Unsolved Problems in Number Theory

Preface to the First Edition Preface to the Second Edition Preface to the Third Edition Glossary of Symbols A. Prime Numbers. A1. Prime values of quadratic functions. A2. Primes connected with factorials. A3. Mersenne primes. Repunits. Fermat numbers. Primes of shape k * 2n + 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 graph. A15. Congruent products of consecutive numbers. A16. Gaussian primes. Eisenstein-Jacobi primes. A17. Formulas for primes. A18. The Erd1/2os-Selfridge classi.cation of primes. A19. Values of n making n - 2k prime. Odd numbers not of the form +-pa +- 2b. 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. B11. Solutions of mo(m) = no(n). B12. Analogs with d(n), ok(n). B13. Solutions of o(n) = o(n + 1). B14. Some irrational series. B15. Solutions of o(q) + o(r) = o(q + r). B16. Powerful numbers. Squarefree numbers. B17. Exponential-perfect numbers B18. Solutions of d(n) = d(n + 1). B19. (m, n + 1) and (m+1, n) with same set of prime factors. The abc-conjecture. B20. Cullen and Woodall numbers. B21. k * 2n + 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 U 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. B31. Binomial coeffcients. B32. Grimm's conjecture. B33. Largest divisor of a binomial coeffcient. B34. If there's an i such that n - i divides _nk_. B35. Products of consecutive numbers with the same prime factors. B36. Euler's totient function. B37. Does o(n) properly divide n - 1? B38. Solutions of o(m) = o(n). B39. Carmichael's conjecture. B40. Gaps between totatives. B41. Iterations of o and o. B42. Behavior of o(o(n)) and o(o(n)). B43. Alternating sums of factorials. B44. Sums of factorials. B45. Euler numbers. B46. The largest prime factor of n. B47. When does 2a -2b divide na - nb? B48. Products taken over primes. B49. Smith numbers. C. Additive Number Theory C1. 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. C10. Modular di.erence sets and error correcting codes. C11. Three-subsets with distinct sums. C12. The postage stamp problem. C13. The corresponding modular covering problem. Harmonious labelling of graphs. C14. Maximal sum-free sets. C15. Maximal zero-sum-free sets. C16. Nonaveraging sets. Nondividing sets. C17. The minimum overlap problem. C18. The n queens problem. C19. Is a weakly indedendent sequence the .nite union of strongly independent ones? C20. Sums of squares. C21. Sums of higher powers. D. Diophantine Equations D1. 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. Di.erence of two powers. D10. Exponential diophantine equations. D11. Egyptian fractions. D12. Marko. numbers. D13. The equation xxyy = zz. D14. ai + bj made squares. D15. Numbers whose sums in pairs make squares. D16. Triples with the same sum and same product. D17. Product of blocks of consecutive integers not a power. D18. Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square. D19. Rational distances from the corners of a square. D20. Six general points at rational distances. D21. 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 E1. 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. Szemer'edi's theorem. Partitioning the integers into classes; at least one contains an A.P. E11. Schur's problem. Partitioning integers into sum-free classes. E12. The modular version of Schur's problem. E13. Partitioning into strongly sum-free classes. E14. Rado's generalizations of van der Waerden's and Schur's problems. E15. A recursion of Gobel. E16. The 3x + 1 problem. E17. Permutation sequences. E18. Mahler's Z-numbers. E19. Are the integer parts of the powers of a fraction in.nitely often prime? E20. Davenport-Schinzel sequences. E21. 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. E31. 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 F1. 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 di.erences are quadratic residues. F9. Primitive roots F10. Residues of powers of two. F11. Distribution of residues of factorials. F12. How often are a number and its inverse of opposite parity? F13. Covering systems of congruences. F14. Exact covering systems. F15. A problem of R. L. Graham. F16. Products of small prime powers dividing n. F17. Series associated with the a-function. F18. Size of the set of sums and products of a set. F19. Partitions into distinct primes with maximum product. F20. Continued fractions. F21. 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 all distinct. F31. Miscellaneous digital problems. F32. Conway's RATS and palindromes. Index of Authors Cited General Index Glossary of Symbols