Unsolved Problems in Number Theory

Title page
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Glossary of Symbols
Introduction
A. Prime Numbers
Al. Prime values of quadratic functions
A2. Primes connected with factorials
A3. Mersenne primes. Repunits. Fermat numbers. Primes of shape k.2^n+1
A4. The prime number race
A5. Arithmetic progressions of primes
A6. Consecutive primes in A.P
A7. Cunningham chains
A8. Gaps between primes. Twin primes
A9. Patterns of primes
A10. Gilbreath's conjecture
A11. Increasing and decreasing gaps
A12. Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes
A13. Carmichael numbers
A14. "Good" primes and the prime number graphe
A15. Congruent products of consecutive numbers
A16. Gaussian and Eisenstein-Jacobi primes
A17. Formulas for primes
A18. The Erdös-Selfridge classification of primes
A19. Values of n making n-2^k prime. Odd numbers not of the form +-p^a+-2^b
A20. Symmetric and asymmetric primes
B. Divisibility
B1. Perfect numbers
B2. Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers
B3. Unitary perfect numbers
B4. Amicable numbers
B5. Quasi-amicable or betrothed numbers
B6. Aliquot sequences
B7. Aliquot cycles. Sociable numbers
B8. Unitary aliquot sequences
B9. Superperfect numbers
B10. Untouchable numbers
Bll. Solutions of mσ(m) = nσ(n)
B12. Analogs with d(n), σ_k(n)
Bl3. Solutions of σ(n) = σ(n + 1)
Bl4. Some irrational series
Bl5. Solutions of σ(q)+σ(r) = σ(q+r)
Bl6. Powerful numbers. Squarefree numbers
Bl7. Exponential-perfect numbers
Bl8. Solutions of d(n) = d(n + 1)
Bl9. (m,n+1) and (m+1,n) with same set of prime factors. The abc-conjecture
B20. Cullen and Woodall numbers
B2l. k.2^n+1 composite for all n
B22. Factorial n as the product of n large factors
B23. Equal products of factorials
B24. The largest set with no member dividing two others
B25. Equal sums of geometric progressions with prime ratios
B26. Densest set with no l pairwise coprime
B27. The number of prime factors of n+k which don't divide n+i, 0< i< k
B28. Consecutive numbers with distinct prime factors
B29. Is x determined by the prime divisors of x+1,x+2,...,x+k?
B30. A small set whose product is square
B3l. Binomial coefficients
B32. Grimm's conjecture
B33. Largest divisor of a binomial coefficient
B34. If there's an i such that n-i divides (^n_k)
B35. Products of consecutive numbers with the same prime factors
B36. Euler's totient function
B37. Does φ(n) properly divide n-1?
B38. Solutions of φ(m) = σ(n)
B39. Carmichael's conjecture
B40. Gaps between totatives
B4l. Iterations of φ and σ
B42. Behavior of φ(σ(n)) and σ(φ(n))
B43. Alternating sums of factorials
B44. Sums of factorials
B45. Euler numbers
B46. The largest prime factor of n
B47. When does 2^a-2^b divide n^a-n^b?
B48. Products taken over primes
B49. Smith numbers
B50. Ruth-Aaron numbers
C. Additive Number Theory
Cl. Goldbach's conjecture
C2. Sums of consecutive primes
C3. Lucky numbers
C4. Ulam numbers
C5. Sums determining members of a set
C6. Addition chains. Brauer chains. Hansen chains
C7. The money-changing problem
C8. Sets with distinct sums of subsets
C9. Packing sums of pairs
Cl0. Modular difference sets and error correcting codes
Cll. Three-subsets with distinct sums
Cl2. The postage stamp problem
Cl3. The corresponding modular covering problem. Harmonious labelling of graphs
Cl4. Maximal sum-free sets
Cl5. Maximal zero-sum-free sets
Cl6. Nonaveraging sets. Nondividing sets
Cl7. The minimum overlap problem
Cl8. The n queens problem
Cl9. Is a weakly indedendent sequence the finite union of strongly independent ones?
C20. Sums of squares
C2l. Sums of higher powers
D. Diophantine Equations
Dl. Sums of like powers. Euler's conjecture
D2. The Fermat problem
D3. Figurate numbers
D4. Waring's problem. Sums of l kth Powers
D5. Sum of four cubes
D6. An elementary solution of x² = 2y⁴-1
D7. Sum of consecutive powers made a power
D8. A pyramidal diophantine equation
D9. Catalan conjecture. Difference of two powers
D10. Exponential diophantine equations
Dll. Egyptian fractions
Dl2. Markoff numbers
Dl3. The equation x^x y^y = z^z
Dl4. a_i + b_j made squares
Dl5. Numbers whose sums in pairs make squares
Dl6. Triples with the same sum and same product
Dl7. Product of blocks of consecutive integers not a power
Dl8. Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square
Dl9. Rational distances from the corners of a square
D20. Six general points at rational distances
D2l. Triangles with integer edges, medians and area
D22. Simplexes with rational contents
D23. Some quartic equations
D24. Sum equals product
D25. Equations involving factorial n
D26. Fibonacci numbers of various shapes
D27. Congruent numbers
D28. A reciprocal diophantine equation
D29. Diophantine m-tuples
E. Sequences of Integers
El. A thin sequence with all numbers equal to a member plus a prime
E2. Density of a sequence with l.c.m. of each pair less than x
E3. Density of integers with two comparable divisors
E4. Sequence with no member dividing the product of r others
E5. Sequence with members divisible by at least one of a given set
E6. Sequence with sums of pairs not members of a given sequence
E7. A series and a sequence involving primes
E8. Sequence with no sum of a pair a square
E9. Partitioning the integers into classes with numerous sums of pairs
E10. Theorem of van der Waerden. Szemerédi's theorem. Partitioning the integers into classes; at least one contains an A.P
Ell. Schur's problem. Partitioning integers into sum-free classes
El2. The modular version of Schur's problem
El3. Partitioning into strongly sum-free classes
El4. Rado's generalizations of van der Waerden's and Schur's problems
El5. A recursion of Gobel
El6. The 3x+1 problem
El7. Permutation sequences
E18. Mahler's Z-numbers
El9. Are the integer parts of the powers of a fraction infinitely often prime?
E20. Davenport-Schinzel sequences
E2l. Thue-Morse sequences
E22. Cycles and sequences containing all permutations as subsequences
E23. Covering the integers with A.P.s
E24. Irrationality sequences
E25. Golomb's self-histogramming sequence
E26. Epstein's Put-or- Take-a-Square game
E27. Max and mex sequences
E28. B₂-sequences. Mian-Chowla sequences
E29. Sequence with sums and products all in one of two classes
E30. MacMahon's prime numbers of measurement
E3l. Three sequences of Hofstadter
E32. B₂-sequences from the greedy algorithm
E33. Sequences containing no monotone A.P.s
E34. Happy numbers
E35. The Kimberling shuffle
E36. Klarner-Rado sequences
E37. Mousetrap
E38. Odd sequences
F. None of the Above
Fl. Gauss's lattice point problem
F2. Lattice points with distinct distances
F3. Lattice points, no four on a circle
F4. The no-three-in-line problem
F5. Quadratic residues. Schur's conjecture
F6. Patterns of quadratic residues
F7. A cubic analog of a Bhaskara equation
F8. Quadratic residues whose differences are quadratic residues
F9. Primitive roots
F10. Residues of powers of two
Fll. Distribution of residues of factorials
Fl2. How often are a number and its inverse of opposite parity?
Fl3. Covering systems of congruences
Fl4. Exact covering systems
Fl5. A problem of R.L. Graham
Fl6. Products of small prime powers dividing n
Fl7. Series associated with the ζ-function
Fl8. Size of the set of sums and products of a set
Fl9. Partitions into distinct primes with maximum product
F20. Continued fractions
F2l. All partial quotients one or two
F22. Algebraic numbers with unbounded partial quotients
F23. Small differences between powers of 2 and 3
F24. Some decimal digital problems
F25. The persistence of a number
F26. Expressing numbers using just ones
F27. Mahler's generalization of Farey series
F28. A determinant of value one
F29. Two congruences, one of which is always solvable
F30. A polynomial whose sums of pairs of values are aIl distinct
F3l. Miscellaneous digital problems
F32. Conway's RATS and palindromes
Index of Authors Cited
General Index