Book:Joe Roberts/Lure of the Integers

Contents

 * Preface


 * Integers 2, 3
 * 1 Equidecomposability
 * 2 Cubing Cubes
 * 3 Sets Congruent to a Proper Part
 * 4 Random Walks


 * Integer 3
 * 1 Sums
 * 2 Beatty's Theorem
 * 3 Iterates
 * 4 $\map \sigma {p^4}$
 * 5 Chaos -- An Ex-"Property" of $3$"
 * 6 Repeating Blocks
 * 7 Sectioning a Circle
 * 8 $k$-transposable Integers
 * 9 Kuratowski's Theorem


 * Integers 3, 4
 * 1 Lattice Points in Tetrahedra
 * 2 A Problem of Mnich
 * 3 Polynomials Sums of Squares


 * Integer 4
 * 1 Davenport-Baker Theorem
 * 2 Arithmetic Progressions in the Fibonacci Sequence
 * 3 Factorials
 * 4 The Next to Last Case of a Factorial Diophantine Equation
 * 5 Hall's Theorem
 * 6 Residue Classes
 * 7 Diophantine Representation
 * 8 Lattice Cornered Polygons
 * 9 von Neumann Algebras
 * 10 A Few Diophantine Equations
 * 11 Four Fifth Powers
 * 12 Stufe of a Real Quadratic Field


 * Integers 4, 5
 * 1 A Problem of Znám
 * 2 An Awful Problem


 * Integer 5
 * 1 Euler Polynomials
 * 2 Primes a Sum and a Difference of Primes
 * 3 Figurate Numbers
 * 4 Consecutive Primitive Roots
 * 5 Five Moduli
 * 6 A Conjecture Based on Goldbach's Conjecture
 * 7 A Square the Sum of Three Squares
 * 8 Primes as Range of a Polynomial
 * 9 A Recursive Sequence
 * 10 Numbers of the Form $n! + k$
 * 11 Duffinian Numbers
 * 12 Digital Roots
 * 13 The Banach-Tarski Paradox
 * 14 Euclidean Algorithm
 * 15 Sieve Approaches to Twin Primes
 * 16 An Extremal Property of $5$
 * 17 Volumes of $n$-Dimensional Spheres
 * 18 Regular Polyhedra


 * Integer 6
 * 1 Sum of Distinct Primes
 * 2 Values of $\phi$
 * 3 Divisors of $a^n - b^n$
 * 4 Square Permutations
 * 5 Shrinking Labelled Squares
 * 6 Irreducible Polynomials
 * 7 Bilinear Transformations
 * 8 Polyhedra
 * 9 Perfect Numbers
 * 10 Mod $N$ Weak Uniform Distribution
 * 11 Sophie Germain's Theorem
 * 12 A Curious Sequence


 * Integer 7
 * 1 Incomparable Rectangles
 * 2 Minimal Numbers
 * 3 A Square
 * 4 Iterates
 * 5 Influence of Computing
 * 6 Fibonacci Numbers of the Form $k^2 + 1$
 * 7 A Conjecture of Ramanujan


 * Integer 8
 * 1 Square Identities
 * 2 Parallelizability of the Spheres
 * 3 Regular Lattice $n$-Simplices
 * 4 A Multiplicative Function
 * 5 Fibonacci Cubes


 * Integer 9
 * 1 Waring for Cubes
 * 2 Sums of Distinct Odd Primes
 * 3 An Identity
 * 4 Public Key Cryptosystems
 * 5 Partitions of a Set


 * Integer 10
 * 1 $1$, $10$ and Primes
 * 2 Number of Relations
 * 3 A Parlor Problem
 * 4 Quasi-normal Sets of Integers
 * 5 An Exponential Diophantine Equation


 * Integer 11
 * Sum of Distinct Primes


 * Integer 12
 * 1 Prime Factors of Fibonacci Numbers
 * 2 Sums of Squares
 * 3 Values of $\map \sigma n$ and $\map \tau n$
 * 4 On the Prime Factorization of Binomial Coefficients


 * Integer 13
 * 1 A conjecture
 * 2 Simultaneous Quadratic Equations
 * 3 A Peculiar Property of $13$
 * 4 Three Thirteens
 * 5 The "Thirteen Spheres Controversy"


 * Integer 14
 * A Kuratowski Result


 * Integers 15, 54, 88
 * 1 Universal and Almost Universal Forms
 * 2 Totitives
 * 3 Waring for Fourth Powers


 * Integer 16


 * Integer 17
 * 1 Sums of Relatively Prime Numbers
 * 2 A Ramanujan Type Congruence
 * 3 Consecutive Integers
 * 4 The Equation $x^3 - y^2 = -17$
 * 5 A Steinhaus Problem


 * Integer 18
 * 1 Meeting Circles
 * 2 More Totitives
 * 3 Sums of Mixed Powers


 * Integer 19
 * Waring's Problem


 * Integers 20, 21
 * A Matter of Purity


 * Integer 21
 * Squared Squares


 * Integer 22
 * A Special Congruence


 * Integer 23
 * 1 LCM of Binomial Coefficients
 * 2 Pseudoprimes
 * 3 Filling Boxes
 * 4 Hilbert's List
 * 5 Birthday Problems
 * 6 The $\tau$ Function


 * Integer 24
 * 1 Sum of Consecutive Squares a Square
 * 2 A Property of $24$
 * 3 Fibonacci Periodicities
 * 4 Characters
 * 5 Divisors of $24$
 * 6 $\map \sigma {24 n - 1}$
 * 7 Self-dual Lattices in $R^n$
 * 8 Some Formulae Containing $24$
 * 9 A Fermat-like Equation


 * Integer 25
 * A Mystery


 * Integer 27
 * 1 Powers as Sums of Digits
 * 2 Langford Sequence


 * Integer 28
 * A Special Perfect Number


 * Integer 29
 * Consecutive Integers


 * Integer 30
 * 1 Property of $30$
 * 2 A Spurious Property


 * Integer 32
 * A Conjecture of Carmichael


 * Integer 33
 * Complete Sequences


 * Integer 36
 * Sums of Digits


 * Integer 38
 * Sums of Composites


 * Integer 47
 * Cutting a Cube into Cubes


 * Integer 52
 * A Linear Recurrence


 * Integer 53
 * $3$-Solubility


 * Integer 67
 * An Inequality on $\map \pi x$


 * Integer 70
 * A Curious Property'''


 * Integer 77
 * 1 Unit Fractions
 * 2 Sums of Squares


 * Integer 84
 * Automorphisms of a Curve


 * Integer 86
 * A School Girl


 * Integer 87
 * Prime Proof


 * Integer 88
 * Quadratic Forms


 * Integer 90
 * Bi-unitary Perfect Numbers


 * Integer 96
 * Prime Subsequence of Primes


 * Integer 100
 * 1 Fibonacci Mod $100$
 * 2 $100$ Digits of pi
 * 3 Recursive Sets


 * Integer 105
 * 1 Cyclotomic Polynomials
 * 2 A Question of Erdős
 * 3 A Decimal Digit Problem
 * 4 An Assertion of Cseh


 * Integer 113
 * Functions of Prime Numbers


 * Integer 117
 * A Diophantine Equation


 * Integer 128
 * Sum of Unequal Squares


 * Integer 143
 * Clockhands


 * Integer 144
 * Fibonacci Square


 * Integer 163
 * 1 Approximate Integers
 * 2 Unique Factorization Domains
 * 3 Quadratic Prime Giving Polynomials


 * Integer 239
 * 1 Sums of Cubes


 * Integer 283
 * Triplets for Primes


 * Integer 495
 * Kaprekar Constants


 * Integer 561
 * Carmichael Numbers


 * Integer 563
 * Wilson Remainders


 * Integer 645
 * Sum of Consecutive Integers


 * Integer 654
 * Gaps in Primes


 * Integer 691
 * A Divisibility Property


 * Integers 714 and 715
 * $714$ and $715$


 * Integer 720
 * Factorials


 * Integer 1093
 * Fermat's Conjecture


 * Integer 1621
 * $7$, $991$, $1141$, $1621$


 * Integer 1729
 * Hardy's Taxi


 * Integer 1794
 * The Binomial Coefficient $\binom {k^2} k$


 * Integer 1848
 * Numeri Idonei


 * Integer 2719
 * $\mathrm {MR} \ 56 \ \# \ 15557$


 * Integer 3304
 * Smith Numbers


 * Integer 4181
 * Converse Numbers


 * Integer 17163
 * Sums of Squares of Primes


 * Integer 20161
 * Abundant Numbers


 * Integer 63001
 * Ramanujan's $\tau$ Function


 * Integer 720720
 * A Divisibility Problem


 * Integer 99925853


 * Integer 906180359


 * Integer 10513222720
 * A Method of Factoring


 * Integer 608981813029
 * A Chebyshev Observation


 * Integer 2935363331541925531
 * Two Sequences of Composite Integers'''


 * Integer 1123595505617977522809
 * A Multiple of $99$


 * Integer 357686312646216567629137
 * Truncatable Primes


 * Word/Phrase Index


 * Integer Index