Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
1,250
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what a number really is? [closed]
https://youtu.be/dKtsjQtigag?si=-7F6L380uJhs2dd2**strong text**
in this video this guy took set theory as the foundation and my doubt is that in this video this guy tells that an empty set exists but ...
1
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1
answer
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Do all finite-cycle-free permutations of $\mathbb{N}$ have square roots?
This answer nicely categorises which permutations of a finite set have square roots. This prompts the following question:
Does every permutation $\sigma:\mathbb{N}\to\mathbb{N}$ that
contains no ...
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0
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28
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Is there a continuous function that is worth -1 in the negative naturals and 1 in the naturals? [closed]
Is there a continuous function that is worth -1 in: -1,-2,-3,... and 1 in: 1,2,3,...?
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A question about the overall characteristics of GCD statistical series
Problem
Define a sequence $a$ with $a_i=\sum_{j=1}^{n}\sum_{k=1}^{n}[gcd (j,k)=i]$. We define another sequence $b$ about $a$, $b_i=\frac{a_i}{\sum_{j=1}^{n}a_i}$. In $n \to \infty$, what ...
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Comparing two pairs of natural numbers function
Is there a fucntion such that:
$$F: \mathbb{N}^2 \rightarrow \mathbb{N}$$
$$C_1: \mathbb{N}^4 \rightarrow \mathbb{N},\, C_2: \mathbb{N}^4 \rightarrow \mathbb{N}$$
$$
\left.
\begin{matrix}
x_1 - a_x &...
0
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1
answer
54
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Does this expression tend to infinity?
Define the functions $$f_0(x,y)=\begin{cases}x! & \text{if } y=1\\f_0(x,y-1)! & \text{otherwise}\end{cases}$$ and
$$
f_1(x,y,z)=\begin{cases}
f_0(x,y) & \text{if } z=1\\
f_0(x,f_1(x,y,z-1))...
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1
vote
0
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34
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Large sets and Erdős-discrepancy
Large Sets
Erdos conjecture
I have a conjecture that is stronger than the Erdos discrepancy conjecture, can someone think of a counter example?
Let $S$ be any large set and let $(x_1,x_2,...)$ be any ...
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0
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23
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Is this proof of the well-ordering principle using induction correct?
This question is about proving the well-ordering principle of the natural numbers using the principle of induction. This other question uses a different induction argument than the one below. I'd like ...
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1
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1
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125
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How to prove or disprove this number theory proposition?
Problem
Define a function $f(\prod_{i=1}^{n}p
_i^{k_i})=\prod_{i=1}^{n}p_i$ where the elements of a sequence $p$ are distinct primes and $k_i \ge 1$. Define the function
$$
g(x)=
\begin{cases}
f(x)+1 &...
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0
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Some questions about set of bijections from $\mathbb{N} \times \mathbb{N}$ to $\mathbb{N}$
Let $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$ the set of bijective functions from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$. I have some questions about this set.
Note that $S(\mathbb{N}\times\...
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2
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0
answers
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Natural density of thickly syndetic set.
A syndetic set $S$
is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
1
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0
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How many functions from $\mathbb{N}\cup\{0\}$ to $\mathbb{N}\cup\{0\}$ have $\phi(ab) = \phi(a) + \phi(b)$?
How many maps $\phi : \mathbb{N}\cup\{0\} \to \mathbb{N}\cup\{0\}$ are there, with the property that $\phi(ab) = \phi(a) + \phi(b)$, $\forall a, b$ $\in \mathbb{N} \cup \{0\}$?
Can I get a hint to ...
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1
answer
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Is there a smallest large set?
A set $A = \{a_1, a_2 ,..\}$ of positive integers is called large if $\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + ...$ diverges. A small set is any set of the positive integers that is not large
...
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1
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How to define addition and multiplication of natural numbers using category theory?
Is there a way to define addition and multiplication of natural numbers using the language of category theory? Like, one could say that "Addition is the unique functor that satisfies..." and ...
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1
answer
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Can we partition the reciprocals of $\mathbb{N}$ so that each sum equals 1
Let $S = \{1, 1/2,1/3,\dots\}$
Can we find a partition $P$ of $S$ so that each part sums to 1, e.g.
$$P_1 = {1}$$
$$P_2 = { 1/2,1/5,1/7,1/10,1/14,1/70}$$
$$P_3 = {1/3,1/4,1/6,1/9,1/12,1/18}$$
$$P_4 = \...
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1
answer
43
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listing all finite subsets of natural numbers [duplicate]
is there a computable algorithm which lists all the finite subsets of natural numbers ?... i know that such a set is atleast countable... but can't determine if we can list every such subset in a ...
2
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Alternative formation/ notation for axiom of mathematical induction
I hope to clarify the difference between the following two statements:
$S\subset \mathbb{N}$ (set of natural numbers),
$\forall n \in \mathbb{N}, \text{ if } n\in S \text{, then } n+1\in S$
$\forall ...
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1
vote
1
answer
130
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Why isnt $|\mathbb{R}| = |\mathbb{N}|$?
Question: To show that 2 sets have the same cardinality, there needs to be atleast one bijective mapping between them. So given the below proof of a bijective mapping below, why can't we say that $\...
0
votes
1
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Find the numbers of the form $\overline{abcd}$ for which the relations are checked simultaneously.
my question
Find the numbers of the form $\overline{abcd}$ for which the relations are checked simultaneously:
i) $\overline{ab}$ and $\overline{cd}$ are consecutive natural numbers
ii) $(2*\overline{...
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-1
votes
1
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What is the solution of this algebraic problem? [closed]
Let $a, b, n, X$ and $Y \in \mathbb{N}$. Find $X$ and $Y$ such that
$(a+b)^{n} = X + Yb$.
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1
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0
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Categorical notion of finiteness
TLWR : I'd like to define either $\operatorname{Hom}_F : \text{Set} \times \text{Set}_* \to \text{Set}_*$ (set of functions with finite support) or simply $\mathcal{P}_F : \text{Set} \to \text{Set}$ (...
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1
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57
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For Infinites $A,B\subset\mathbb{N}$ s.t exists $n \in A$ and $r,l < n$ s.t the sets $B\cap(n\mathbb{N}+r)$ and $B\cap(n\mathbb{N}+l)$ are infinite.
We note that for every subset $B$ of $\mathbb{N}$ and for every $n \in \mathbb{N}$ we have the following
$$
B = \bigcup_{0 \leq r < n} (n\mathbb{N} + r) \cap B,
$$
where $n\mathbb{N} + r = \{ n \...
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0
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49
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How are numbers assigned to a group of real life objects?
I hope I don't come off as dense but suppose I constructed (informally) the decimal number system assuming the existence of symbols, 1-9 and defining the operation of addition on it with usual ...
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1
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2
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N is infinite (a proof)
In the book Introduction to Set Theory (Third Edition, CRC Press, 1999), by Karel Hrbacek and Thomas Jeck, the following appears on page 70:
2.2 Lemma. -- If n ∈ N , then there is no one-to-one ...
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1
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How do I solve this velocity equation? [closed]
$$t-t_0=m\int\frac{\mathrm dv}{F}=-\frac{m}{b}\int\frac{\mathrm dv}{v}$$
$$t-t_0=-\frac{m}{b}\ln\frac{v}{v_0}$$
$$v=v_0 \cdot e^{-\frac{b}{m}(t-t_0)}$$
Can you please help me? How do I convert it from ...
3
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5
answers
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Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?
Check if there are nonzero natural numbers $n,x,y$ such that:
$$\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}. $$Thank you in advance!
My ideas
So we can simply show that $4n+5,5n+1,9n+4$ are ...
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1
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Trouble manipulating quotients in $\mathbb{N}^\times$
As part of a proof about multinomial expansions, I am encountering difficulty in showing the following equality:
$$\frac{k!}{\alpha!} = \frac{(k-\alpha_1)!}{\alpha'!}\frac{k!}{\alpha_1!(k - \alpha_1)!}...
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1
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1
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Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$? [closed]
Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$? Justify!!
I literally have no idea for this problem. I thought of doing the last digit but it doesnt help at all. I wonder if we ...
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1
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Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$
I am trying to solve the following problem (Amann & Escher Analysis I, Exercise I.6.3):
Show that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$.
I emphasize that the ...
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0
votes
0
answers
37
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On the uniqueness of the addition operation on $\mathbb{N}$
My textbook (Amann and Escher, Analysis I) gives a theorem which says that the operations of addition and multiplication (and a partial order $\leq$) exist and are uniquely defined by a whole host of ...
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1
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45
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How to Show that cartesian product of finite sets is a finite set
It asks to show it by using definitions and theorem of sets, but not using cardinality.
Suppose $A$, $B$ is finite sets, by definition it means
There is a bijective function $f : \mathbb{N}_n \...
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votes
3
answers
124
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choosing a random integer from $\mathbb{N}$
I heard that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $∞$.
I have an interesting question: if we assume that we had an infinite computer which had ...
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11
votes
3
answers
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How do we compare the set of natural numbers in different models of ZFC
In a model of ZFC, an inductive set is a set $A$ satisfying $\emptyset\in A$ and $n\cup\{n\}\in A$ for every $n\in A$. Suppose that $X$ is the inductive set given by the axiom of infinity. In my last ...
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Confusion regarding the definition of $\omega$ (the set of natural numbers) in a model of ZFC
Suppose that ZFC is consistent. In a model of ZFC, an inductive set is a set $A$ satisfying $\emptyset\in A$ and $n\cup\{n\}\in A$ for every $n\in A$. Suppose that $X$ is the inductive set given by ...
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1
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Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively [closed]
I have concluded the reading of second chapter of Prof. Tao’s Analysis books in which he covers natural numbers and defines addition and multiplication operation on them,
He states the following ...
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0
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50
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Proving the well ordering principle starting from the axiom of completeness. Is this topological proof valid?
While reading this SE thread, I saw in the comments someone say "the proof [that the completeness axiom implies the well ordering principle] will take some work".
However, this other thread ...
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1
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1
answer
90
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What is 'increment' in Peano Axioms?
I am reading Tao's book on Analysis in which the first two axioms apropos natural numbers are,
0 is a natural number.
If n is a natural number, then n++ is also a natural number.
As a motivation ...
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votes
1
answer
93
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Why does Shilov exclude $0$ from natural numbers?
In the book "Linear Algebra" by Georgi E. Shilov, Chapter I the exclusion of $0$ from the numbers considered is justified by a note stating:
Given two elements $N$ and $E$, say, we can ...
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Why does this sequence switch from odds to evens?
Consider a function: $f: \Bbb N \to \Bbb N$ defined as follows:
$$f(n)=\bigg\lbrace\#\lbrace a_t(x) \rbrace >\#\lbrace b_t(x) \rbrace: \# \pitchfork \mathrm{id}\bigg \rbrace.$$
This notation means ...
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How to prove natural number addition using induction? [duplicate]
I am a self learner so excuse me if I am asking a seemingly easy question , But I ve been stuck at this point for couple of days , I think I understand mathematical induction and what the author is ...
1
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0
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60
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Why does ZF natural number construction not simply use n+1={n} instead of n+1=n $\cup$ {n}?
From Set-theoretic definition of natural numbers
n+1=n $\cup$ {n}
i.e.
0 = {}
1 = {{}}
2 = {{},{{}}}
etc
It seems to me that a simpler, equally valid definition would be
n+1={n}
i.e.
0 = {}
1 = ...
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0
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Difference between N & N+ domain
$\{0,1,2,...\}$ is $\mathbb{N}$, and $\{1,2,3,...\}$ is $\mathbb{N}^+$ if I'm not wrong.
Does that mean this author does not want to include $0$ in the function?
Why some authors include $0$ in ...
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2
votes
1
answer
141
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What is the proper definition of a "Factor"?
The definition I found on most websites was "A natural number $x$ is a factor of a natural number $y$ if $\frac{y}x$ leaves no remainder."
This definition seemed correct until I searched &...
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votes
1
answer
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Peter winkler's "Numbers" puzzle "Zeroes, Ones, and Twos"
I have a problem with the solution for the (b) part of the problem. The problem is as follows:
Let $n$ be a natural number. Prove that $2^n$ has a multiple whose representation contains only ones and ...
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0
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Induction principle in its set formulation and in its property formulation: which one to use in a well-redacted Induction Step of an induction?
I have read this answer about the well ordering principle and the induction principle. It especially says that "any proper axiomatization of $\mathbb N$ in modern logic does not involve set-...
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1
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45
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Confusion regarding definition of Natural Numbers (from book Numbers, english version of Zahlen)
In the book "Numbers" by Ebbinghaus et. al, the Natural numbers are defined as:
The natural numbers form a set $\mathbb N$, containing a distinguished element $0$, called zero, together with ...
1
vote
1
answer
58
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find the supremum and infimum of a set [closed]
find the inf/sup of the set A= { n $ \in \mathbb{N} $ | $n^2-3n +1 $}
before finding the inf and sup i checked for the first terms of this set, for n $\in $ {0, . . . 6} we have {1 , -1 , -7 ,...
1
vote
0
answers
84
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Is there a sensible first order expression meaning that every set in $\omega$ is some set-theoretical natural number?
Motivation: With the following I want to understand the relationship between the set-theoretical natural numbers 0,1,2,... and the set of natural numbers $\omega$ better.
Let 0 := $\emptyset$, 1 := {$\...
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votes
2
answers
144
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Can we define natural numbers starting from another set other than empty set?
I'm new to set theory and I'm studying how natural numbers are defined.
I know that the natural numbers are defined as
$0$ (zero) = $\emptyset$
$n^+$ = $n \cup$ {$n$}
And one of the most important ...
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votes
0
answers
83
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Large gaps between small primes
Does there exist a positive integer $n>5$ such that the sum of the two largest primes less than $n$ equals $n$? If yes, lovely! If not, what is the largest prime gap possible between the two ...
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