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Questions tagged [natural-numbers]

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

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of and , respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to , such as the distribution of , are studied in . Problems concerning counting and ordering, such as partition enumeration, are studied in .

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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, it seems as though putting the digits of the…
Pro Q
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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. Pseudocode: Consider that 1 is the starting index of…
mr-fotev
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Can you make all natural number from 3?

Can you make all natural number from 3 with only four function that $$x!,\; \sqrt{x},\;\lceil x\rceil,\;\lfloor x \rfloor $$ ? ex) $1=\lfloor \sqrt3 \rfloor$ $\;\;\;\;\;2= \lceil \sqrt3…
Sintist
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Has this conjecture been proven/disproven/found before?

My 11-year-old son has made the following conjecture, and wants to know if it's already known to be true or false: if n is a whole number, define an operation #(n) = 1+2+3…..+n (like factorial but with addition) The conjecture: if n is odd,…
Mike Levin
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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 $m>n$. Then $m^m > n^n$. Raising this inequality to the…
element
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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 logic and learning what are first-order formal…
Patrick Da Silva
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Are there any interesting examples of subsets of $\mathbf{N}$ that are known to be nonempty, but of which no elements are known?

There are many results in mathematics that establish the existence of some object without actually constructing said object. I am wondering if there are any interesting properties of the natural numbers such that it is known that there exists a…
ajd
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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 squares , when p - some natural number ?
Andrew
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What's the digit "a" in this number?

a and b are digits in a four-digit natural number 7a5b. If 7a5b is divisible by 18, how many different possible values can "a" have?
Ally
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Is every infinite set of natural numbers a sum of two infinite sets of natural numbers?

If $A,B\subseteq\{0,1,2,\ldots\},$ then $A+B=\{a+b:a\in A,b\in B\}.$ Is it true that for any $X\subseteq \{0,1,2,\ldots\}$ infinite, there exist infinite sets $A,B\subseteq\{0,1,2,\ldots\}$ such that $A+B=X?$
Bartek
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What is a definitive explanation for the following property of natural numbers?

(I want to clear out the fact that I'm no professional mathematician. Please feel free to correct me if needed. I have no mathematical background. I am 13 years old, this question is originated from pure math curiosity) Let $\mu$ be any number…
guilherme1771
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$a+b+c+d+e=abcde$ What is $\max(a)?$

The question from my nephew is this: If $a, b, c, d, e \in N^+$ and $a+b+c+d+e=a\times b\times c\times d\times e$, then what's the the maximum possible value of $a$? Thanks ahead:)
Paul
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Unique element m in N

Let $0 < x$. Show that there is a unique $m \in \mathbb{N}$ such that $m-1 \leq x < m$. Hint: Consider the set $\{ n \in \mathbb{N} : x < n\}$ and use the well-ordering of $\mathbb{N}$. The textbook definition says $\mathbb{N}$ is well ordered and…
user176071
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Isn't zero natural enough to be included in the set of natural numbers?

I always define $\mathbb{N}$ to include $0$ but some authors don't. Since the elements of $\mathbb{N}$ are used for counting, shouldn't $0\in\mathbb{N}$? $0$ is the number of cows in a classroom for example. Moreover, $0\in\mathbb{N}$ is a…
user5402
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How to solve $at + b = 0 \pmod {(a-t)}$?

Is there any way (except trying for $t=0,1,2,\ldots,a-1)$ to solve the following equation for $t$ when $a$ and $b$ are known? $$ at +b = 0 \pmod{(a-t)} \text{ with } a,b,t \in N $$
vauge
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