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Questions tagged [natural-numbers]

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

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1answer
16 views

Supremum equals maximum on a subset of natural numbers

I'm wondering if $\sup_{x \in M} f(x) = \max_{x \in M} f(x)$ holds when $f$ is some arbitrary function and $M = \{0,1,\dots,n\}$ for some $n \in \mathbb N$. My idea is that $M$ is closed and bounded ...
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4answers
67 views

If $a,b,c$ and $d$ non-zero natural number such that $ab=cd$

Question : If $a,b,c$ and $d$ non-zero natural number such that $ab=cd$ Show that : $a^2+b^2+c^2+d^2$ is not prime number My try : Call $m$ : $\gcd$ of $a,b$ then $m|_a$ and $m|_b$ Then $\...
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0answers
58 views

If $xy57z$ is written in the decimal system, is divisible by $729$, then find $x,y,z$ [on hold]

The number $xy57z$ written in the decimal system, is divisible by $729$. Find $x,y$ and $z$ I find this problems in book and I need solution by step to understand it Problems :
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1answer
75 views

prove that $2^{n}+1$ is divisible by $n=3^k$ for $k≥1$ [closed]

prove that : $2^n+1$ is divisible by all number from : $n=3^k$ for $k≥1$ I find this problems in book and I need ideas to approach it Problems :
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2answers
52 views

Let : $A=\frac{2^{4n+2}+1}{5}$ , $n>1$

Prove that the number A is not primary Such that : $A=\frac{2^{4n+2}+1}{5}$ $n≥2$ n=2 then $A=205$ Please I need some ideas to approach it
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1answer
30 views

Equalities with sum of squares

Does anyone have an idea about the following exercises? I have tried both of them by using induction, but I haven't got it. Prove that for every $k\in{\mathbb{N}}$, if $4k=m_1^2+\dots m_{3+k}^2$ with ...
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1answer
76 views

Why is the definition of inductive set well defined?

I've been studying from Enderton's Mathematical Introduction to Logic in which he defines an inductive set as follows: To simplify our discussion, we will consider an initial set $B \subseteq U$ ...
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0answers
12 views

convergence velocity

I've been given some succession to work on, in particular I've been asked the variation of the velocity of the convergence of the following succession, which obviously depends on $\alpha$: $$a_n=\Big(...
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1answer
36 views

Can even/odd classes be applied in triples, quadruples, etc., and have any uses?

The classes of even and odd numbers has many uses, and we can find rules about combining them. An odd number added to another odd number always yields an even number. Even + even = even. Odd + even = ...
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2answers
36 views

Quadratic equation with natural number coefficients

Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2-bx+c=0$ are distinct and both lie in the interval (0,1) (1,2) (2,3) (Brackets signify open interval, roots are $IN BETWEEN $ ...
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0answers
17 views

General Solution of Linear Diophantine Equation with 2 and more than 2 variable

We know the theorem of linear diophantine equation from Bezout's identity that the solution is in ordered pair form: $\left(x + m \dfrac{b}{\text{gcd}(a,b)}\,,\,y-m \dfrac{a}{\text{gcd}(a,b)}\right)$ ...
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0answers
35 views

Properties of this notion of density in $\Bbb{N}$?

For a given $S \subseteq \Bbb{N}$, the asymptotic density of $S$ is defined as $$d_\text{asy}(S) := \lim_{n \to \infty} {\#(k \in S : k \le n) \over n}$$ If the limit exists. Wikipedia says this is ...
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1answer
41 views

In Non-standard analysis, is the number of natural numbers a hyperreal number?

In Non-standard analysis, is the number of natural numbers a hyperreal number? In other words, if $H$ is the hyperreal infinite unit, does the sum $\sum\limits_{n=1}^H 1$ yield the number of natural ...
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5answers
79 views

Is it true that $n \leq 2^{n-1}$ for all natural numbers $n > 0$? [closed]

Seems to be true but I want to make sure: $n \leq 2^{n-1}, n > 0$
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2answers
47 views

Show that a number $n$ is divisible by 6 if and only if it can be written as a sum of three distinct divisors.

If $6|n$ then $n=6k=3k+2k+k$. And $3k|n$, $2k|n$ and $k|n$. Now let $p,q$ and $r$ be three distinct divisors of $n$ so that : $$n=p+q+r$$ Because $p|n $, $ q|n$ and $r|n$ I figured that $p|q+r $, $ ...
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2answers
57 views

Prove that when writing up all even numbers in a column, then chaining $2n+1$ from them, we get all natural numbers exactly once.

Here is the idea: Write up all even numbers in the first column, then get numbers in the second column by taking the number to the left ($n$), and calculating $2n+1$. And keep repeating this. Here is ...
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1answer
37 views

Proof of equivalence between two methods of binary to decimal conversion.

I have two binary to decimal conversion methods and want a proof - or an intuition at least - of why they are equivalent. The first method is quite intuitive to me and seems to be more popular: $[...
1
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1answer
52 views

Generalised Divisibility

I have a following question: Can we find for any natural number $n \in \mathbb{N}$, a sequence of only $\{0,1\}$ as elements such that the sequence has exactly $n\ 1's$ and is divisible by $n$ when ...
1
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1answer
52 views

Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection [duplicate]

Question: Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection. From what I've read about infinite families, I need to ignore those who have the properpty $...
2
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1answer
64 views

An apparently harmless exercise concerning induction

I'm trying to solve Exercise 12 at page 14 from "A Concrete Introduction to Higher Algebra" by L. Childs. The text is the following. Let $b \in \mathbb{R}, b \ge 2$. Prove that $$(b^n - 1)(b^n - b)(b^...
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1answer
62 views

sum of the first $n^2$ natural numbers closed form

Before I get downvoted I am still a beginner so please bare with me. I know the summation of the first n are $\frac{n(n+1)}{2}$. Does that imply the sum of the first $n^2$ is $\frac{n^2(n^2+1)}{2}$?
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1answer
46 views

Generations of a counting sequence

A counting sequence is a sequence whose terms are also sequences. To be specific, its terms are sequences of natural numbers. [And if you are familiar with the look and say sequence, its very similar ...
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0answers
717 views

Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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0answers
28 views

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$ for all natural numbers $X,Y$ where $X>Y$. Further, there exist integers $...
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2answers
119 views

Prove that $\frac{(72!)}{(36!)^2}-1$ is divisible by 73 [closed]

Prove that $\frac{(72!)}{(36!)^2}-1$ is divisible by 73. My approach is as follow $73n=\frac{(72!)}{(36!)^2}-1$ I tried remainder theorem but could not prove it.
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2answers
55 views

How to determine the smallest value of $N=n^4+6n^3+11n^2+6n$ if 13 and 19 both divide N?

I tried to solve for an integer solution by making N equal to multiples of 247 but this is not leading me anywhere. I then tried using the tests for divisibility which did not seem to lead me anywhere ...
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1answer
31 views

Is this problem solvable with positive integer linear programming?

I have the unknowns $w,x,y,z$ that are all in $\mathbb{N}$ and $\gt0$. The known parameters $\alpha,\beta,\gamma,\delta$ are all in $\mathbb{N}$ and $\gt0$ too. Given $\alpha,\beta,\gamma,\delta$, I ...
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0answers
137 views

Proving that addition in $\mathbb{N}$ is associative and commutative

I would like to prove the associative and commutative property of the natural numbers by using Peanos axioms only. Did I justify every step in my proofs correctly? Definition. $\forall n,m \in \...
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1answer
54 views

Existence of 3 natural numbers that divide each other when squared and have 1 taken away from them

Does there exist natural numbers, $a,b,c > 1$, such that; $a^2 - 1$ is divisible by $b$ and $c$, $b^2 - 1$ is divisible by $a$ and $c$ and $c^2 - 1$ is divisible by $a$ and $b$.
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1answer
28 views

formal definition of operations in the ring of integers $\mathbb{Z}$

How do I compute step by step this formal definition of the multiplication operation $\cdot_{\mathbb Z}$ in $\mathbb Z$ $$(m,n)\cdot_{\mathbb Z}(p,q)=(m\cdot p+n\cdot q,m\cdot q+n\cdot p) ?$$ What ...
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1answer
36 views

Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite.

Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite. I have: Let $I_a = \{i \in n:f(i)=a\}$ for $a \in A$. Since $f$ is onto $A$, $I_a$ is nonempty, and by the well-...
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1answer
42 views

Does the Axiom schema of Replacement imply the Axiom of Infinity? [duplicate]

The axiom of infinity says that the set of natural numbers exists, while the axiom of replacement says that if an object (a member of a set) exists, then all definable mappings of that object yield ...
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0answers
30 views

Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto.

Question: Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto. Then show that if $i\in m$ and $j \in n$, then $p(i,j) \in p(m,n)$. The book offers a ...
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1answer
98 views

For any natural $m \neq n$ show that $|\sqrt[n]{m} - \sqrt[m]{n}| > \frac{1}{mn}$.

Here's my try. The inequality above is equivalent to $$|m^{\frac{1}{n}} - n^{\frac{1}{m}}|> \frac{1}{mn}$$ First, I want to get rid of the absolute value. Assume without loss of generality that $...
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1answer
30 views

$f,g: \mathbb{N} \rightarrow \mathbb{N} $, where $f(n)=g(2n)$. Prove that if $f$ is surjective, $g$ is not injective. [duplicate]

Trying this question for a while. $f,g: \mathbb{N} \rightarrow \mathbb{N} $ $\forall n \in \mathbb{N}: f(n)=g(2n)$ I need to prove that if $f$ is surjective $g$ is not injective. Now I see that $g$...
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1answer
38 views

Q: Prove that for any natural number, there exists a multiple that (in decimal form) only uses digits 0 and 1 [duplicate]

I'm supposed to prove the theorem in the title for a combinatorics class (continuation of discrete structures class). At the moment I have no idea how to aproach this question. I would appreciate ...
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0answers
27 views

$3n = 8x+4$ control flag inside of a function

I want to be able to evaluate wether the result of a function satisfies an expression. Is this possible to do? Example I have an expression, a linear polynomial: $$8x+4$$ for $x\in\mathbb{N}$ If I ...
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0answers
16 views

Proof that square of sum equal to sum of cubes of N. [duplicate]

Does exist a proof for this: $$(1 + 2 + 3 + 4)^2=1^3 + 2^3 + 3^3 + 4^3$$
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1answer
52 views

Prove that for every $n$, the binary representation of $n + 1$ contains exactly one bit that flips from $0$ to $1$

I know since $n$ is a binary representation it can be represented $ \sum_{i = 0}^{p}b_{i}\cdot 2^{i}$, where $b_{i}\in \{0,1\}.$ I have the intuition for this problem, i think. If $n$ is odd then the ...
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1answer
23 views

Maximum standard deviation of $n$ positive integers with set mean

Suppose the set $S$ contains $n$ positive integers. If the mean $\mu$ of the elements is known, is there a method to finding the maximum possible value of the standard deviation $\sigma$ for $S$? I ...
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2answers
39 views

Positive integers that are equal to the product of their figures [closed]

How many are the natural numbers (whole positive integers ≥ 1) that are equal to the product of their figures?
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3answers
54 views

Why is absolute value of negative exponent equal to positive value?

I am asked to integrate the following: $\int_{-\infty}^{0}e^{-\left\lvert 3x\right\rvert}dx$ And I am told that $e^{-\left\lvert x\right\rvert}=e^{x}$ How is it that an absolute value (the exponent)...
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3answers
54 views

Why is the derivative of $3^x$ equal to $3^x \cdot \ln 3$

Our teacher tells us to convert it this way $ 3^x = e^{\ln 3^x}= e^{x\cdot\ln 3}$ and then use the rule $e^u\cdot u'$ but I can't understand where $\ln$ comes from and how $\ln 3^x$ = $x\cdot \ln 3$.
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1answer
36 views

Let $F:A \to A$ be a function and let $Y \subseteq A$. Prove that the class $S = \{B:Y\subseteq B \subseteq A$ and $F[B] \subseteq B\}$ is a set.

Let $F:A \to A$ be a function and let $Y \subseteq A$. (a) Prove that the class $S = \{B:Y\subseteq B \subseteq A$ and $F[B] \subseteq B\}$ is a set. Can I use the theorem that states: Let $\phi(x)$ ...
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2answers
29 views

Prove that $n\not=n^+$ for all $n \in \omega$

Prove that $n\not=n^+$ for all $n \in \omega$. Where $\omega$ is the set of natural numbers, and $n^+ = n ∪ \{n\}$. Attempt: Assume $n = n^+$ for some $n \in \omega$. Then $n=n∪\{n\}$. This is ...
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1answer
32 views

Unique form to write any natural number

I want to know if this is true, I have done many examples, but don't know how to prove it: If $m \geq 1$ and $0 \leq k <m$, then, for any $n \in \mathbb{N}$, there is a unique $m$ and $k$ such ...
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0answers
10 views

If $y$ is an integer and $y^n= k^{\gamma}\beta$, $\gcd(\beta,k)=1$, then $k^{\gamma/n}$ and $\beta^{1/n}$ are integers [duplicate]

Proof: Since $\gcd(\beta, k)=1$, we get $\gcd(\beta, k^\gamma)=1$. Now, $y$ is an integer. So, $(y^n)^{1/n}=k^{\gamma/n}{\beta}^{1/n}$ must be an integer. But, there is no common prime between $k^{\...
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0answers
53 views

Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
2
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1answer
37 views

What does $(a_n)_n \in A^{\mathbb{N}}$ mean?

What does $(a_n)_n \in A^{\mathbb{N}}$ mean? What kind of sequence is that? How does the indexing work? What's the $A$ to natural numbers power?
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2answers
100 views

Official name of “sized integers”, a kind of number where $00$ is not equal to $0$?

Hierarchical identifiers, labels and indexes... All can use digits as character-strings, differenciating $0$ and $00$, $1$ and $001$, but preserving all other numeric interpretations, like order ($...