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

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 opposed to friendly) numbers occur more…
cbmanica
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Notion of "closeness of two numbers", made precise

In everyday life we have this intuitive idea of two (e.g. natural) numbers being close to each other. We say things like "$365$ and $360$ are close to each other". I was wondering if this informal notion could be made fully precise, and I came up…
alex811
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Sum/Product of two natural numbers is a natural number

I wanted to prove that the sum and the product of two natural numbers is a natural number. Intuitively it's clear to my why that is true, however I couldn't prove it. So our lecturer first defined what an inductive set is. Then he defined the…
Charles Carmichael
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What is the sum of natural numbers to an even power?

We know $\sum_{0}^{N}m^2=\frac{N(N+1)(2N+1)}{6}$. Is there a generic expression for $\sum_{0}^{N}m^n$ where $n$ is an even number?
Ary
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Why is the sum over all positive integers equal to -1/12?

Recently, sources for mathematical infotainment, for example numberphile, have given some information on the interpretation of divergent series as real numbers, for example $\sum_{i=0}^\infty i = -{1 \over 12}$ This equation in particular is said to…
Andreas Grapentin
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Discrete mathematics proof relating to Fermat's Theorem

Assuming the Fermat Theorem, show that there is no natural number $x$, $y$, and $z$ and $n\geq3$ such that $$\frac{1}{x^n} + \frac{1}{y^n} = \frac{1}{z^n}. $$ So far I think proof by contradiction may be the best route, but I cannot find any place…
mrQWERTY
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Closed natural numbers

I am reading this paper about interleavers for turbo code design, and when it describes the so called block interleavers, it says that To obtain a block interleaver function it is necessary to factorize its length: $N_b=X\times Y$ where $X$ and $Y$…
Josu Etxezarreta Martinez
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Prove that for each natural number n, any set with n elements has $\frac{n(n-1)}{2}$ two-element subsets

Prove that for each natural number n, any set with n elements has $\frac{n(n-1)}{2}$ two-element subsets. I'm just confused by what this is asking, I haven't learned about two-element subsets yet.
Claire
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Cancellation law for multiplication of natural numbers

I'm trying to prove the following cancellation law for multiplication of natural numbers: if $xz=yz$ for natural numbers $x,y$ and $z$, where $z$ is non-zero, then $x=y$. I'm working with the peano-axioms and I've already proven elementary…
Owl Tree
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Is there a difference in the rate of decrease between $f(x)$ and $g(x)$ for increasing $x$?

I have the following two functions of $x$: $ f(x) = \frac{c}{c + (N-1)o + Nd + xl}$ $g(x) = ae + (1-a)\frac{1}{x+2N}$ with $0 \leq a, e, c, o, d, l \leq 1$ and $N, x \in \mathbb{N}^+$. For both functions increasing $x$ is obviously associated with a…
Henrik
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Questions regarding early natural numbers.

Consider the real number $0.123456789101112\dots$, where you concatenate the digits of the natural numbers. Certain natural numbers are "early", meaning, they appear earlier as a substring of digits than they are supposed to. For example: $12$, $23$…
user107952
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Example of set of naturals without asymptotic density

I read from Wikipedia that there are sets of naturals whose asymptotic density is undefined. How can this possibly be? Can anyone show me an example?
user107952
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natural number reorder problem

Suppose the original natural numbers are sorted as 1, 2, ..., N. The distances of two neighbors are 1. Is there any method to reorder the natural number list to maximize the distance of ALL neighbors? Furthermore, is it possible to find the list to…
Tang Laoya
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Why is this map order-preserving?

Can somebody explain to me why $f: \mathbb N \rightarrow \{1,1+1,1+1+1,...\}$ where 1 is an identity element of ordered field, preserves order? Intuitivelly I understand that it does, but I don't know how to justify it?
user62136
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How does it work? In the nested case.

I cannot understand how double/nested induction works. I am supposed to prove that the addition operation is both commutative (x + y = y + x) and associative (x+(y+z)) = ((x+y)+z) from the following definitions of the naturals and addition (no other…
Kliker
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