Book:Joe Roberts/Lure of the Integers

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Joe Roberts: Lure of the Integers

Published $\text {1992}$, MAA Spectrum

ISBN 978-0883855027


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