For question about natural numbers

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-1
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2answers
76 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
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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: ...
1
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4answers
221 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
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1answer
47 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
82 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
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7answers
553 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
388 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
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0answers
85 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
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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
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4answers
97 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
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1answer
67 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
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0answers
54 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
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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
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4answers
80 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
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2answers
67 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
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0answers
36 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
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0answers
50 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 ...
1
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2answers
30 views

How is natural log integration broken up into this range? (equation is contained the script)

When I was reading a paper, I found an strange derivation like $$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$ when $w$ is ...
1
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1answer
40 views

First-order Peano Axioms and order-completeness of $\mathbb{N}$

Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo. Notation: We denote the system of first-order Peano Axioms (along with ...
1
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1answer
79 views

Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since ...
0
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1answer
28 views

Position of 20096 in triangular array of all natural numbers

Write the set of all natural numbers in a triangular array as Find the row number and column number where $20096$ occurs. For example, $8$ occurs on row: $3$, column: $2$ Now, the upper row is ...
0
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0answers
41 views

Abstract Algebra- Functions [duplicate]

Give an example of a function $f : \mathbb N \to \mathbb N$ with the property that there exists a function $g : \mathbb N \to \mathbb N$ such that the composition $g ∘ f$ is the identity function on ...
-1
votes
2answers
89 views

On compositions of functions and identity functions [duplicate]

Give an example of a function $f\colon\mathbb N\to\mathbb N$ with the property that there exists a function $g\colon\mathbb N\to\mathbb N$ such that the composition $g \circ f$ is the identity ...
1
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1answer
44 views

Relation between positive integers greater than 2

$x_1, x_2,\ldots, x_{96}$ are positive integers greater than 2, which satisfy the relation: $$ \frac{1}{x_1^4}+\frac{1}{x_2^4}+\cdots+\frac{1}{x_{96}^4}=\frac{1}{6}$$ I have two questions: 1. At least ...
-5
votes
3answers
127 views

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists [closed]

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists a function $g: \mathbb{N} \rightarrow \mathbb{N}$ such that the composition $g \circ f$ is the ...
0
votes
1answer
25 views

How to prove these problems? (Operation on N)

If $m+n = m+k$, then $n=k$. If $mk = nk$ and $k\neq0$, then $m=n$. I'm not sure what t do. Do I have to use math induction or not? (Base on Peano system)
0
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0answers
29 views

Alternative Bijection from $\mathbb{N}$ to its finite subsets

There are different proofs for the fact that there's a bijection from $\mathbb{N}$ to the set of all its finite subsets, but I would like to know if this explicit bijection does work, too. ...
5
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3answers
82 views

Are there any interesting examples of subsets of $\mathbf{N}$ that are known to be nonempty, but of which no elements are known?

There are many results in mathematics that establish the existence of some object without actually constructing said object. I am wondering if there are any interesting properties of the natural ...
-1
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2answers
49 views

Discrete Mathematics Proof Question

Prove or disprove that there are infinitely many $x, y, z \in \mathbb N$ such that $$\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2}$$ Currently, I tried to substitute $x, y,$ and $z$ with $2n$ and $n$ ...
2
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2answers
52 views

Discrete mathematics proof relating to Fermat's Theorem

Assuming the Fermat Theorem, show that there is no natural number $x$, $y$, and $z$ and $n\geq3$ such that $$\frac{1}{x^n} + \frac{1}{y^n} = \frac{1}{z^n}. $$ So far I think proof by contradiction ...
1
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1answer
26 views

Proof $\prod_{i = 1}^n \frac{n + i}{2i-3} = 2^n(1-2n)$ using inducction

i'm trying to solve this, using induction. The base step is easy, there's no difficult there. The problem comes in the inductive step, I got to demonstrate that: $$\prod_{i = 1}^{n+1} \frac{n+ 1 + ...
3
votes
2answers
89 views

solve equation in positive integers

Can anybody help me with this equation? Solve in $\mathbb{N}$: $$3x^2 - 7y^2 +1=0$$ One solution is the pair $(3,2)$, and i think this is the only pair of positive integers that can be a solution. ...
1
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1answer
50 views

Properties of Natural Numbers and Mathematical Induction

When working with natural numbers how to check that the property we consider is "permissible" to speak about? And not like the property "The smallest positive integer not definable in under eleven ...
2
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2answers
117 views

Natural numbers in Set Theory

We seem to accept the fact $(\omega,+,\times,<,0,1)^{V}$, where $V:=x=x$ is the set theoretic universe, properly reflects what is intuitively understood to be the set of natural numbers, i.e. we ...
0
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1answer
37 views

How Can I find the summation of divisors of $n^p$.

For Example $n=8$ and $p=2$. So $n^p=64$. And the summation of divisors is $1+2+4+8+16+32+64=127$. But the problem arises when $n=10^6$ and $p=10^6$. Remember u can modulus the result by $100$.
0
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1answer
38 views

map sum of square integers to a contiguous range of integers

Given a list $a$ of integers, $$n_{a_1}, n_{a_2}, ..., n_{a_d}$$ have $$N_a = \sum_{j=1}^d n_{a_j}^2.$$ The various $N_a$, $N_b$ etc. are integers, but are not contiguous: for example, if $d=2$, ...
1
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0answers
43 views

Two Perfect Squares--$(3n+1) \& (4n+1)$. [duplicate]

Assume $n$ is a Natural Number which satisfies the following 2 properties simultaneously: $01$ . $(3n+1)$=$a$12 for some Natural Number $a$1. $02$ . $(4n+1)$=$a$22 for some Natural Number $a$2. ...
1
vote
1answer
70 views

Prove any $k\in\mathbb{N}$ can be created using sum of $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$

I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition. My attempt was just bad... I've ...
8
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7answers
2k views

Prove that $4$ is the only solution to $2+2$. [duplicate]

This question was featured on Saturday Morning Breakfast Cereal and I haven't been able to find a proof. Can anyone help?
4
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3answers
93 views

What's the digit “a” in this number?

a and b are digits in a four-digit natural number 7a5b. If 7a5b is divisible by 18, how many different possible values can "a" have?
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1answer
100 views

Modulo over rational numbers?

Consider two irreducible fractions: $r_1 = \frac{p_1}{q_1}$ $r_2 = \frac{p_2}{q_2}$ with $r_1 \ge 0$ and $r_2 \ge 0$. How the modulo $\%$ is defined over rational numbers (I think that is $r_3$ ...
1
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1answer
29 views

Why is it true that Max(X) = Min(-X) for a group X composed of numbers in Z

I'm looking for a formal explanation that won't involve calculus (if possible) that would explain why for every group of numbers X (such that all numbers in X are integers), Max(X) = Min(-X) (where -X ...
2
votes
0answers
60 views

yes, why does a negative times a negative make a positive? [duplicate]

for a while I have been interested in the details of the construction of the integers from the natural numbers. credit to the software, for as soon as I began writing this, for it drew my attention ...
2
votes
2answers
124 views

Landau's “Foundations of Analysis” - Addition of natural numbers

At the beginning of his Foundations of Analysis book (translated from German), Landau writes in his Preface for the Teacher : Peano defines $x+y$ for fixed $x$ and all $y$ as follows : $$x+1 = ...
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1answer
50 views

Dimension of the rationals over the integers

What is the dimension of the $Q$, when they are seen as a vector space over the integers $Z$ (with the usual definitions of addition and multiplication)? Initially I thought that the dimension ought ...
4
votes
3answers
178 views

Prove commutativity of product in natural numbers

The objective of this excersise it's to demostrate the result known as commutativity in $\mathbb{N}.$ For that it's defined the function p: $\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} $ $ ...
0
votes
2answers
44 views

Best constant integer inequality

Suppose $a_1,\dots,a_n$ are positive integers. Trivially one has that $$ \sum_{i=1}^n a_i^2 \leq \left (\sum_{i=1}^n a_i \right)^2 $$ I am wondering whether it is possible to make it somehow sharper, ...
3
votes
2answers
169 views

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 ...
3
votes
5answers
177 views

Product of all primes

Is the product of all primes a natural number? In other words, is this true: $$ \prod\limits_{\text{primes}} p_i \in \mathbb{N} $$ And if so, what about just some of them: $$ ...
1
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0answers
44 views

Is it correct? $\lim_{n\rightarrow \infty} \frac{c^{n}}{n!^{\frac{1}{k}}}$

This is what we have $$\lim_{n\rightarrow\infty} \frac{c^{n}}{n!^{\frac{1}{k}}},$$ $$n \in N, k>0, c>0$$ If n->inf ...