Book:David Wells/Curious and Interesting Numbers

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David Wells: The Penguin Dictionary of Curious and Interesting Numbers

Published $1986$.


Contents

Introduction
A List of Mathematicians in Chronological Sequence
Glossary
Bibliography
The Dictionary
Tables
The First $100$ Triangular Numbers, Squares and Cubes
The First $20$ Pentagonal, Hexagonal, Heptagonal and Octagonal Numbers
The First $40$ Fibonacci Numbers
The Prime Numbers less than $1000$
The Factorials of the Numbers $1$ to $20$
The Decimal Reciprocals of the Primes from $7$ to $97$
The Factors of the Repunits from $11$ to $R_{40}$
The Factors, where Composite, and the Values of the Functions $\map \phi n$, $\map d n$ and $\map \sigma n$
Index


Next


Further Editions


Errata

Positive Integer is Divisible by Sum of Consecutive Integers iff not Power of 2

$2$:

An integer is the sum of a sequence of consecutive integers if and only if it is not a power of $2$.


Decimal Expansion of $\pi$

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$


Notation for Pi

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

Euler, who first used the Greek letter $\pi$ in its modern sense, ...


Leonhard Paul Euler

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

Euler ... gave an even more impressive demonstration of the power of these new methods by calculating $\pi$ to $20$ decimal places in just one hour.


Pi: Modern Developments

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

... in $1983$ a Japanese team of Yoshiaki Tamura and Tasumasa Kanada produced $16,777,216$ ($= 2^{24}$) places.


Tamura-Kanada Circuit Method: Example

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

Here are the values of $\pi$ after going round just $3$ times on a pocket calculator. It is already correct to $5$ decimal places!
\((1):\quad\) \(\displaystyle \) \(\) \(\displaystyle 2 \cdotp 91421 \, 35\) $\quad$ $\quad$
\((2):\quad\) \(\displaystyle \) \(\) \(\displaystyle 3 \cdotp 14057 \, 97\) $\quad$ $\quad$
\((3):\quad\) \(\displaystyle \) \(\) \(\displaystyle 3 \cdotp 14159 \, 28\) $\quad$ $\quad$


Pythagorean Triangle with Sides in Arithmetic Progression

$5$:

The $3-4-5$ triangle is the only Pythagorean triangle whose sides are in arithmetical progression.


No 4 Fibonacci Numbers can be in Arithmetic Progression

$5$:

(Incidentally, no four terms of the Fibonacci sequence can be in arithmetic progression.)


Perfect Number is Sum of Successive Odd Cubes except 6

$6$:

[ $6$ ] is the only perfect number that is not the sum of successive cubes.


Historical Note on the St. Ives Problem

$7$:

Pierce comments that it seems to be of the same origin as the House that Jack Built, and that Leonardo uses the same numbers as Ahmes and makes his calculations in the same way.


Definition of Deltahedron

$8$:

A deltahedron is a polyhedron each of whose faces are triangular.


Product of Two Triangular Numbers to make Square

$15$:

For every triangular number, $T_n$, there are an infinite number of other triangular numbers, $T_m$, such that $T_n T_m$ is a square. For example, $T_3 \times T_{24} = 30^2$.


Triangular Number Pairs with Triangular Sum and Difference

$15$:

$15$ and $21$ are the smallest pair of triangular numbers whose sum and difference ($6$ and $36$) are also triangular. The next such pairs are $780$ and $990$, and $1,747,515$ and $2,185,095$.


Palindromic Triangular Numbers

$15$:

There are $40$ palindromic triangular numbers below $10^7$.


Stronger Feit-Thompson Conjecture

$17$:

The only known prime values for which $p^q - 1$ and $q^p - 1$ have a common factor less than $400,000$ are $17$ and $3313$. The common factor is $112,643$.


Magic Hexagon

$19$:

There is only one way in which consecutve integers can be fitted into a magic hexagonal array ... The numbers $1$ to $19$ can be so arranged, a fact first discovered by T. Vickers.


Semiperfect Number

$20$:

$20$ is the $2$nd semi-perfect number or pseudonymously pseudoperfect number, because it is the sum of some of its own factors: $20 = 10 + 5 + 4 + 1$.


Squares Ending in 5 Occurrences of 2-Digit Pattern

$21$:

If a square ends in the pattern $xyxyxyxyxy$, then $xy$ is either $21$, $61$ or $84$.