Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
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How to prove $a ≤ b$ OR $b ≤ a$ for all $a, b$ in $\mathbb{N}$?
I'm currently reading Terence Tao's "Analysis I" and I'm at the beginning where he defines the natural numbers and proves some of their basic properties using the Peano axioms.
I've nearly ...
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Pairs of infinite subsets of natural numbers that intersect on an infinite set give the same property to triples?
Let $A \subseteq P(\mathbb{N})$ such that $A$ is infinite and for each pair $B,C \in A$ then $B \cap C$ is infinite, this implies that for any triple $B,C,D \in A$, $B\cap C \cap D \neq \varnothing$? ...
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Why does $\forall n \in \mathbb{N} \vdash P(n)$ not imply $\forall n \in \mathbb{N}P(n)$?
I am trying to understand why $\forall n \in \mathbb{N} \vdash P(n)$ doesn't imply $\forall n \in \mathbb{N}P(n)$ by studying the concepts in this answer, and had 2 questions about the difference ...
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Find all $(a,d)\in\{1, 2, ..., 9\}^2$ such that $ad$ is a square [duplicate]
Find all six-digit numbers of the form $abcbcd$ for which $a \cdot b^2 \cdot c^2 \cdot d$ is a perfect square of a natural number (note: 0 is not a natural number, so obviously $a, b, c, d \neq 0$, ...
user1340018
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Consecutive multiplication of natural numbers problem [duplicate]
Prove that the product of any three consecutive natural numbers is not a perfect square. If there were four numbers, I know how to solve the problem, as I would somehow mention the number $ n(n+1)(n+2)...
user1340018
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Find all pairs of natural numbers $m$ and $n$ that satisfy the following conditions: [closed]
Find all pairs of natural numbers $m$ and $n$ that satisfy the following conditions:
$m$ and $n$ are two-digit numbers,
$m - n = 16$,
the last digit of the numbers ...
user1315539
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Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$
This is a trivial task:
Given $a \in \mathbb{N}$ and $$a=4$$
Show $$a > 1$$
Part of the challenge for newcomers like me is that "easy" tasks actually make it harder to think about the ...
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Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I)
In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms:
Axiom 3.4 If $a$ is an object, then there exists a set
$...
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$f((x,y))=(x*gcd(x,y),y) $is injective and surjective?
The function $f:\mathbb{N^+}\times\mathbb{N^+} \rightarrow \mathbb{N^+}\times\mathbb{N^+} $ definide as $f((x,y))=(x*gcd(x,y),y) $is injective and surjective?
Let $(a,b)\in\mathbb{N^+}\times\mathbb{N^+...
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Characterizing cofinite submonoids of $\langle \mathbb{N}, + \rangle$?
A set $S \subseteq \mathbb{N}$ is a submonoid of $\langle \mathbb{N}, + \rangle$ when $0 \in S$ and $S$ is closed under addition (that is, $m+n \in S$ whenever $m$ and $n$ are). For example, $\mathbb{...
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how to find expression of $a\text{ mod } n$ using polynomial of $e^{\frac{2a\pi i}{n}}$?
I tried to find a polynomial for $a\text{ mod } n$ (for $a,n\in N$) by using powers of $\exp\left(\frac{2a\pi i}{n}\right)$
which mean find the coefficients $f(n,k)$ in
$$ a\text{ mod } n=\sum_{k=0}^{...
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Do we have complete understanding of $\mathbb N$?
We have some understanding of natural numbers. We have PA theory and we believe that $\mathbb N$ is one of the PA models. But PA can't prove some statements about $\mathbb N$ even though they are true ...
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These constrained alternating series always satisfy an inequality.
Let $x_0 \in \Bbb{N}$ and suppose that $x_1 \lt \frac{x_0}{2}$, while $0 = x_n \leq \dots \leq x_3 \leq x_2\leq x_1$.
Then is it possible that:
$$
\sum_{i = 0}^n (-1)^i x_i \gt 1
$$
no matter what ...
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proving the set of natural numbers is infinite (Tao Ex 2.6.3)
Tao's Analysis I 4th ed has the following exercise 3.6.3:
Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational
numbers.
$f$ is strictly increasing in both
arguments.
Can $f$ be one-to-one?
This question is related to many ...
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