For question about natural numbers
3
votes
2answers
54 views
How to solve $at + b = 0 \pmod {(a-t)}$?
Is there any way (except trying for $t=0,1,2,\ldots,a-1)$ to solve the following equation for $t$ when $a$ and $b$ are known?
$$ at +b = 0 \pmod{(a-t)} \text{ with } a,b,t \in N $$
4
votes
1answer
70 views
Property of natural numbers involving the sum of digits
How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k.
Example:
$$
...
0
votes
0answers
26 views
number property [duplicate]
How can you prove that every natural number M or M+1 can be written as k + Sum(k), where Sum(k) represents the digits sum of number k.
Example:
248 = 241 + Sum(241) = 241 + 2 + 4 + 1
2
votes
1answer
40 views
What model do you get from PA without addition and multiplication?
I have the feeling that this question is trivial, but I cannot figure the answer by myself nor from the stuff I have read. So the question is if addition (and multiplication) can be shown as a theorem ...
9
votes
4answers
197 views
Should $\mathbb{N}$ contain $0$? [closed]
This is a classical question, that has led to many a heated argument:
Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$?
It is immediately obvious that the question is ...
10
votes
2answers
254 views
A matrix w/integer eigenvalues and trigonometric identity
Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated:
Let $n$ be a natural number.
(a) Consider the following Toeplitz/circulant symmetric matrix:
...
0
votes
4answers
89 views
Can you have a numeral system with infinite digits?
If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that ...
5
votes
1answer
72 views
Functional Equation help
Came across this problem a little while ago but can't seem to get beyond a certain point.
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$
for all $n$.
...
-4
votes
1answer
57 views
I need help with the proof of Theorem 4, Chapter I in E. Landau's “Foundations of Analysis”
I need help with the proof of Theorem 4, Chapter I in E. Landau's "Foundations of Analysis"
To every pair of natural numbers $x$, $y$, we may assign in exactly one way a natural number, called ...
6
votes
2answers
130 views
Dividing a number by zero [duplicate]
Why can't you divide a number by zero?
It is possible to say $\sqrt{-1}$ is an imaginary number $i$, but why can't you say $\frac{1}{0}$ is also an imaginary number $z$ (for example)?
2
votes
3answers
119 views
Proving the so-called “Well Ordering Principle”
Is there anything wrong with the following proof?
Theorem. Every non-null subset $B \subset\mathbb{N}$ has a least member.
Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
0
votes
3answers
80 views
Find the smallest natural number that satisfy $13^N = 1 \pmod {2013}$
Moderator Note: This is a current contest question on Brilliant.org.
Find the smallest natural number that satisfy:
$$13^N = 1 \pmod {2013}$$
My idea is to use the Fermat's little theorem ...
3
votes
1answer
68 views
Proof showing there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
Given that $n\in\Bbb N$, show that there exists a $k\in\Bbb N$ such that for all $m\ge k$, there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
1
vote
3answers
141 views
Creating the set of natural numbers
I am not a mathematician but an engineer, so I can read some basics of the language proofs are written in. Second I am bad in probability and infinity and my question covers both. So I like to ...
2
votes
1answer
54 views
associativity of multipication of natural numbers
I am trying to prove by induction the associativity of natural numbers. It is easy to see that if $n,m\in \mathbb{N}$, then $(mn)1=m(n1)$. If $p\in \mathbb{N}$ is such that $(mn)p= m(np)$, then ...
0
votes
2answers
69 views
Why this happens with numbers of Mersenne type that are prime?
If we define the function $f$ on $\mathbb N$ as $f(n)$ is the "iterative sum of the digits of $n$ until we arrive at the single digit", for example $f(123456789)=f(45)=f(9)=9$ then if $M_p$ is a ...
9
votes
3answers
206 views
Existence of a sequence that has every element of $\mathbb N$ infinite number of times
I was wondering if a sequence that has every element of $\mathbb N$ infinite number of times exists ($\mathbb N$ includes $0$). It feels like it should, but I just have a few doubts.
Like, assume ...
2
votes
0answers
53 views
Sequences of Natural Numbers without Arithmetic Subsequences
Let's call a sequence $k^+$-free if it is monotonic and contains no arithmetic sequence of length $k$. Define the $\bf{density}$ of a sequence of natural numbers $s_n$ as $$\lim_{n\to \infty} ...
0
votes
3answers
101 views
What is the relationship between any natural number and two other natural numbers?
By this I mean, you could take any natural number, apply some kind of operation (arithmetic or other), and end up with two natural numbers. Then, you can apply an inverse operation on the two produced ...
0
votes
0answers
41 views
Parametric Recursion Theorem
Let $A$ and $P$ be sets and let $a:P\to A$ and $g:P\times A\times \Bbb N\to A$ be functions. Then there is only one function $f:P\times\Bbb N$ such that:
(a) $f(p,0)=a(p), \forall p\in P$.
(b) ...
0
votes
2answers
38 views
Determine with a proof the largest number which can be written as a product of natural numbers which have sum 2012
I'm having trouble finding the largest number and proving it.
2
votes
2answers
56 views
How to prove that $+$ is commutative on the natural numbers?
Let $N$ be a non empty set. Let $s:N\to N$ a function satisfying:
there is only one element in $N-s(N)$ (denoted by $1$);
$s$ is injective;
for any subset $X\subset N$, if $1\in X$ and $(n\in N ...
5
votes
2answers
178 views
$a+b+c+d+e=abcde$ What is $\max(a)?$
The question from my nephew is this:
If $a, b, c, d, e \in N^+$ and $a+b+c+d+e=a\times b\times c\times d\times e$, then what's the the maximum possible value of $a$? Thanks ahead:)
2
votes
1answer
53 views
Connecting functional identity of a function with its image set
Okay, now I really need real help, maybe the task is not too heavy but I do not know the easy way to solve this. Let´s start with the problem, now.
Suppose that we have some function $f: \mathbb N ...
0
votes
3answers
96 views
Proof for the divisibility of natural numbers by at least one prime
Like any other natural number, N is divisible by at least one prime number (it is possible that N itself is prime).
Is there a proof for this?
4
votes
1answer
70 views
Is there a set theoretic construction of the natural numbers or integers such that the product of two numbers is their Cartesian product?
Is there a set theoretic construction of the natural numbers or integers such that the product of two numbers is their Cartesian product? What I mean is, e.g., if $25 = A$ and $2 = B$ then $50 = ...
34
votes
9answers
2k views
Given real numbers: define integers?
I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following:
Integer numbers are just special cases (a subset) of real numbers. ...
1
vote
4answers
146 views
Does such a natural number exist, that it would be divisible by every other natural number
I've got to prove (or disprove) the following statement:
$\exists x \in \mathbb{N} \; \forall y \in \mathbb{N}: y \mid x$,
which translates into "It exists such $x$ from the set of natural numbers, ...
1
vote
2answers
54 views
Function to convert each number in a M..N to another number in the same range
I'm trying to find a function that can convert each number (I mean natural numbers) in the range M..N to another number in the same range.
Later I need to convert ...
2
votes
2answers
122 views
What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?
The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
1
vote
1answer
77 views
Cardinality natural numbers
Is the cardinality of the natural numbers a natural number?
$|\mathbb{N}| \in \mathbb{N}\text{ or } |\mathbb{N}| \notin \mathbb{N}$, that is the question.
4
votes
5answers
290 views
Prove 24 divides $u^3-u$ for all odd natural numbers $u$
At our college, a professor told us to prove by a semi-formal demonstration (without complete induction):
For every odd natural: $24\mid(u^3-u)$
He said that that example was taken from a high ...
2
votes
1answer
110 views
multiple approaches/ways to prove that $1000^N - 1$ cannot be a divisor of $1978^N - 1$
Am interested in learning to do multiple proofs for the same problem, and hence I chose this problem:
Prove that for any natural number $N$,
$1000^N - 1$ cannot be a divisor of $1978^N - 1$. ...
2
votes
1answer
105 views
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 ...
0
votes
1answer
205 views
Peano's Postulates Proofs
How can I prove the following two questions:
Prove using Peano's Postulates for the Natural Numbers that if a and b are two natural numbers such that a + b = a, then b must be 0?
Prove using Peano's ...
2
votes
1answer
343 views
Recurrences that cannot be solved by the master theorem
I am given this problem as extra credit in my class:
Propose TWO example recurrences that CANNOT be solved by the Master Theorem.
Note that your examples must follow the shape that $T(n) = ...
1
vote
4answers
138 views
Expression that only result a natural number
There's any math trick that can turn any negative number to 0?
I'm not sure if this is possible without conditional logic
f(x>=0) = x
f(x<0) = 0
1
vote
1answer
67 views
Estimation of a polynomial
I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2).
The claim translated to polynomials is the following:
Assume $n\geq 3, c\geq 1, ...
4
votes
4answers
191 views
Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?
Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?
I had also got a clue: it's related to e.
Please help! ...
0
votes
2answers
225 views
How to define a bijection between natural numbers and the set of all polynomials with natural coefficients and finite variables?
Is there an explicit algorithm which establish a bijection between polynomials with finite variables and natural coefficients and natural numbers. Does anyone have one of these?Thanks.
3
votes
2answers
191 views
Interesting or non-obvious finite subsets of the natural numbers
I was recently explaining to someone how to prove that there are infinitely many prime numbers, and I mentioned to them that it's not immediately obvious, upon first encountering the natural numbers, ...
5
votes
3answers
263 views
Field with natural numbers
To make sure that we are talking about the same, I would like to post the relevant definitions I know first.
Definitions:
A pair $(G, +)$ where $G$ is a set and
$+: G \times G \rightarrow G$
is ...
3
votes
4answers
602 views
Consistency of Peano axioms (Hilbert's second problem)?
(Putting aside for the moment that Wikipedia might not be the best source of knowledge.)
I just came across this Wikipedia paragraph on the Peano-Axioms:
The vast majority of contemporary ...
14
votes
2answers
223 views
Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?
2
votes
1answer
228 views
Proof that $1$ is the least natural number using the Principle of Well Ordering
I'm doing a question from a PDF on Abstract Algebra.(The pdf can be found here http://abstract.pugetsound.edu/ 2012 edition). I have to show that the Principle of Well-Ordering implies that $1$ is the ...
1
vote
4answers
137 views
Euler's theorem for powers of 2
According to Euler's theorem,
$$x^{\varphi({2^k})} \equiv 1 \mod 2^k$$
for each $k>0$ and each odd $x$. Obviously, number of positive integers less than or equal to $2^k$ that are relatively ...
21
votes
14answers
2k views
How can we produce another geek clock with a different pair of numbers?
So I found this geek clock and I think that it's pretty cool.
I'm just wondering if it is possible to achieve the same but with another number.
So here is the problem:
We want to find a number ...
1
vote
3answers
202 views
Perfect square with digit-sum 15
Prove that there is not a single natural number $N$ with sum of digits equal to 15 that is the square of an integer.
1
vote
3answers
289 views
Proof that binomial coefficient is a natural number [duplicate]
Possible Duplicate:
Proof that a Combination is an integer
What is the proof that the binomial coefficient is a natural number?
$$k\ge0,n\ge k \implies {n \choose k} \in N,$$
I guess ...
3
votes
1answer
71 views
Prove that $\mathbb{N}$ is nonwhere dense in $\mathbb{R}$
Prove that the set $ \displaystyle{\mathbb{N} =\{1,2,3, \cdots \} }$ is nonwhere dense in metric space $ \displaystyle{ \left( \mathbb{R} ,|\cdot| \right)}$ .
I have found a solution in two steps:
...
