain a new $N$-digit number. What is the probability (as $N\...
0
votes
1answer
37 views

Definition of the Number Two in ZF Set Theory

This page shows how natural numbers may be defined as unions of sets. What is a straightforward way to appeal to intuitive notions to dispel the misconception that {Ø} + {Ø} = {{Ø}, {Ø}, Ø} and ...
0
votes
2answers
46 views

The theory of natural number is complete? [closed]

Let be T the first-order theory of $(\mathbb{N}, <, +, 0, 1)$ with language $L = \{<, +, 0, 1\}$. Is T complete?
3
votes
2answers
34 views

Help proving there is a sequence of rational numbers

I'm trying to prove the following: Let $\Bbb Q$ be the countable set of rational numbers and $\{x_n\}_{n=1}^\infty$ be a sequence such that for every q $\in$ $\Bbb Q$ there is a $n \in \Bbb N$ with $...
0
votes
0answers
52 views

Boolean algebras/ Shelah/ Unclear step in the proof

Here on the page $10$, what does the displayed formula in the $6$th line $$\text {lg}(\eta_\ell)<\omega\Rightarrow \bigcup\{\text{Rang}(\nu_\ell(k):k<\text{lg}(\nu_\ell))\}\cap\bigcup_{k<\...
30
votes
7answers
4k views

Are there any two numbers such that multiplying them together is the same as putting their digits next to each other?

I have two natural numbers, A and B, such that A * B = AB. Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018. From trying out a lot of different combinations,...
0
votes
2answers
44 views

$1 < a$ and $b\ne0$ imply $1<a^b$

$1 < a$ and $b\ne0$ imply $1<a^b$ when $a,b$ are arbitrary nonnegative integers. I've tried to prove it by induction. I've assumed that $b < a$ (Is valid my assumption?) I'm using this ...
2
votes
2answers
40 views

$a = d$ implies $a^b = d^b$

Prove that $a = d$ implies $a^b = d^b$, where $a, d$ are arbitrary nonnegative integers and $b$ is any positive integer. If I could use division I think it could be something like that: $a^b / d^b ...
1
vote
1answer
81 views

$a < b$ and $c<d$ imply $a+c < b+d$

$a < b$ and $c<d$ imply $a+c < b+d$ when $a,b,c,d$ are arbitrary nonnegative integers. I know that (assuming we include zero) $$\begin{align*} a<b \Leftrightarrow (\exists x\in \mathbb ...
1
vote
2answers
64 views

Product $\sigma$-algebra of power sets

Let $X,Y$ be any two sets. In general, what is the product $\sigma$-algebra $\mathcal{P}(X) \times \mathcal{P}(Y)$? In the context (using Fubini's Theorem to prove that one can reverse the order of ...
-1
votes
1answer
52 views

Complicated index models and Boolean algebras/ Shelah/ Unclear step in the proof

Here on the page $10$ in the $5$th line (the proof of lemma $1.10$), Shelah defines $n_*$ as $\omega$: $$n_*=\omega,$$ and then he continues: be such that $n_*\geq\text{max}\{n(0),...,n(m-1)\}<\...
1
vote
1answer
58 views

Natural numbers as a subset of integer numbers: $\mathbb{N}\subset\mathbb{Z}$.

Within set theory, having the natural numbers $\mathbb{N}$ built as the minimal inductive set with the corresponding additive and multiplicative operations defined, integers $\mathbb{Z}$ can be set as ...
0
votes
0answers
13 views

Are the natural numbers definable in the (2nd-order) theory of complete ordered fields?

I read that the natural numbers are not definable in the theory of real closed fields (RCF), which captures the 1st-order properties of the real numbers. That's why RCF doesn't contradict Gödel's ...
2
votes
1answer
27 views

Proving $n \leq 3^{n/3}$ for $n \geq 0$ via the Well-Ordering Principle

I'm attempting to prove: $$n \leq 3^{n/3} \quad \text{for }n \geq 0$$ I'm having a little trouble continuing. This is what I have so far: Suppose for a contradiction there is a subset of ...
0
votes
1answer
36 views

Existence of $T$ such that $[(T\circ T)(f)](n)=f(n+1)$

Defined $A$ as the set of all the functions $f:\mathbb Z\rightarrow \mathbb R$ exists a function $T:A\rightarrow A$ such that $$ (T^2f)(n)=f(n+1)\text{ for all }n\in\mathbb Z\text{ and }f\in A $$ or ...
0
votes
0answers
7 views

Existence part of Iteration theorem

Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → ...
0
votes
0answers
39 views

Why does $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$ exhibit a bijection with the pairs (x,y)?

Consider $$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$ I discovers it essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So intuitively, given any P(x,y) we follow ...
-1
votes
1answer
33 views

$f(x,y)=(x/2^y) \mod 16$ a Bivariate function?

I have a two-input function on the integers or naturals. Is a bivariate function a function that takes two inputs or is there anything more to it? For example take the function: $f(x,y)=(x/2^y) \mod ...
1
vote
6answers
63 views

Let $k\in\Bbb{N}$. Prove that $0<\frac{1}{k}-\ln(1+\frac{1}{k})<\frac{1}{2k^2}$

Not sure how to approach it, tried with basic algebraic manipulation but got no where. We are learning Mean Value Theorem and Taylor's Theorem so I would believe maybe we use one of those two theorems,...
-2
votes
2answers
71 views

Prove using induction that $n^6 < 3^n$,for all $n > 18$

Prove using induction (or using any other elementary precalculus techniques) that $$n^6 < 3^n, \forall n \geq 19.$$ I have no idea how to do this. Writing the induction step, I get that I need to ...
0
votes
1answer
65 views

Question about an inequality.

$$\forall i\in \{1,2,\cdots, k\}, n_i\in\mathbb{N}$$ $$\sum_{i=1}^k n_i =n$$ then $$\sum_{i=1}^k n_i^2\leq n^2-(k-1)(2n-k)$$ Like comment, If we apply induction, $i)\ k=2$ $n_1+n_2=n\land n_1,n_2\...