Book:David Wells/Curious and Interesting Numbers

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



Historical Note on Doubling the Cube
$1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$:

Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2
$2$:

Tamref's Last Theorem
$2$:

Decimal Expansion of $\pi$
$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

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

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

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

Tamura-Kanada Circuit Method: Example
$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

Pythagorean Triangle with Sides in Arithmetic Sequence
$5$:

Sam Loyd's Missing Square
$5$:

Fibonacci Number as Sum of Binomial Coefficients
$5$:

No 4 Fibonacci Numbers can be in Arithmetic Sequence
$5$:

Perfect Number is Sum of Successive Odd Cubes except 6
$6$:

Set of $3$ Integers each Divisor of Sum of Other Two
$6$:

Only Number which is Sum of $3$ Factors is $6$
$6$:

Historical Note on the St. Ives Problem
$7$:

Definition of Deltahedron
$8$:

Product of Two Triangular Numbers to make Square
$15$:

Triangular Number Pairs with Triangular Sum and Difference
$15$:

Palindromic Triangular Numbers: $1$
$15$:

Stronger Feit-Thompson Conjecture
$17$:

Magic Hexagon
$19$:

Sum of Sequence of Alternating Positive and Negative Factorials being Prime
$19$:

Semiperfect Number
$20$:

Squares Ending in 5 Occurrences of 2-Digit Pattern
$21$:

$23$ is Largest Integer not Sum of Distinct Powers
$23$:

Apothecaries' Ounce
$24$:

24 is Smallest Composite Number the Product of whose Proper Divisors is Cube
$24$:

Sociable Chain: $12,496$
$28$:

Historical Note on Definition of Perfect Number
$28$:

Sequence of Prime Primorial minus 1
$29$:

Schatunowsky's Theorem
$30$:

Pascal's Rule
$35$:

Hilbert-Waring Theorem/Particular Cases/5
$37$:

46/Historical Note
$46$:

Prime between n and 9 n divided by 8
$48$:

Definition:Highly Composite Number
$60$:

Kaprekar's Process for 2-Digit Numbers
$63$:

Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
$64$:

Prime Numbers which Divide Sum of All Lesser Primes
$71$:

$4$ Positive Integers in Arithmetic Sequence which have Same Euler Phi Value
$72$:

Reciprocal of 89
$89$:

Integers such that Difference with Power of 2 is always Prime
$105$:

Reciprocals of Odd Numbers adding to 1
$105$:

Triperfect Number
$120$:

Multiply Perfect Number of Order 8
$120$:

Carmichael's Theorem
$144$:

Smallest Prime Magic Square with Consecutive Primes from $3$
$144$:

Sum of 2 Squares in 2 Distinct Ways: $145$
$145$:

3-Digit Numbers forming Longest Reverse-and-Add Sequence
$187$:

Multiplicative Magic Square/Examples/Order 3/Smallest/Historical Note
$216$:

Plato's Geometrical Number
$216$:

Fermat Pseudoprime to Base 4
$217$:

Prime Decomposition of 7th Fermat Number
$257$:

Product of Sequence of Fermat Numbers plus 2
$257$:

297
$297$:

492 is Sum of 3 Cubes in 3 Ways
$492$:

Products of 2-Digit Pairs which Reversed reveal Same Product
$504$:

Prime Decomposition of 5th Fermat Number
$641$:

Consecutive Integers whose Product is Primorial
$714$:

Period of Reciprocal of $729$ is $81$
$729$:

Triangular Number Pairs with Triangular Sum and Difference: $T_{39}$ and $T_{44}$
$780$:

Multiple of 999 can be Split into Groups of 3 Digits which Add to 999
$999$:

Integer both Square and Triangular
$1225$:

Squares whose Digits can be Separated into 2 other Squares
$1444$:

Gregorian Calendar
$3333$:

Product with Repdigit can be Split into Parts which Add to Repdigit
$6666$:

6667
$6667$:

Mersenne Number whose Index is Mersenne Prime
$8191$:

9801
$9801$:

Smallest Pandigital Square
$11,826$:

Smallest Fourth Power as Sum of 5 Distinct Fourth Powers
$50,625$:

Kaprekar's Process on 5 Digit Number
$99,954$:

Recurring Part of Fraction times Period gives 9-Repdigit
$142,857$:

Reciprocal of $142 \, 857$
$142,857$:

Integer whose Digits when Grouped in $3$s add to Multiple of $999$ is Divisible by $999$
$142,857$:

$147 \, 852$
$147,852$:

Properties of Family of 333,667 and Related Numbers
$333,667$:

Palindromic Triangular Numbers: $2$
$828,828$:

Triangular Number Pairs with Triangular Sum and Difference: $T_{1869}$ and $T_{2090}$
$1,747,515$:

Factorial as Product of Consecutive Factorials
$3,628,800$:

Archimedes' Cattle Problem
$4,729,494$:

Hardy-Ramanujan Number: $87 \, 539 \, 319$
$87,539,319$:

Pandigital Integers remaining Pandigital on Multiplication
$123,456,789$:

Right-Truncatable Prime
$739,391,133$:

$555,555,555,555,556$
$555,555,555,555,556$:

Square of Small Repunit is Palindromic
$1,111,111,111,111,111,111$:

Probability of All Players receiving Complete Suit at Bridge
$2,235,197,406,895,366,368,301,560,000$:

General Fibonacci Sequence whose Terms are all Composite
$1,786,772,701,928,802,632,268,715,130,455,793$:

$180 \times \paren {2^{127} - 1} + 1$ is Prime
$180 \times \paren {2^{127} - 1} + 1$:

Upper Bound for Number of Grains of Sand to fill Universe
$10^{51}$:

Mersenne Prime $M_{521}$
$2^{521} - 1$:

Ackermann Function
$2^{65536}$:

Mersenne Prime $M_{86 \, 243}$
$2^{86243} - 1$:

Speed of Light
$2^{86243} - 1$:

Horace Scudder Uhler
$9^{9^9}$:

Number of Primes up to $n$ Approximates to Eulerian Logarithmic Integral
$10^{10^{10^{34}}}$: