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Questions tagged [natural-numbers]

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

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1answer
33 views

Number of nonnegative solutions of equation ax+by=n

If $a,b$ are natural numbers and $\gcd(a,b) = 1$ then number of nonnegative solutions of equation $ax+by=n$ is equal to $\lfloor $$\frac{n}{ab}$$ \rfloor$ or $\lfloor $$\frac{n}{ab}$$ \rfloor$ + 1 ...
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1answer
29 views

5x+6y=n, x,y,n∈N, find the smallest n for which there exist four different pairs of solutions (x,y)

I found this solution using simple brute force algorithm but I have no idea how to find answer using mathematical tools. Solution from brute force algorithm: n = 101 x1 = 1, y1 = 16 x2 = 7, y2 = 11 ...
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0answers
9 views

Is there a specialized formula for Lagrangian interpolation on equispaced points?

If we know $f(0),f(1),f(2),\cdots f(n)$, is there a specialized version of the Lagrangian interpolation formula and a shortcut to compute the coefficients ? (Stability is not a concern.)
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2answers
60 views

Finite Ordinals and Natural Numbers

I'm studying set theory and I'm focusing on von neumann ordinals. I've built an understanding of the reasoning that brings to the set-theoretic construction of the natural numbers whose soundness I'm ...
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1answer
42 views

Differently-representable naturals in bases $b>1$

In this book that I read, the basis representation theorem is stated like this: Let $k$ be any integer larger than $1$. Then, for each positive integer $n$, there exists a represenation $$n=a_0k^s+...
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1answer
25 views

Examples of commutative semirings satisfying $kb = hb + r \ \; \text{ iff } \; k = h \text{ and } r = 0$.

Let $\Bbb M$ be a commutative semiring. Setting $b = 1 + 1$, $\, \Bbb M$ also satisfies P-1: For every $k,h \in \Bbb M$ and $r \in \{0,1\}$ $$\tag 1 kb = hb + r \ \; \text{ iff } \; k = h \text{ and ...
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1answer
72 views

For $m,n\in \omega, m \leq n$ imply $\exists ! p\in \omega\ s.t\ m+p=n$

For a set $A$, we define $A^+:=A\cup\{A\}$ When we define, $$0=\emptyset,\ 1=0^+,\ 2=1^+,\ \cdots$$ set of natural number $\omega$ is defined as $$\omega=\{0,1,2,\cdots\}$$ The order $"\leq"$ is ...
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2answers
121 views

Elementary diophantine equations with unknown solutions [closed]

Solvability of a general diophantine equation has been proved undecidable. As a famous example of knowledge, we know that $x^n+y^n=z^n$ has no solutions (in $\mathbb{N}$) for $n>2$. As a famous ...
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1answer
30 views

Question about singleton & inclusion

I am working on the ZFC construction of $\mathbb{N}$ : Axiom of infinity : there exists a set E such that $\varnothing \in \text{E} \,\wedge \forall s \in \text{E}, s\cup \{s\} \in \text{E}. $ ...
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0answers
52 views

Is the NNO in Cat isomorphic to $\mathbb {N}$ viewed as a discrete, skeletal category?

Suppose N is any category which satisfies following axioms: There exists a distinguished object, z. There exists a distinguished functor, $\sigma': N\rightarrow N$. To any category $X$ with a ...
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1answer
32 views

Proving commutativity of multiplication

I am trying to prove Lemma 2.3.2 in Tao's analysis text: that for any two natural number, $n$ and $m$, we have $n \times m = m \times n$. I only have the properties of the natural numbers and addition ...
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1answer
43 views

Proof that $a < b$ if and only if $a+ + \leq b$

I am trying to prove part (e) of Proposition $2.2.12$ in Tao's analysis textbook that for natural numbers $a, b$, $a < b$ if and only if $a ++ \leq b$, where $a++$ is the successor of $a$. I am ...
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1answer
80 views

Inverse of a bijective function involving cases

In continutation to a question that i asked earlier and got answered here :Discretizing a mathematical equation This is a bijective mapping from the set of ordered tuples $(x,y,z)$ where each $x,y,z\...
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1answer
29 views

Proof that order is reflexive

I am trying to rove Proposition $2.2.12$ in Terence Tao's analysis text. I am a bit stick on part a, which states that for any $a \in \mathbb{N}$, $a \geq a$, a fact I need to prove using only the ...
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2answers
44 views

Do such unique representations of positive integers exist?

It is well known that every positive integer $n>0$ can be represented uniquely in the form $$ n=2^k(2m+1), $$ for positive integers $k,m\geq0$. Does there exist one or more constants $c>1$ such ...
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1answer
24 views

Is it possible to know a priori if ab is bigger, equal or smaller than cd without actually computing the multiplication? [closed]

Given a, b, c and d belonging to N, is it possible to know if ab is bigger, equal or smaller than cd without actually multiplying the terms and comparing the products? For instance, I found this rule:...
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1answer
62 views

Does this equation yield only primes?

Interested in solving this equation for $x$: $\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ for $n=1,2,3,...$ For $n=1$ up to $n=9,$ I got $x=5,11,13,19,29,37,47,59,73.$ $\pi(x)$ is the prime ...
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1answer
19 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
69 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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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
31 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
40 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
21 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
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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
43 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
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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
48 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
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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: $[...
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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 ...
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1answer
53 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 $...
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1answer
65 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
65 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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735 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
130 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
56 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
33 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
140 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
29 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
31 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 ...