For question about natural numbers

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3
votes
1answer
64 views

What would provable existence of a non-standard natural number imply about ZFC?

Assume that in ZFC we can define a set $x$ in some way and that the following are provable: $x$ is a finite ordinal, (i.e. $x\in\omega$) $x\neq0$ $x\neq1$ $x\neq2$ etc. So to say that the ...
1
vote
2answers
42 views

Number of ways to write a natural as a sum of naturals [duplicate]

Problem: Let $n$ be a natural number, and $S(n)$ be the number of ways $n$ can be written as a sum of naturals. For instance, $S(3) = 4$ because $3 = 2+1 = 1+2 = 1+1+1$ and these are four different ...
0
votes
0answers
21 views

Is $f(i)=n-i, f:\mathbb{N_n}\rightarrow \mathbb{N_n}$ a bijection?

My text states that $f:\mathbb{N_n}\rightarrow \mathbb{N_n}$ where $f(i) = n-i$ is a bijection. I am not convinced about this because if $i=n$, then $f(n)=n-n=0$ but $0 \notin \mathbb{N_n}$. ...
0
votes
2answers
106 views

Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?

This is a bit of a philosophical question. According to "Set Theory" by Jech, the set $\omega$ of natural numbers is defined as the least nonzero limit ordinal. After thinking about this definition ...
3
votes
5answers
575 views

The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each ...
1
vote
2answers
93 views

How do I go about solving this derivative?

I have the function $$f(x)=\ln\sqrt{8+\cos^2x}$$ so $$1.f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$so$$2.f(x)=\frac{1}{2}\ln(8+\cos^2x)$$so $$3.f'(x)=\frac{1}{2}\left[\frac{-2 \cos x^{\sin x}}{8+\cos ...
2
votes
2answers
49 views

I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
1
vote
1answer
41 views

Four coplanar points in $\mathbb{N}^3$ space

Is it possible to write out natural number coordinates of four three-dimensional points $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} \in \mathbb{N}^3$, with the following determinant zero? ...
0
votes
2answers
30 views

$\mathbb{Z}^{+}$ includes zero or not?

Does $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer.
6
votes
1answer
50 views

Lattice-Theoretic Interpretation of the Fundamental Theorem of Arithmetic

When equipping $\mathbb{N}^\ast=\mathbb{N}\setminus \{0\}$ with the divisibility relation, it forms a lattice with minimum 1, supremum given by the least common multiple, and infimum given by the ...
3
votes
3answers
57 views

Ways to sum to $n$ with $m$ integers that are $\le k$

Given three natural numbers $n$, $m$ and $k$, how many ways are there to write $n$ as the sum of $m$ natural numbers in the set $\{0, 1, \ldots, k\}$, where order does matter? I've seen the "Ways to ...
6
votes
3answers
127 views

Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?

One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ ...
1
vote
2answers
54 views

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 ...
0
votes
2answers
103 views

The rule of operator “+”

1.It is natural for us now to see the natural number $1,2,\cdots$ and the operator "+", but for me it is hard to see how we define "+", i.e. I can't see the rule of $a+b$. 2.Another question is how ...
2
votes
3answers
78 views

One-element model of first-order PA

The First-Order axiomatisation of PA is: $\forall x. x = x$ $\forall x, y. x = y \rightarrow y = x$ $\forall x, y, z. x = y \land y = z \rightarrow x = z$ $\forall x. 0 \ne S(x)$ $\forall x, y. S(x) ...
1
vote
3answers
35 views

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 ...
0
votes
1answer
24 views

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?
1
vote
0answers
46 views

Is the proof of non-totality of a function $f$ trivial if $f$ is recursively defined only on a subset of $\Bbb N$?

I define a fuction $*$ in a recursive way on a subset of natural numbers (with subset I mean that is $*$ define in the second argument only for $\Bbb N-\{0,1\}$ in other words we have $*:\Bbb N ...
0
votes
1answer
23 views

How to show that for any $n\in\Bbb N\;,n^+=n+1$

Show that for Any $n\in\Bbb N\;,n^+=n+1$ We know for $ n=0 , 0^+=1+0$ for $n\in\Bbb N$ supposing that $n^+=1+n$ then have to show $(n+1)^+=n+1+1=n^++1$
2
votes
1answer
59 views

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?
5
votes
2answers
60 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
-1
votes
3answers
110 views

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
2
votes
2answers
600 views

Three consecutive integers which are power of prime but not prime

Does there exist three consecutive positive integers such that each of them is the power of a prime i.e., is there exist $n \in \mathbb{N}$, such that $n=p^i$, $n+1 = q^j$ and $n+2 = r^k$, where $p$, ...
1
vote
0answers
85 views

Zero divided by zero [duplicate]

I'm relearning mathematics by reading through What is Mathematics?. It begins by explaining the natural numbers, introducing the following property: $a.0=0$ To make $a$ the subject, I would divide ...
1
vote
1answer
44 views

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.
6
votes
2answers
125 views

Sum of sum of $k$th power of first $n$ natural numbers.

I was working on a problem which involves computation of $k$-th power of first $n$ natural numbers. Say $f(n) = 1^k+2^k+3^k+\cdots+n^k$ we can compute $f(n)$ by using Faulhaber's Triangle also by ...
1
vote
1answer
33 views

Number of unique products of two integers of bounded size

If $S,T$ are two sets of integers, define $S*T$ to be the set $S*T = \{st \mid s \in S, t \in T\}$. Let $[1,n]$ denote the set of integers in the range from $1$ to $n$, i.e., $[1,n] = ...
1
vote
3answers
28 views

Proving an inequality for all natural numbers involving parameters

The problem asks to prove the following inequality: $$\forall a,b>0,a\ne b\;\forall n\in \Bbb N, n>1:2^{n-1}(a^n+b^n)>(a+b)^n$$ I'd appreciate a hint, because I tried induction but it lead ...
0
votes
2answers
93 views

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 ...
0
votes
1answer
43 views

Notation regarding the maximum function over a list of naturals

So I'm trying to write down the maximum function(with a precise mathematical notation) over a set of integers by utilizing the generic maximum function which takes two integers, $max: \mathbb{N} ...
0
votes
2answers
49 views

Biggest 8 digit number following two specific rules

Which is the biggest 8 digit number of the form "abcdefgh" which is made up only of 1, 2, 3 and 4 and which follows the rule: the digit 1 is one digit away from another 1, the digit 2 is two digits ...
1
vote
2answers
30 views

$3^a\mid s(n) \Rightarrow 3^a\mid n$

This is not a homework question, neither a championship problem (as far as I've searched in the net), and it came up noticing a singular pattern, involving the powers of $3$: "Prove or disprove that ...
0
votes
1answer
31 views

If $\frac1\alpha+\frac1\beta=1$, irrational, then $\{\lfloor n\alpha\rfloor:n\in\Bbb N\}\uplus\{\lfloor n\beta\rfloor:n\in\Bbb N\}=\Bbb N$

Let $\alpha,\beta\in\Bbb R\setminus\Bbb Q$ such that $\frac1\alpha+\frac1\beta=1$, and define $S(x)=\{\lfloor nx\rfloor:n\in\Bbb N\}$. (Note that my convention takes $0\notin\Bbb N$.) The claim is ...
-1
votes
2answers
104 views

Why is the sum over all positive integers equal to -1/12? [duplicate]

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 ...
1
vote
0answers
13 views

Random Algebra Problem 2

Prove that if a, b, c are integers and x, y, z are non-integer real numbers and $\alpha$ is a real number, for every given set of x, y, z, the number $\alpha$ obtained from the following equation: ...
0
votes
4answers
376 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
1
vote
1answer
55 views

Random Algebra Problem

Prove that if a, b, c, x, y, z, and $\alpha$ are natural numbers. For every given set of x, y, z, the number $\alpha$ obtained from the following equation: $$\frac{a^2}{x^2} + \frac{b^2}{y^2} + ...
3
votes
2answers
89 views

Is $\lim\limits_{n \to \infty} n$ “equal” to $\mathbb{N}$?

In set theory, the natural numbers are defined by means of inductive sets and the successor operation $S(n+1) = n \cup \{n\}$ As such, we have $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$, ...
4
votes
7answers
612 views

How do I show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab?

I really need help with this question. I am required to show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab, and I'm not quite sure how. Any work/help is ...
7
votes
3answers
411 views

Can the natural number have an uncountable set of subsets?

Let $\mathbb{N}$ be the set of natural numbers. Let $X_{i},i\in I$ be an uncountable sequence of subsets such that $$ \bigcup_{i\in I}X_{i}=\mathbb{N} $$ and $$ \bigcup_{i\in J}X_{i}\subsetneq ...
2
votes
0answers
126 views

The set of all natural numbers is closed under addition

I'm trying to prove the theorem described in the title, but my proof is so obvious I doubt it is sufficient. Here's my way of proving it: Definition of addition: Let a, b, and c be natural numbers. ...
-1
votes
1answer
18 views

Turn formula with remainder

How do I turn these formulas: $$\begin{align} y &= \left\lfloor\frac{ x \mod 790}{10}\right\rfloor + 48 \\ z &= (x \mod 790) \mod 10 + 10\left\lfloor\frac{x}{790}\right\rfloor + 48 ...
0
votes
4answers
120 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors and it is presumed that $a'>a$ and $b'>b$. How can we show that the only solution to this ...
1
vote
1answer
82 views

Is this a correct definition of the natural numbers in ZF?

Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$ $y\in{s}\rightarrow(y$ is transitive$)$, and if $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is ...
1
vote
0answers
65 views

Natural Numbers Equation

I am trying to find the $(k_1,k_2,...,k_N)$ tuples solutions to an all natural numbers equation in the following form : Given $n\in\mathbb{N}^{*}$, $N\in\mathbb{N}^{*}$ and $n_i\in\mathbb{N}^{*}\leq ...
1
vote
1answer
26 views

Proof that there are at the most two numbers of exactly six digits that squared end with the same six digits

Written in a more formal way, proof that there are at the most $2$ numbers $n$ of six digits, that $$n^2 \equiv n \mod 10^6$$ Research effort: if $n^2 \equiv n \mod 10^6$ this means $10^6\mid ...
6
votes
4answers
82 views

Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research ...
0
votes
2answers
76 views

Prove $a_n = 2^n$ by strong induction [closed]

Given the sequence $a_n = a_{n-1} + ... + a_0 + 1$, prove by strong induction that for any $n ∈ \mathbb{N}, a_n = 2^n$
1
vote
0answers
38 views

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? ...
0
votes
0answers
61 views

Natural numbers, a proof for the divisibility of any 3 given numbers?

I'm following EdX "Effective Thinking Through Mathematics" and they posed the following question: "If $x, y, z$ are natural numbers other than 1, and you multiply them together and add 1, ($x ...