# 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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17 votes
2 answers
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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 (...
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 $1$...
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8 votes
0 answers
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7 votes
0 answers
136 views

### 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. ...
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7 votes
0 answers
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 $q$...
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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 ...
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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, ...
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 ...
• 18.1k
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 ...
• 2,405
6 votes
0 answers
10k views

### 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? ...
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6 votes
1 answer
305 views

### 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 $c$ &...
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6 votes
0 answers
166 views

3 votes
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### $p=2^n+3$ 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 ...
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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 ...
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