Unsolved Problems in Number Theory

Description

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