Questions tagged [arithmetic]

Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag. Questions about number theory (sometimes called "arithmetic") should not use this tag and should instead use (number-theory) or (elementary-number-theory).

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Is there a numeral system that makes both addition and multiplication easy?

Decimal positional notation, the system for writing numbers we all use every single day, makes addition very easy by transforming it from a computation to a repeated operation on individual digits (...
user avatar
8 votes
0 answers
263 views

A geometrical analogy between $1/x$ and $\sqrt{x}$

In a question on geometrical constructions of numbers, two constructions appear to be related by analogy. In the first construction a specific straight line $L$ (blue) is constructed (going through $1$...
Hans-Peter Stricker's user avatar
8 votes
0 answers
143 views

About the regularity of the decomposition in prime factors

Definition of $\rho$. Let's consider a function $\rho$, acting on the prime decomposition of an integer $n$: $$\begin{matrix} \rho\colon & \mathbb N_{\geqslant 1}&\to &\mathbb N_{\...
E. Joseph's user avatar
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8 votes
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To what extent can the fondamental theorem of arithmetic be used to give a canonical form to non-integer numbers?

The fundamental theorem of arithmetic gives us a unique way of writing any non-zero integer. For any $n \in \mathbb{Z}^*$, we have a unique decomposition : $$n = (-1)^\epsilon \prod\limits_{i \in \...
Esperluet's user avatar
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7 votes
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Interpretation of 1089-number trick in terms of symmetric group action on Ext-group?

The well known "1089 trick" (see e.g. here) says that if you take a three-digit number, subtract its reverse and to this answer add the reverse of the answer again, then you always get 1089. ...
user221483's user avatar
7 votes
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181 views

Could a nice principe, or at least a simpler alternative proof, be found regarding a lemma of Gauss

To prove the quadratic reciprocity law, Gauss needed the following lemma: If $p$ is a prime number congruent to 1 modulo 8, then there exists a prime $q<p$ such that $p$ is a non residue modulo $q$...
MikeTeX's user avatar
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7 votes
1 answer
150 views

Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
user282321's user avatar
7 votes
0 answers
222 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
Daniel Gerigk's user avatar
6 votes
0 answers
292 views

Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...
Hans-Peter Stricker's user avatar
6 votes
0 answers
190 views

Shepherdson's model for Open Induction

In the paper "A Non-Standard Model for a Free Variable Fragment of Number Theory", Shepherdson constructs a recursive model for a fragment of arithmetic known as "Open Induction". I would like to ...
russoo's user avatar
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A runs 7/4 times as fast as B. If A gives B a start of 84m, how far must the winning post be...?

The problem statement in the book is: $A$ runs $7/4$ times as fast as $B$. If $A$ gives $B$ a start of $84$m, how far must the winning post be so that $A$ and $B$ might reach it at the same time? ...
user103816's user avatar
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6 votes
1 answer
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Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ &...
atif93's user avatar
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6 votes
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Regarding identities with sums of consecutive squares

This comic http://abstrusegoose.com/63 points out an interesting identity with sums of consecutive squares. Let us take positive integers $k, p, q$, with $p < q$, and ask if $k^2 + \ldots + (k + p)^...
Pedro's user avatar
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5 votes
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Proving that $\Sigma_{i=0}^{n-1} \alpha^i = 0$ for $k\mid n$ and $\alpha$ of order $k$

I've "found" the following theorem: If $n$ is a composite number, $k$ is a divisor of $n$, and $\alpha$ an element of $(\mathbb Z/n)^\times$ of order $k$, then $$\sum_{i=0}^{n-1} \alpha^i = ...
MikeTeX's user avatar
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5 votes
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Here is another "$e$-$\pi$-without -calculating" comparison

When playing in a Python console I observed $\,\pi^\pi\approx 36.46\,$ and $\,e^e\approx 15.15$, and that their ratio is close to $\dfrac{12}{5} =$ a fraction with small numerator and denominator, ...
Hanno's user avatar
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If $\forall n \in \mathbb{N}^*, \ a^n - 1 \mid b^n - 1$ therefore $\exists p \in \mathbb{N}^*,\ b=a^p$.

Let $a,b \in \mathbb{N}$ such that $2 \le a \le b$. Suppose $\forall n \in \mathbb{N}^*, \ a^n - 1 \mid b^n - 1$, show that $\exists p \in \mathbb{N}^*,\ b=a^p$. I saw a solution using calculus (...
Michelle's user avatar
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5 votes
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About the number of solutions of $\varphi(n)=m$

Let's denote for all $m\geqslant 2$, $N_m$ the number of solutions of $$\varphi(n)=m$$ where $\varphi$ is Euler totient function, i.e. $$N_m:=\#\{n\in \mathbb N,\ \varphi(n)=m\}.$$ We can prove ...
E. Joseph's user avatar
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5 votes
0 answers
363 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 ...
JonHerman's user avatar
  • 2,901
5 votes
1 answer
134 views

Square and reverse reading of an integer

For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$, we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= \sum_{i=0}...
francis-jamet's user avatar
4 votes
0 answers
81 views

Find a positive integer $n$,$m$, $p$, $q$ ($p \neq q$, $m>1$) such that $p$ and $q$ divide $mn^2 - 1$ and $mn$ divides $p - q$.

Our programming teacher asked us to find triple positive integers n,p,q (m=2) such that: $p$ and $q$ divide $2n² - 1$ and $2n$ divides $p - q$ With the program below, I didn't find such an integer ...
Etanche's user avatar
  • 320
4 votes
0 answers
126 views

Asymptotic density of an infinite union of subgroups

Let $1 < a_1 < a_2 < a_3 <{} ...$ be a sequence of integers. For a subset $A \subset \Bbb Z$, denote by $d(A)$ its natural density (if it exists). Is it true that $$ \lim_{N \to +\infty} ...
Alphonse's user avatar
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4 votes
0 answers
108 views

I found a pattern in consecutive squares: $(a^2-b^2)-(b^2-c^2)$ is always $2$.

I was working on squares of numbers then found out that the difference of difference between two consecutive numbers is $2$. Saying this with an example like $$2^2=4\qquad\qquad 3^2=9\qquad\qquad 4^2=...
Dhananjay Batwal's user avatar
4 votes
0 answers
106 views

Is there a binary operation $\otimes:\mathbf N^2\to\mathbf N$ such that $n\otimes n=0$ and $0\otimes n=n\otimes 0=n$ other than xor?

Is there an associative binary operation $\otimes:\mathbf N^2\to\mathbf N$ such that $0\otimes n=n\otimes 0=n$ and $n\otimes n=0$ other than xor? Basically I am trying to find something that is ...
user avatar
4 votes
0 answers
144 views

Two numbers with a given difference having the same number of divisors

So, it is required to prove that for each natural $k$ there are two natural numbers with a difference $k$ having the same number of divisors. For example, for the case $k=27$, the pair $(18,45)$ is ...
QLimbo's user avatar
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60 views

Is there any operation on real numbers, defined using only the basic arithmetic operations, which is diassociative, but not associative?

By operation I mean binary function from pairs of reals to reals. By diassociative I mean that the subset generated by any two numbers is associative - or, any string composed only of two types of ...
Seth Schmidt-O'Hainle's user avatar
4 votes
0 answers
73 views

Is there an infinity of $k \in \mathbb{N}$ s.t. $2 {k \choose k/2} - 1$ is prime?

my question is basically in the title. Is there a way to answer it, or are there any theorems or conjectures regarding it? Thank you.
laku's user avatar
  • 57
4 votes
0 answers
123 views

Write integers up to 100 using $2,3,4$ and $5$.

I plan on giving this challenge to my students but I can't seem to solve it myself. Challenge : write all the integers from $0$ to $100$ using the numbers $2,3,4$ and $5$ exactly once and in this ...
krirkrirk's user avatar
  • 2,027
4 votes
1 answer
100 views

Can bounded addition and multiplication be computable in a non-standard model of arithmetic?

Let $M = (N, \oplus, \otimes, <_M, 0_M, 1_M)$ be a nonstandard model of peano arithmetic. $\oplus$ and $\otimes$ are uncomputable due to Tennenbaum's theorem. For $c \in N$, let $\oplus_{<c}, \...
Christopher King's user avatar
4 votes
2 answers
245 views

Recursive sequence modulo 4

For all odd positive integers $k$, I define a recursive sequence by $$ d_k=2+ {k\choose 1}d_{k-2} + {k\choose 2}d_{k-4} + \dots +{k\choose \frac{k-1}{2}}d_1\\ d_1=2 $$ I want to study this sequence ...
A. GM's user avatar
  • 328
4 votes
1 answer
550 views

Proving addition of rational numbers is well defined

I am trying to show that the addition of rational numbers is well defined. Does anyone know if this is a legitimate strategy? Also, I am quite unfamiliar with coding on here. Q is defined as $Q = \{m/...
James Snell's user avatar
4 votes
0 answers
150 views

Algebraic number that exponentiated with algebraic number give $\pi$

I'm not sure if an algebraic number elevated with an algebraic exponent can give rise to a transcendental number. If that's the case does anybody know a closed form for an algebraic number that ...
Dac0's user avatar
  • 9,114
4 votes
0 answers
93 views

Sorting prime numbers on two sets of equals weights

Lets denote $(p_n)$ the sequence of all prime numbers $(p_1=2, p_2=3,\ldots)$. The conjecture is the following. For $n$ odd and greater than $2$, $$\exists I \subset \{1,\ldots ,n\} \quad \...
E. Joseph's user avatar
  • 14.8k
4 votes
2 answers
326 views

Square root of continued fraction

Assuming I've been given a number in the form of a continued fraction. Is there a general way to write the square root of that number as continued question? For example, consider $$1+\sqrt{2} = 2+\...
celtschk's user avatar
  • 43.2k
4 votes
0 answers
657 views

Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
mathema's user avatar
  • 41
4 votes
0 answers
103 views

$d$ and $d+1$ both dividing certain integers

\begin{align} & 1\cdot 72 \\ & 2\cdot 36 \\ & 3\cdot 24 \\ & 4\cdot 18 \\ & 6\cdot 12 \\ & 8\cdot 9 \end{align} When the divisors of a number are listed in this way, let us ...
Michael Hardy's user avatar
4 votes
1 answer
246 views

Find prime numbers $p,q$ such that $p^n+p^{n-1}+...+p+1=q^2+q+1$

Let $p,q$ are prime numbers and $n$ is a even number such that : $p^n+p^{n-1}+...+p+1=q^2+q+1$ Find $p,q$? I think : $p^n+p^{n-1}+...+p+1=q^2+q+1\Rightarrow p^n+p^{n-1}+...+p=q(q+1)\Rightarrow p|...
Lê Tấn Khang's user avatar
4 votes
0 answers
618 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has $...
Gabriel Soranzo's user avatar
3 votes
2 answers
94 views

The number of people in an audthorium is between 80 and 100, knowimg that the number of men is 7/8 the number of women, what is the number of people?

I would like some help solving that problem. My approach: I've shown that x is between 128/3 and 160/3 but that didn't help me and now I'm stuck. Thanks in advance
Pritchard's user avatar
  • 107
3 votes
0 answers
109 views

Solution for fractional operation "N-ation of N by N"

A lot of articles seems to be concerned with "unimaginably" large numbers, leading to a conclusion that we just do not have sufficiently good notation for dealing with such numbers. So ...
alamar's user avatar
  • 131
3 votes
0 answers
73 views

Is there a transfinite version of Post's Theorem?

Let $\emptyset^{(n)}$ denote the $n$th Turing jump of the empty set. Post's theorem states: A set $B$ is $\Sigma^0_{n+1}$ if and only if $B$ is recursively enumerable by an oracle Turing machine with ...
Andreas Tsevas's user avatar
3 votes
0 answers
168 views

Prove the Gödel theorem using the Busy Beaver function

Assume we know the following for a fact: $L$ is an arithmetic language and contains a formula (abbreviated “$BB(x_i, x_j)$”) expressing the Busy Beaver function, i.e. $BB(k, n)$ is true if and only ...
ASA's user avatar
  • 400
3 votes
0 answers
68 views

Is there any way to predict aliquot sequences?

Recently I learned about aliquot sequences. Here's how they work: you start with any number. For an example, I'll use $10$. Then you find all the factors of that number, besides itself. So for $10$, ...
Mathemagician314's user avatar
3 votes
0 answers
168 views

Does the set $\emptyset^\emptyset$ exist?

If $A$ and $B$ are two set then $B^A$ is the set of all function from $A$ to $B$, that is $$ B^A:=\{f\in\mathcal P(A\times B):f\,\,\text{function}\} $$ Now if $A$ and $B$ are finite it is not hard to ...
Antonio Maria Di Mauro's user avatar
3 votes
1 answer
298 views

riddle: two fuses, how to measure 10 seconds

I have this riddle: You have two fuses. Both fuses will burn completely in one minute, but they burn uneven and differently. Can you with the help of these two fuses measure exactly 45 seconds? Can ...
user394334's user avatar
  • 1,230
3 votes
0 answers
46 views

Find the expectation of time that a leaf falls on the ground moving with Brownian motion

I came across a question related to Brownian motion. May someone gives me a hint on this? Suppose a leaf is falling from 1 metre(100 cm) above the ground. It is moving with Brownian motion vertically ...
junyaozheng98's user avatar
3 votes
0 answers
114 views

On Fermat's Last theorem

Fermat's Last theorem states that for $n>2$ there is no solution to the equation $c^n = a^n + b^n$ where $a,b,c$ are positive integers. Now, let $d_n = \inf_{a,b,c \in \mathbb{N}^*} |c^n - a^n - b^...
W.314's user avatar
  • 348
3 votes
0 answers
232 views

What's a simple, natural function that is "halfway" between addition and multiplication?

What's a simple, natural function that is "halfway" between addition and multiplication, in some sense? Can you think of a way to make that precise? $$ x + y \quad\longrightarrow\quad ? \...
Bjørn Kjos-Hanssen's user avatar
3 votes
0 answers
42 views

How to obtain the mean and standard deviation of 3 numbers that form an arithmetic sequence with the sum of 12 ($4$ and $\frac{2\sqrt{3}}{3})$?

The sum of 3 numbers that form an arithmetic sequence is 12. Determine the mean and standard deviation. The answers are $4$ and $\frac{2\sqrt{3}}{3}$ respectively. My attempt: $U_1=a-d$ $U_2=a$ $U_3=...
Fiorella Susanto's user avatar
3 votes
0 answers
69 views

$p=2^n+3$ prime $\implies$ $(p\bmod3)=2\;$ or $(p\bmod5)\in\{2,3,4\}$?

Let $n$ be such that $p=2^n+3$ is prime (see A057732). Does it hold at least one of $\gcd(3,p-1)=1$ or $\gcd(5,p-1)=1$, which is equivalent to the proposition in title? Motivation is construction of a ...
fgrieu's user avatar
  • 1,758
3 votes
0 answers
41 views

Expression for number of unique outcomes algebraic sequences (24-game)

Intro Let there be a game in which participants are presented with a set of $n$ numbers $x = [x_1, x_2, \dots, x_n]$, a set $O$ containing $k$ arithmetic operators, and a target number $T$. The goal ...
Joost's user avatar
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