Questions tagged [natural-numbers]

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

574 questions
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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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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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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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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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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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Is it true that the first n prime numbers can be divided into two sets with sums differing at most by 1 for all n>1?

Example ($n=6$): $$2+5+13=20$$ $$3+7+11=21$$
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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-...
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 ...