1
vote
3answers
85 views

Sum of the digits

Let $N$ be the greatest number that will divide $1305,4665$ and $6905$, leaving the same remainder in each case. Then what is the sum of the digits in $N$?
4
votes
0answers
94 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
0
votes
0answers
29 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
3
votes
4answers
144 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
1
vote
1answer
35 views

Euler product of Dirichlet series

Let $f$ be an arithmetic function such $f(n_1n_2)=f(n_1)f(n_2)$ for all $n_1,n_2 \in \mathbb{N}$ with $\gcd(n_1,n_2)=1$. Suppose we know that the Dirichlet series $$F(s) = \sum_{n=1}^{\infty}f ...
7
votes
5answers
691 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
0
votes
2answers
120 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
3
votes
0answers
60 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
2
votes
1answer
70 views

Prove this simple arithmetic relation

Prove that if $$a \mid b$$ and $$a \mid c$$ then $$a \mid bx+cy$$ for any integers $x$ and $y$. Here's my proof: $$b = ak$$ $$c = am$$ $$bx+cy = akx+amy = a(kx+my)$$ Notice that $kx+my$ is an ...
6
votes
0answers
125 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
1
vote
0answers
21 views

Determine an arithmetic relation

Let $f$ be an arithmetic function. Let $p$ be a prime number, $\chi(n)=\left(\frac{n}{p}\right)$ be a primitive Dirichlet character modulo p, where here $~\left(\frac{n}{p}\right)$ is the ...
3
votes
1answer
67 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
0
votes
2answers
55 views

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$?

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$ for some $k\in \mathbb{Z}$ ? or we can prove that this never belongs to $\mathbb{Z}$ ?
4
votes
2answers
120 views

Why is the language of arithmetic usually $(+, \cdot, 0, s)$, not $(+, \cdot, 0, 1)$?

The formalized theory of arithmetic has usually $(+, \cdot, 0, s)$ as its language. However, from what we usually do in ring theory, it seems natural to use $(+, \cdot, 0, 1)$ as the language of ...
2
votes
5answers
56 views

For any prime $p>3$ show that 3 divides $2p^2+1$

Does anyone know how to show this preferable without using modular For any prime $p>3$ show that 3 divides $2p^2+1$
1
vote
4answers
71 views

Show that if $p$ is a prime number $> 3$ then $24 \mid p^2-1$ [duplicate]

Hi guys can someone help me with this ?(Without using Modular arithmetic) Show that if $p$ is a prime number $>3$ then $24$ $\mid$ $p^2-1$
0
votes
1answer
52 views

Explain theorem in Number theory

can some one explain with a clear example this theorem for me, Let ($A_1$, $A_2$, $A_3$,..., $A_n$) be integars and $p$ a prime number. if $p|(A_1A_2A_3...A_n)$ then there exist some $1 \leq k \leq ...
2
votes
1answer
47 views

consecutive prime power

I'm interesting on consecutive prime power numbers. I see that there is the Mersenne primes and the Fermat Primes that give solutions and $(8,9)$. In Sloane collection it is referred on A006549 and it ...
1
vote
1answer
36 views

Reducing simultaneously a pair of fractions $\frac{a^2}{b},\frac{ a^3}{c}$ using only gcds

Given three positive integers $a,b,c$ and I want to find the smallest positive integers $a', b', c'$ such that $$ \frac{a^2}{b} = \frac{a'^2}{b'} \quad \text{and} \quad \frac{a^3}{c} = ...
0
votes
1answer
75 views

An easy question with integer numbers

I have an easy question of arithmetic. Let $a, b, N$ be integer numbers such that $\mathrm{gcd}(a,b,N) = 1$. Is it true that there exists an integer number $x \in \mathbb{Z}$ such that ...
3
votes
3answers
40 views

Divisibility in base $7$ problem

Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.
0
votes
1answer
47 views

How can I determine least common multiple for a given number and all numbers before it?

The wikipedia article on least common multiples only talks about determining the least common multiple between 2 numbers. I'm looking for an algorithm that will determine it for a set of numbers 1 .. ...
1
vote
1answer
38 views

Is it possible to extract any encoded $x, y \in \mathbb{N^*}$ from $z=ax + by$

Is there any specific $a, b \in \mathbb{R}$, $\forall x,y \in \mathbb{N^*}$, take $z=a\cdot{}x+b\cdot{}y$ (then $z\in\mathbb{R}$), we can always extract $a,b$ from $z$. Here below are some trials I ...
6
votes
1answer
51 views

Defining natural numbers without $0$ or $1$.

Let's define Peano's axioms having $2$ as the first number: $\newcommand\Nt{\mathbb N''}2\in\Nt$. $\newcommand\next{\mathop{\mathrm{next}}}\forall n\in\Nt:\next n\in\Nt$ (or $\next:\Nt\to\Nt$). ...
0
votes
1answer
51 views

What are the smallest possible theories?

Im wondering how we could define a general form for the smallest possible theory in some formal language. In other words, if we have the formal language of first order logic, what is the smallest set ...
1
vote
1answer
53 views

Two variable integer equation

I have the following equation: $$ p^q(2^{q-1}-1)=9p^7q $$ I need to solve for $p$ and $q$. $p$ and $q$ are integers. I think I could take the case $p=0$ separately and for that one $q$ could be ...
0
votes
3answers
89 views

Number of digits of $2^{1000}$ [duplicate]

A friend asks me to find the number of digits of $2^{1000}$. I tried to look for a pattern by calculating the first powers of $2$ but I didn't find it. How should I proceed? Thanks.
0
votes
0answers
50 views

Density of numbers for which binary addition chains are shortest

Relative to natural powers of two, how often does it happen that the shortest length of addition chain for a natural number $n$ agrees with that of the chain used in binary exponentiation? ...
0
votes
2answers
356 views

Simple algorithm to get the square of an integer using only addition?

This problem was mentioned in passing in a reading and it piqued my curiosity. I'm not sure where to start. Any pointers? (perhaps square root was meant?)
2
votes
2answers
129 views

Do Hyperreal numbers include infinitesimals?

According to definition of Hyperreal numbers The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + 1 ...
0
votes
3answers
64 views

How to prove some statements about divisibility and the $\gcd$ function

Struggling with some number theory homework. Could use a helping hand. The two statements are as follows $\gcd(c, ab) \mid \gcd(c,a)\gcd(c,b)$ If $c \mid ab$ and $\gcd(a,b)=1$, then ...
12
votes
2answers
200 views

Special subdivision of numbers from 1 to 99

I've been lately working on a problem I still can't solve. The problem is: Can we divide numbers from 1 to 99 into 33 groups of three numbers, such that in every group one number is the sum of the ...
0
votes
1answer
64 views

Natural number n-Divisibility

The number of natural number $n$ in the interval $[1005,2010]$ for which the polynomial $$1+x+x^2+x^3\dots +x^{(n-1)}$$ divides the polynomial $$1+x^2+x^4\dots+x^{2010}$$ is: I could realize that ...
7
votes
4answers
397 views

Mere coincidence? (prime factors) [closed]

Whether some things in mathematics are mere coincidences might keep philosophers busy for 100,000 aeons, but maybe when such a coincidence gets exploited then it's not a "mere" coincidence any more. ...
3
votes
3answers
168 views

Finding the sum of two numbers knowing only the primes

Pretend $N_1$ is the prime factorization of 30 and $N_2$ is the prime factorization of 8. Is there a way, using only $N_1$ and $N_2$, to get the prime factorization of the sum, 38? It is easy to do ...
2
votes
1answer
60 views

Maximal non averaging subset Algorithm [closed]

Given a set of positive integers, not greater than 100000, how can we find the maximal subset such that no three numbers form an Arithmetic progression? I am looking for some algorithmic approach. If ...
0
votes
2answers
180 views

A proof of $n*0=0$?

The only proof I've seen for this assumes that $0$ follows all the rules of arithmetic. How can we make that assumption when dividing by $0$ is a problem? I know that some people don't agree that all ...
82
votes
1answer
10k views

Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
1
vote
2answers
135 views

Are the propreties of arithmetic unproven?

For example, the property which says that $$a(b+c)=ab+ac$$ This is very clear for integers, but is it actually provable for all real numbers (and complex maybe). Or the commutative property which says ...
5
votes
2answers
63 views

How do I append an integer to the left of another integer?

For example: . is my append operator f(x,y) = |x| . |y| f(1,45) = 145 f(233,10) = 23310 f(8,2) = 82 f(0,1) = 1 This is a trivially easy problem to ...
2
votes
3answers
62 views

Constrictions on A.P with factorials.

There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression. Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
16
votes
2answers
546 views

How to define addition through multiplication?

One might define multiplication $\bullet$ on $\mathbb Z$ as follows: $\bullet: \mathbb Z\times \mathbb N\ni (a,b) \mapsto a+\cdots+a\in \mathbb Z$ where we add $b$ times. But suppose we are in a ...
17
votes
1answer
480 views

Are the real numbers ever needed to prove a property of the natural numbers?

Suppose no one had invented/discovered the real numbers yet (so e.g., no calculus), would this constrain the possible theorems or knowledge we could have about the natural numbers?
0
votes
1answer
85 views

Generate unique integer from $n$ integers and solve to get the integers from result

What could be the best way to generate a unique integer from $n$ integers in order $(n_1,n_2,\ldots)$? Further, from $n$, we should be able to get back each $n_1, n_2,\ldots $ etc. For example, from ...
8
votes
2answers
172 views

Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$

I'm looking for numbers of the form $$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$ where $p_{i}$ are prime numbers, ...
2
votes
3answers
347 views

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so ...
2
votes
2answers
138 views

Sum set fixpoint, how many iterations?

I want to approach linear equations of the following form over the integers $\mathbb{Z}$: $$x_1 + \cdots + x_n = 0.$$ I stepped over the sum set, which is defined as follows: $$S + T = \{ x + y ...
2
votes
3answers
123 views

Find the remainder in the following case.

Find the remainder when $444^{444^{444}}$ is divided by $7$. My approach : $E(7) = 6 $ $444^{444} \pmod 6 = 0$ so , $444^0 \pmod 7 = 1$
1
vote
2answers
126 views

Find the total number of digits in the following case.

Two numbers $2^k$ and $5^p$ are expanded first then written side by side i.e. adjacent to each other. Find the total number of digits in that case if $k = p = 2004$. My approach : ...
4
votes
5answers
162 views

Find the possible value from the following.

Find the possible value from the following. I'm not able to end up on a concrete note, as I'm unable to get the essence of question, still not clear to me. $x$, $y$, $z$ are distinct reals such ...