Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
1,407 questions
4 votes
1 answer
170 views
A really weird limit: $\lim_{n \to \infty, \, n \in \mathbb{N}} \frac{\tan(n^2+1)}{n!}$
I am trying to evaluate the following limit: $$\lim_{n \to \infty} \frac{\tan(n^2+1)}{n!}, \qquad n \in \mathbb{N}.$$ It seems to me that this limit should be $0$, but I would like to understand how ...
- 126
0 votes
1 answer
111 views
How to distinguish $\mathbb{N},\mathbb{N}^2,\cdots ,\mathbb{N}^n$ with the help of some properties? [closed]
This question arose while attempting to prove the proposition in the provided link. Because the cardinality of a set formed by the Cartesian product of finitely many sets of natural numbers is always ...
- 161
9 votes
1 answer
696 views
A riddle involving the difference between lcm and GCD
I just read this riddle which i find quite interesting: Two friends are staying at the same hotel. While chatting in the lobby, one says to the other: "The difference between the least common ...
- 189
3 votes
2 answers
558 views
What do model theorists mean when they talk about set sizes?
I am investigating this claim "The Peano axioms don't imply a countable set of natural numbers". I think the truth of it depends if we you use First order logic or second order logic. In ...
- 14.5k
0 votes
2 answers
104 views
Large numbers estimation
For commonly discussed enormous yet finite numbers, such as Graham's Number or TREE(3), is there any computation of their order of magnitude that can be expressed like $\log(N)$ or $\log(\log(\log(...(...
- 123
-2 votes
1 answer
151 views
Show that there are two natural numbers of the same parity $a$ and $b$ such that $(1+\sqrt{3})^{2026}=\sqrt{a}+\sqrt{b}$ [duplicate]
the problem Show that there are two natural numbers of the same parity $a$ and $b$ such that $(1+\sqrt{3})^{2026}=\sqrt{a}+\sqrt{b}$ my idea I might fear solving the problem, we might have to find the ...
- 508
0 votes
1 answer
46 views
Prove the Commutative Property of Addition for Finite Sums
I am trying to solve the problem $2.2.11$ in the book A Course in Mathematical Analysis by D. J. H. Garling. I will restate the problem as follow: "Suppose that $\left(a_1, a_2, \cdots, a_n\right)...
0 votes
0 answers
68 views
On the Necessity of Constructing Number Systems from First Principles in Real Analysis
When studying real analysis, is it necessary to go as deeply as Terence Tao does in Analysis I, for example by constructing the natural numbers, integers, and rationals from first principles? Or is it ...
- 182
0 votes
0 answers
104 views
Finding the inverse of all binomial coefficient polynomials
Let $f_k(x)$ $=$ $x \choose k$. How can I find all $g_k(x)$ such that $f_k(g_k(x)) = x$, where $k>0 \in ℕ, x>=0\in ℝ$? Obviously, $f_k(x)$ is factorable: $$ f_k(x) = {x \choose k} = \frac{1}{k!}\...
- 314
1 vote
1 answer
75 views
Show that for every $a \in \mathbb N^*$ there exists $b \in \mathbb N$ for which $a^m < b^n < (a+1)^m$, where $m>n\geq 1$ are natural numbers.
the problem Let the natural numbers $m>n \geq 1$. Show that for every $a \in \mathbb N^*$ there exists $b \in \mathbb N$ for which $a^m < b^n < (a+1)^m$. My idea: So we can also show that ...
- 508
3 votes
0 answers
256 views
solution-verification | determine the number of solutions of $[ \sqrt{k} ] + [ \sqrt{k+1} ] =n $
the problem Let $n\in \Bbb{N}$ Determine how many numbers $k\in \Bbb{N}$ verify $[ \sqrt{k} ] + [ \sqrt{k+1} ] =n $ my idea So, for a start, I use the fact that every natural number can be bonded ...
- 508
2 votes
1 answer
230 views
Proving that there is a bijection between any two Peano systems
I want to prove that if two given 'models' satisfy the Peano axioms, there must be a bijection between them. In other words, there is only one version of the natural numbers in set theory. The ...
- 163
0 votes
1 answer
134 views
Is this a correct proof for whether the cardinality of the set of odd positive numbers is the same as the cardinality of the set of natural numbers? [closed]
I intended to post this proof as my own answer to my previous question. But I thought that it would be more beneficial to formally ask (= post a question) whether the proof is correct. After all, I ...
1 vote
2 answers
175 views
Proving that a recursive function is unique - Tao's Analysis
In Exercise $3.5.12$ in Tao's Analysis $1$, he asks the following: Let $X$ be a set, and $f:\mathbb{N} \times X \to X$ and let $c \in X$. Prove that there exists an $a:\mathbb{N} \to X$, such that (i)...
- 163
2 votes
2 answers
423 views
Is the cardinality of the set of odd numbers really the same as the cardinality of the set of natural numbers?
The Wikipedia article on Hilbert's paradox of the Grand Hotel reads that However, in Hilbert's Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of ...