# Tagged Questions

87 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 ...
27 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 ...
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### 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.
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### 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 ...
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.
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### 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 .. ...
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### 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 ...
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 $\next:\Nt\to\Nt$). ...
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 ...
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### 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 ...
87 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.
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### 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? ...
319 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?)
125 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 ...
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### 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 ...
198 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 ...
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### 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 ...
393 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. ...
160 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 ...
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 ...
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
523 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 ...
478 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?
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### 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 ...
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, ...
341 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 ...
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 ...
121 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 , $444^0 \pmod 7 = 1$
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### 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 : ...
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 ...