Questions tagged [natural-numbers]
For question about natural numbers $\Bbb N$, their properties and applications
159
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Prove $3^n \ge n^3$ by induction
Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$.
I can do this myself, but can't figure out any kind of "beautiful" way to do it.
The way I do it is:
Assume $3^n \ge n^3$
Now,
$(n+1)^3 = n^3 + 3n^...
- 222
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How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.
While answering a question here I noticed that:
$$n=1+\sum_{k=1}^{n}{\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor}$$
for every natural number $n$.
I tried to demonstrate it using Legendre's ...
- 2,103
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Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?
Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
- 45.9k
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What is the mathematical definition of "standard arithmetic/standard natural numbers"?
As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-...
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Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$
It seems obvious that $$\forall n \in \Bbb Z^+, \sum_{i=1}^n{1} = n $$
However, I'm having trouble coming up with a formal proof for this.
Given a concrete number like $4$, we can say that $$\sum_{i=...
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proof by induction that every non-zero natural number has a predecessor
I am trying to prove by induction that every non-zero natural number has at least one predecessor. However, I don't know what to use as a base case, since 0 is not non-zero and I haven't yet ...
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Prove the commutativity property of addition of natural numbers by induction
the background I'm allowed to deal with to solve this problem is as follows:
Definition of +:
\begin{equation} m+0=m\quad \text{for all}\quad m \in \mathbb{N} \\
m+(k+1) = (m+k)+1
\end{equation}
in ...
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proof of commutativity of multiplication for natural numbers using Peano's axiom
How do you prove commutativity of multiplication using peano's axioms.I know we have to use induction and I have already proved n*1=1*n.But I cant think of how to prove the inductive step.
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Combinatorics - How many numbers between 1 and 10000 are not squared or cubed?
Simple question.
How many numbers between 1 and 10000 can't be written as $n^2$ or $n^3$ when $n \in \mathbb N$?
I know the way to solve this is with inclusion-exclusion. but for that I need to find ...
- 12.8k
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Better representaion of natural numbers as sets?
Natural numbers can be represented as
$0=\emptyset$
$1=\{\emptyset\}$
$2=\{\{\emptyset\}\}$
$...$
or as
$0=\emptyset$
$1=\{0\}=0\cup\{0\}$
$2=\{0,1\}=1\cup\{1\}$
$...$
What are the names of ...
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0
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Defining Addition of Natural Numbers as the Algebra of 'Push-Along' Functions
Let $N$ be a set containing an element $1$ and $\sigma: N \to N$ an injective function satisfying the following two properties:
$\tag 1 1 \notin \sigma(N)$
$\tag 2 (\forall M \subset N) \;\text{If } ...
- 11.5k
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Surjectivity of floor of harmonic sequence
Define
$$H_n := \displaystyle\sum_{k=1}^n \dfrac 1k $$
The problem asks to prove that the map $\phi:\mathbb{N}^\star \longrightarrow \mathbb{N}^\star $ defined by
$$\phi(n) := \lfloor H_n \rfloor $$...
- 1,283
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An apparently harmless exercise concerning induction
Let $b \in \mathbb{R}, b \ge 2$. Prove by induction that $$(b^n - 1)(b^n - b)(b^n -b^2)\cdots(b^n - b^{n-2}) \ge b^{n(n-1)}-b^{n(n-1)-1}$$ for all $n \in \mathbb{N}, n \ge 1$.
For the case $n = 2$, I ...
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To prove that $(\mathbb{P(N)},d)$ is a metric space
To prove that $(\mathbb{P(N)},d)$ is a metric space where $\mathbb{P(N)}$ is the power set of $\mathbb{N}$. And $d:\mathbb{P(N)}\times \mathbb{P(N)} \mapsto\mathbb{R}$ is defined by
$d(A,B)=\begin{...
- 2,620
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Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set?
Consider the following set-theoretic definition of natural numbers:
$0$ is defined as $\emptyset$
If $n$ is defined, then the successor of $n$ is defined as $n^+ = \{n\} \cup n$
Thus $1 = \{0\}$, $2 ...
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