Book:David Wells/Curious and Interesting Numbers/Second Edition

Contents

 * Introduction
 * Acknowledgements
 * 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
$1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$:

Brun's Constant
$1 \cdotp 90216 \, 054 \ldots$:

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

Tamref's Last Theorem
$2$:

Continued Square Root of 1, 2, 3, 4, ...
$3$:

Notation for Pi
$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$:

Pi: Modern Developments
$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$:

Pythagorean Triangle with Sides in Arithmetic Sequence
$5$:

Sam Loyd's Missing Square
$5$:

Fibonacci Number as Sum of Binomial Coefficients
$5$:

Corollary to Euler-Binet Formula
$5$:

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

Number of Fibonacci Numbers with Same Number of Decimal Digits
$5$:

Sequence of Fibonacci Numbers ending in Index
$5$:

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 $6$
$6$:

Divisibility of Elements of Pythagorean Triple by 7
$7$:

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

Definition of Deltahedron
$8$:

Relation between Square of Fibonacci Number and Square of Lucas Number
$11$:

Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes
$13$:

Solutions of Ramanujan-Nagell Equation
$15$:

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

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

Integers with Prime Values of Divisor Sum
$16$:

Smallest Odd Number not of form $2 a^2 + p$
$17$:

Stronger Feit-Thompson Conjecture
$17$:

Only Number Twice Sum of Digits is 18
$18$:

Prime to Own Power minus $1$ over Prime minus $1$ being Prime
$19$:

Semiperfect Number
$20$:

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

Smallest Integer not Sum of Two Ulam Numbers
$23$:

Numbers with Square-Free Binomial Coefficients
$23$:

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: Mistake 1
$28$:

Historical Note on Definition of Perfect Number: Mistake 2
$28$:

Sequence of Prime Primorial minus 1
$29$:

Schatunowsky's Theorem
$30$:

Smallest Positive Integer not of form $\pm 4 \pmod 9$ not representable as Sum of Three Cubes
$30$:

Giuga Number
$30$:

Smallest Set of Weights for Two-Pan Balance
$31$:

Integer as Sum of 5 Non-Zero Squares
$33$:

Triplets of Products of Two Distinct Primes
$33$:

Prime Factors of 35, 36, 4734 and 4735
$35$:

Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards
$35$:

Square Numbers which are Divisor Sum values
$36$:

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

Euler Lucky Number/Examples/41
$41$:

Non-Palindromes in Base 2 by Reverse-and-Add Process
$43$:

Subfactorial/Examples/5
$44$:

Definition:Kaprekar Triple/Sequence
$45$:

$46$: Historical Note
$46$:

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

Hilbert-Waring Theorem/Particular Cases/4
$53$:

Primes not Sum of or Difference between Powers of 2 and 3
$53$:

Even Integer with Abundancy Index greater than 9
$55$:

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
$72$:

Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value
$100$:

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

Odd Integers whose Smaller Odd Coprimes are Prime
$105$:

Integers whose Divisor Sum equals Half Phi times Divisor Count
$105$:

Reciprocals of Odd Numbers adding to 1
$105$:

Divisor Counting Function of $108$
$108$:

Integers whose Divisor Sum is Cube
$110$:

Difference between Two Squares equal to Repunit
$111$:

Smallest Number which is Sum of 4 Triples with Equal Products
$118$:

Triperfect Number
$120$:

Multiply Perfect Number of Order 8
$120$:

Numbers whose Difference equals Difference between Cube and Seventh Power
$125$:

Triangles with Integer Area and Integer Sides in Arithmetical Sequence
$126$:

Sequence of Quasiamicable Pairs
$140$:

Carmichael's Theorem
$144$:

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

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

Sequence of Square Centered Hexagonal Numbers
$169$:

169 as Sum of up to 155 Squares
$169$:

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

Numbers such that Divisor Count divides Phi divides Divisor Sum
$210$:

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

Plato's Geometrical Number
$216$:

Amicable Pairs with Common Factor 3
$220$:

Amicable Pair with Smallest Common Prime Factor 5
$220$:

Solution of Ljunggren Equation
$239$:

Solutions of Diophantine Equation $x^4 + y^4 = z^2 + 1$ for $x = 239$
$239$:

Prime Decomposition of 7th Fermat Number
$257$:

Pépin's Test
$257$:

Consecutive Powerful Numbers
$288$:

297
$297$:

1,111,111,111
$297$:

Fourth Powers which are Sum of 4 Fourth Powers
$353$:

Lucas-Carmichael Number
$399$:

Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers
$466$:

Solutions to $p^2$ Divides $10^p - 10$
$487$:

Kaprekar's Process on $3$ Digit Number ends in $495$
$495$:

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

Prime Decomposition of 5th Fermat Number
$641$:

Consecutive Integers whose Product is Primorial
$714$:

$720$ is Product of Consecutive Numbers in Two Ways
$720$:

Solutions to Approximate Fermat Equation $x^3 = y^3 + z^3 \pm 1$
$729$:

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

Sum of 4 Consecutive Binomial Coefficients forming Square
$767$:

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
$999$:

Numbers Not Expressible as Sum of no more than 5 Composite Numbers
$1167$:

Integer both Square and Triangular
$1225$:

Square Numbers which are Sum of Sequence of Odd Cubes
$1225$:

1477
$1477$:

$17$ Consecutive Integers each with Common Factor with Product of other $16$
$2185$:

$2601$ as Sum of $3$ Squares in $12$ Different Ways
$2601$:

3000
$3000$:

Perfect Digit-to-Digit Invariant: $3435$
$3435$:

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

6667
$6667$:

8712
$8712$:

Number of Different Ways to play First n Moves in Chess
$8902$:

9801
$9801$:

Smallest Pandigital Square
$11,826$:

Smallest Integer which is Product of 4 Triples all with Same Sum
$25,200$:

Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7
$29,351$:

Pentagonal and Hexagonal Numbers
$40,755$:

Carmichael Number with 4 Prime Factors
$41,041$:

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

Ackermann Function: $1$
$65,536$:

Property of $74,162$
$74,162$:

Sets of 4 Prime Quadruples
$99,131$:

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

Numbers whose Cube equals Sum of Sequence of that many Squares
$103,823$:

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 $999$
$142,857$:

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

$517 \, 842$
$147,852$:

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

Definition:Rare Number
$621,770$:

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
$4,729,494$:

Sequence of Triplets of Primitive Pythagorean Triangles with Same Area
$13,123,110$:

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

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

Triangular Numbers which are Product of 3 Consecutive Integers
$258,474,216$:

Polydivisible Number: $381 \, 654 \, 729$
$381,654,729$:

Palindromic Smith Number/Examples/123,455,554,321
$12,345,554,321$:

General Fibonacci Sequence whose Terms are all Composite: 1
$62,638,280,004,239,857$:

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

Sequence of 9 Consecutive Integers each with 48 Divisors
$17,796,126,877,482,329,126,044$:

General Fibonacci Sequence whose Terms are all Composite: 2
$1,786,772,701,928,802,632,268,715,130,455,793$:

Integer which is Multiplied by 9 when moving Last Digit to First
$10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$:

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

Definition:Fischer-Griess Monster: Historical Note
$808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$:

=== Amicable Pair: Examples: $59 \, 554 \, 936 \ldots \, 048 \, 448 - 59 \, 554 \, 936 \ldots 105 \, 472$ ===

$2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$:

=== Factorisation of $\paren {11^{104} + 1} / \paren {11^8 + 1}$ ===

$\paren {11^{104} + 1} / \paren {11^8 + 1}$:

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

Repunit Prime $R_{317}$
$11,111,111, \ldots 111,111$:

Titanic Sophie Germain Prime: 1
$39,051 \times 2^{6001} - 1$:

Titanic Sophie Germain Prime: 2
$39,051 \times 2^{6001} - 1$:

Sequence of Integers whose Factorial plus 1 is Prime
$1477! + 1$:

Primorial $15 \, 877$
$15,877 \# - 1$:

Ackermann Function: $2$
$2^{65,536}$:

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

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

Gigaplex
$1^{\mathrm{billion} }$:

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