None of the Above

  • Richard K. Guy
Part of the Unsolved Problems in Intuitive Mathematics book series (PBM, volume 1)


The first few problems in this miscellaneous section are about lattice points, whose Euclidean coordinates are integers. Most of them are two-dimensional problems, but some can be formulated in higher dimensions as well.


Lattice Point Unsolved Problem Primitive Root Decimal Digit Quadratic Residue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. J. W. S. Cassels, Introduction to the Geometry of Numbers, Springer-Verlag, N.Y. 1972. L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und in Raum, Springer-Verlag, Berlin, 1953.Google Scholar
  2. J. Hammer, Unsolved Problems Concerning Lattice Points, Pitman, 1977.Google Scholar
  3. O. H. Keller, Geometrie der Zahlen, Enzyclopedia der Math. Wissenschaften, Vol. 12, B. G. Teubner, Leipzig, 1954.Google Scholar
  4. C. G. Lekkerkerker, Geometry of Numbers, Bibliotheca Mathematica, Vol. 8, Walters-Noordhoff, Groningen; North-Holland, Amsterdam, 1969.Google Scholar
  5. C. A. Rogers, Packing and Covering, Cambridge Univ. Press, 1964.Google Scholar
  6. Acland-Hood, Bull. Malayan Math. Soc. 0 (1952–53) E 11–12.Google Scholar
  7. Michael A. Adena, Derek A. Holton and Patrick A. Kelly, Some thoughts on the nothree-in-line problem, Proc. 2nd. Austral. Conf. Combin. Math., Springer Lecture Notes 403 (1974) 6–17; MR 50 #1890.Google Scholar
  8. David Brent Anderson, Update on the no-three-in-line problem, J. Combin. Theory Ser A 27 (1979) 365–366.MathSciNetzbMATHCrossRefGoogle Scholar
  9. W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 12th ed. Univ. of Toronto Press, 1974, p. 189.Google Scholar
  10. D. Craggs and R. Hughes-Jones, On the no-three-in-line problem, J. Combin. Theory Ser. A 20 (1976) 363–364; MR 53 #10590.Google Scholar
  11. H. E. Dudeney, The Tribune, 1906: 11: 07.Google Scholar
  12. H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, 94, 222.Google Scholar
  13. Martin Gardner, Mathematical Games: Challenging chess tasks for puzzle buffs and answers to the recreational puzzles, Sei. Amer. 226 #5 (May 1972) 112–117, esp. pp. 113–114.Google Scholar
  14. Martin Gardner, Mathematical Games: Combinatorial problems, some old, some new and all newly attacked by computer, Sci. Amer. 235 #4 (Oct 1976) 131–137, esp. pp. 133–134 also 236 #3 (Mar 1977) 139–140.Google Scholar
  15. Michael Goldberg, Maximizing the smallest triangle made by N points in a square, Math. Mag. 45 (1972) 135–144.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Richard K. Guy, Bull. Malayan Math. Soc. 0 (1952–53) E 22.Google Scholar
  17. Richard K. Guy, Unsolved combinatorial problems, in D. J. A. Welsh, Combinatorial Mathematics and Its Applications, Academic Press, London, 1971, p. 124.Google Scholar
  18. Richard K. Guy and Patrick A. Kelly, The no-three-in-line problem, Canad. Math. Bull. 11 (1968) 527–531.MathSciNetzbMATHCrossRefGoogle Scholar
  19. R. R. Hall, T. H. Jackson, A. Sudbery, and K. Wild, Some advances in the no-threein-line problem, J. Combin. Theory Ser. A 18 (1975) 336–341.MathSciNetzbMATHCrossRefGoogle Scholar
  20. P. A. Kelly, The use of the computer in game theory, M. Sc. thesis, Univ. of Calgary, 1967.Google Scholar
  21. Torleiv Klove, On the no-three-in-line problem II, J. Combin. Theory Ser. A 24 (1978) 126–127; MR 57 #2962; Zbl. 393. 05004.Google Scholar
  22. Torleiv Klove, On the no-three-in-line problem III, J. Combin. Theory Ser. A 26 (1979) 82–83; Zbl. 393. 05005.Google Scholar
  23. Carl Pomerance, Collinear subsets of lattice point sequences an analog of Szemerédi’s theorem, J. Combin. Theory Ser. A 28 (1980) 140–149.MathSciNetzbMATHCrossRefGoogle Scholar
  24. K. F. Roth, On a problem of Heilbronn, J. London Math. Soc. 25 (1951) 198–204, esp. Appendix p. 204; II, III Proc. London Math. Soc. 25 (1972)193–212, 543–549.Google Scholar
  25. K. F. Roth, Developments in Heilbronn’s triangle problem, Advances in Math. 22 (1976) 364–385; MR 55 #2771.Google Scholar
  26. Wolfgang M. Schmidt, On a problem of Heilbronn, J. London Math. Soc. 4 (1971/72) 545–550.Google Scholar
  27. A. Brauer, Über die Verteilung der Potenzreste, Math. Z. 35 (1932) 39–50; Zbl. 3, 339.Google Scholar
  28. H. Davenport, The Higher Arithmetic,Hutchinson’s Univ. Library, 1952, 74–78. Richard H. Hudson, On sequences of quadratic nonresidues, J. Number Theory 3 (1971) 178–181; MR 43 #150.Google Scholar
  29. Richard H. Hudson, On a conjecture of Issai Schur, J. reine angew. Math. 289 (1977) 215–220; MR 58 #16481.Google Scholar
  30. D. H. Lehmer and Emma Lehmer, On runs of residues, Proc. Amer. Math. Soc. 13 (1962) 102–106; MR 25 #2035.Google Scholar
  31. D. H. Lehmer, Emma Lehmer and W. H. Mills, Pairs of consecutive power residues, Canad. J. Math. 15 (1963) 172–177; MR 26 #3660.Google Scholar
  32. P.-A. Barrucand and Harvey Cohn, A rational genus, class number divisibility and unit theory for pure cubic fields, J. Number Theory 2 (1970) 7–21.MathSciNetzbMATHCrossRefGoogle Scholar
  33. H. C. Williams, Improving the speed of calculating the regulator of certain pure cubic fields, Math. Comput. 35 (1980) 1423–1434.zbMATHCrossRefGoogle Scholar
  34. S. L. G. Choi, Covering the set of integers by congruence classes of distinct moduli, Math. Comput. 25 (1971) 885–895; MR 45 #6744.Google Scholar
  35. R. F. Churchhouse, Covering sets and systems of congruences, in Computers in Mathematical Research,North-Holland, 1968, 20–36; MR 39 #1399.Google Scholar
  36. Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975) 79–81.MathSciNetzbMATHCrossRefGoogle Scholar
  37. P. Erdös, Some problems in number theory, in Computers in Number Theory,Academic Press, 1971, 405–414; esp. pp. 408–409.Google Scholar
  38. J. Haight, Covering systems of congruences, a negative result, Mathematika 26 (1979) 53–61; MR 81e: 10003.Google Scholar
  39. J. H. Jordan, Covering classes of residues, Canad. J. Math. 19 (1967) 514–519; MR 35 #1538.Google Scholar
  40. J. H. Jordan, A covering class of residues with odd moduli, Acta Arith. 13 (1967–68) 335–338; MR 36 #3709.Google Scholar
  41. C. E. Krukenberg, PhD thesis, Univ. of Illinois, 1971, 38–77.Google Scholar
  42. A. Schinzel, Reducibility of polynomials and covering systems of congruences, Acta Arith. 13 (1967) 91–101; MR 36 #2596.Google Scholar
  43. N. Burshtein and J. Schönheim, On exactly covering systems of congruences having moduli occurring at most twice, Czechoslovak Math. J. 24 (99) (1974) 369–372; MR 50 #4521.Google Scholar
  44. A. S. Fraenkel, A characterization of exactly covering congruences, Discrete Math. 4 (1973) 359–366.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Bretislav Novak and Stefan Znâm, Disjoint covering systems, Amer. Math. Monthly 81 (1974) 42–45.MathSciNetzbMATHCrossRefGoogle Scholar
  46. S. Znam, On Mycielski’s problem on systems of arithmetical progressions, Colloq. Math. 15 (1966) 201–204; MR 34 #134.Google Scholar
  47. S. Znam, On exactly covering systems of arithmetic sequences, Math. Ann. 180 (1969) 227–232; MR 39 #4087.Google Scholar
  48. Znam, A simple characterization of disjoint covering systems, Discrete Math. 12 (1975) 89–91.MathSciNetCrossRefGoogle Scholar
  49. R. D. Boyle, On a problem of R. L. Graham, Acta Arith. 34 (1978) 163–177.MathSciNetzbMATHGoogle Scholar
  50. E. Z. Chein, On a conjecture of Graham concerning a sequence of integers, Canad. Math. Bull. 21 (1978) 285–287; MR 80d:10024; Zbl. 392. 10002.Google Scholar
  51. P. Erdös, Problems and results in combinatorial number theory, in A Survey of Combinatorial Theory, North-Holland, 1973, 117–138.Google Scholar
  52. R. L. Graham, Advanced problem 5749-, Amer. Math. Monthly 77 (1970) 775.CrossRefGoogle Scholar
  53. J. Marica and J. Schönheim, Differences of sets and a problem of Graham, Canad. Math. Bull. 12 (1969) 635–637.MathSciNetzbMATHCrossRefGoogle Scholar
  54. R.J. Simpson, On a conjecture of R. L. Graham, Acta Arith. (to appear)Google Scholar
  55. William Yslas Vélez, Some remarks on a number theoretic problem of Graham, Acta Arith. 32 (1977) 233–238.MathSciNetzbMATHGoogle Scholar
  56. Gerald Weinstein, On a conjecture of Graham concerning greatest common divisors, Proc. Amer. Math. Soc. 63 (1977) 33–38; Zbl. 369. 10003.Google Scholar
  57. Riko Winterle, A problem of R. L. Graham in combinatorial number theory, Congressus Numerantium I,Proc. Conf. Combin. Baton Rouge, Utilitas Math. Pub. 1970, 357–361; MR 42 #3051.Google Scholar
  58. Louis Comtet, Advanced Combinatorics, Dreidel, Dordrecht, 1974, p. 89.zbMATHCrossRefGoogle Scholar
  59. Alfred van der Poorten, A proof that Euler missed… Apery’s proof of the irrationality of g(3). An informal report, Math. Intelligencer 1 (1979) 195–203.zbMATHCrossRefGoogle Scholar
  60. Alfred J. van der Poorten, Some wonderful formulas…an introduction to poly-logarithms, Proc. Number Theory Conf, Queen’s Univ., Kingston, 1979, 269–286; MR 801: 10054.Google Scholar
  61. P. Erdös, Some recent problems and results in graph theory, combinatorics and number theory, Congressus Numerantium XVII, Proc. 7th S.E. Conf. Combin. Graph Theory, Comput. Boca Raton, 1976, 3–14Google Scholar
  62. J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London Ser. A 248 (1955) 73–96; MR 17–14.Google Scholar
  63. Harold Davenport, Note on irregularities of distribution, Mathematika 3 (1956) 131–135; MR 19, 19.Google Scholar
  64. John E. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington Mass., 1968, 19–20, Problems 5, 6.Google Scholar
  65. N. J. A. Sloane, The persistence of a number, J. Recreational Math. 6 (1973) 97–98.Google Scholar
  66. J. H. Conway and M. J. T. Guy, n in four 4’s, Eureka 25 (1962) 18–19.Google Scholar
  67. Popken and K. Mahler, On a maximum problem in arithmetic, Nieuw Arch. voor Wisk. (3) 1 (1953) 1–15.Google Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Richard K. Guy
    • 1
  1. 1.Department of Mathematics and StatisticsThe University of CalgaryCalgaryCanada

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