Some Diophantine Equations

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

Abstract

This quotation from the preface of Mordell’s book, Diophantine Equations, Academic Press, London, 1969, indicates that in this section we shall have to be even more eclectic than elsewhere. If you’re interested in the subject, consult Mordell’s book, which is a thoroughgoing but readable account of what is known, together with a great number of unsolved problems. There are well-developed theories of rational points on algebraic curves, so we mainly confine ourselves to higher dimensions, for which standard methods have not yet been developed.

Keywords

London Math Unsolved Problem Integer Solution Diophantine Equation Parametric Solution 
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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Reference

  1. B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, II, J. reine angew. Math 218 (1965) 79–108.MathSciNetMATHGoogle Scholar
  2. Andrew Bremner, Pythagorean triangles and a quartic surface, J. reine angew. Math 318 (1980) 120–125.MathSciNetMATHGoogle Scholar
  3. Andrew Bremner, A geometric approach to equal sums of sixth powers, Proc. London Math. Soc (to appear).Google Scholar
  4. Andrew Bremner, A geometric approach to equal sums of fifth powers, J. Number Theory (to appear).Google Scholar
  5. S. Brudno, Some new results on equal sums of like powers, Math. Comput 23 (1969) 877–880.MathSciNetMATHGoogle Scholar
  6. S. Brudno, On generating infinitely many solutions of the diophantine equation A 6 + B6 + C 6 = D 6 + E 6 + F 6, Math. Comput 24 (1970) 453–454.MathSciNetMATHGoogle Scholar
  7. S. Brudno, Problem 4, Proc. Number Theory Conf. Univ. of Colorado, Boulder, 1972, 256–257.Google Scholar
  8. Simcha Brudno, Triples of sixth powers with equal sums, Math. Comput 30 (1976) 646–648.MathSciNetMATHGoogle Scholar
  9. S. Brudno and I. Kaplansky, Equal sums of sixth powers, J. Number Theory 6 (1974) 401–403.MathSciNetMATHGoogle Scholar
  10. V. A. Dem’janenko, L. Euler’s conjecture (Russian), Acta Arith 25 (1973/74) 127–135; MR 50 #12912.Google Scholar
  11. Jan Kubièek, A simple new solution to the diophantine equation A3 + B 3 + C 3 = D 3, (Czech, German summary), Casopis Pést. Mat 99 (1974) 177–178.Google Scholar
  12. L. J. Lander, Geometric aspects of diophantine equations involving equal sums of like powers, Amer. Math Monthly 75 (1968) 1061–1073.MathSciNetMATHGoogle Scholar
  13. L. J. Lander and T. R. Parkin, Counterexample to Euler’s conjecture on sums of like powers, Bull. Amer. Math. Soc 72 (1966) 1079; MR 33 #5554.Google Scholar
  14. L. J. Lander, T. R. Parkin, and J. L. Selfridge, A survey of equal sums of like powers, Math. Comput 21 (1967) 446–459; MR 36 #5060.Google Scholar
  15. R. Norrie, Univ. of St. Andrews 500th Anniv. Mem. Vol, Edinburgh, 1911, 89. Morgan Ward, Euler’s three biquadrate problem, Proc. Nat. Acad. Sci. U.S.A. 31(1945) 125–127; MR 6, 259.Google Scholar
  16. Morgan Ward, Euler’s problem on sums of three fourth powers, Duke Math. J. 15 (1948) 827–837; MR 10, 283.Google Scholar
  17. P. Erdös and R. Oblâth, Über diophantische Gleichungen der Form n! = xP ± y“ and n! ± m! = xP, Acta Litt. Sci. Szeged 8 (1937) 241–255; Zbl 17. 004.Google Scholar
  18. K. Inkeri and A. J. van der Poorten, Some remarks on Fermat’s conjecture, Acta Arith. 36 (1980) 107–111.MathSciNetMATHGoogle Scholar
  19. Wells Johnson, Irregular primes and cyclotomic invariants, Math. Comput 29 (1975) 113–120; MR 51 #12781.Google Scholar
  20. D. H. Lehmer, On Fermat’s quotient, base two, Math. Comput 36 (1981) 289–290. Paulo Ribenboim, 13 Lectures on Fermat’s Last Theorem,Springer-Verlag, New YorkGoogle Scholar
  21. Heidelberg, Berlin, 1979; see Bull. Amer. Math. Soc 4 (1981) 218–222; MR 81f 10023.Google Scholar
  22. J. L. Selfridge, C. A. Nicol and H. S. Vandiver, Proof of Fermat’s last theorem for all prime exponents less than 4002, Proc. Nat. Acad. Sci. U.S.A. 41 (1955) 970–973; MR 17, 348.Google Scholar
  23. Daniel Shanks and H. C. Williams, Gunderson’s function in Fermat’s last theorem, Math. Comput 36 (1981) 291–295.MathSciNetMATHGoogle Scholar
  24. Samuel S. Wagstaff, The irregular primes to 125000, Math. Comput 32 (1978) 583–591; MR 58 #10711.Google Scholar
  25. H. E. Dudeney, Amusements in Mathematics, Nelson, 1917, 26, 167.Google Scholar
  26. Raphael Finkelstein, On a diophantine equation with no non-trivial integral solution, Amer. Math. Monthly 73 (1966) 471–477.MathSciNetMATHGoogle Scholar
  27. W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidskr 34 (1952) 65–72.MathSciNetMATHGoogle Scholar
  28. E. Lucas, Problem 1180, Nouv. Ann. Math. (2) 14 (1875) 336.Google Scholar
  29. G. N. Watson, The problem of the square pyramid, Messenger of Math 48 (1918/19) 1–22.Google Scholar
  30. S. Chowla, The number of representations of a large number as a sum of non-negative nth powers, Indian Phys.-Math. J. 6 (1935) 65–68; Zbl 12. 339.Google Scholar
  31. H. Davenport, Sums of three positive cubes, J. London Math. Soc. 25 (1950) 339–343; MR 12, 393.Google Scholar
  32. P. Erdös, On the representation of an integer as the sum of k kth powers, J. London Math. Soc. 11 (1936) 133–136; Zbl 13. 390.Google Scholar
  33. P. Erdös, On the sum and difference of squares of primes I, II, J. London Math. Soc 12 (1937) 133–136, 168–171; Zbl 16.201, 17.103.Google Scholar
  34. P. Erdös and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc 13 (1938) 134–139.Google Scholar
  35. P. Erdös and E. Szemerédi, On the number of solutions of m = Ek-1 x, Proc. Symp. Pure Math. 24 Amer. Math. Soc., Providence, 1972, 83–90.Google Scholar
  36. G. H. Hardy and J. E. Littlewood, Partitio Numerorum VI: Further researches in Waring’s problem, Math. Z 23 (1925) 1–37.MathSciNetMATHGoogle Scholar
  37. Jean Lagrange, Thèse d’Etat de l’Université de Reims, 1976.Google Scholar
  38. K. Mahler, Note on hypothesis K of Hardy and Littlewoord, J. London Math. Soc 11 (1936) 136–138.MathSciNetGoogle Scholar
  39. K. Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc 39 (1935) 431–466.MathSciNetGoogle Scholar
  40. W. J. Ellison, Waring’s problem, Amer. Math. Monthly 78 (1971) 10–36.MathSciNetMATHGoogle Scholar
  41. Chao Ko, Decompositions into four cubes, J. London Math. Soc 11 (1936) 218–219. V. L. Gardiner, R. B. Lazarus, and P R. Stein, Solutions of the diophantine equation x3 + y 3 = z 3 - d, Math. Comput 18 (1964) 408–413; MR 31 #119.Google Scholar
  42. M. Lal, W. Russell, and W. J. Blundon, A note on sums of four cubes, Math. Comput 23 (1969) 423–424; MR 39 #6819.Google Scholar
  43. A Makowski, Sur quelques problèmes concernant les sommes de quatre cubes, Acta Arith 5 (1959) 121–123; MR 21 #5609.Google Scholar
  44. J. C. P. Miller and M. F. C. Woollett, Solutions of the diophantine equation x3 + y3 + z3 = k, J. London Math. Soc. 30 (1955) 101–110; MR 16, 979.Google Scholar
  45. A. Schinzel and W. Sierpinski, Sur les sommes de quatre cubes, Acta Arith. 4 (1958) 20–30.MathSciNetMATHGoogle Scholar
  46. Edward A. Bender and Norman P Herzberg, Some diophantine equations related to the quadratic form ax e + by 2, Bull. Amer. Math. Soc 81 (1975) 161–162.MathSciNetMATHGoogle Scholar
  47. J. Blass, On the diophantine equation Y2 + K = Xs, Bull. Amer. Math. Soc 80 (1974) 329.MathSciNetMATHGoogle Scholar
  48. J. H. E. Cohn, On square Fibonacci numbers, J. London Math. Soc 39 (1964) 537–540; MR 29 #1166.Google Scholar
  49. J. H. E. Cohn, The diophantine equation y2 = Dx 4 + 1, I, J. London Math. Soc 42 (1967) 475–476; MR 35 #4158; II Acta Arith 28 (1975/76) 273–275; MR 52 #8029; III Math. Scand 42 (1978) 180–188; MR 80a:10031 Google Scholar
  50. N. P. Herzberg, Integer solutions of by e + p“ = x 3, J. Number Theory 7 (1975) 221–234; Zbl 302. 10021.Google Scholar
  51. D. J. Lewis, Two classes of diophantine equations, Pacific J. Math. 11(1961)1063–1076. W. Ljunggren, Zur Theorie der Gleichung x2 + 1 = Dy 4, Avh. Norske Vid. Akad. Oslo, I, 5 (1942) #5, 27 pp; MR 8, 6.Google Scholar
  52. W. Ljunggren, On a diophantine equation, Norske Vid. Selsk. Forh. Trondheim, 18 #32 (1945) 125–128; MR 8, 136.Google Scholar
  53. W. Ljunggren, New theorems concerning the diophantine equation Cx2 + D = y“, Norske Vid. Selsk. Forh. Trondheim 29 (1956) 1–4; MR 17, 1185.Google Scholar
  54. W. Ljunggren, On the diophantine equation Cx2 + D = y“, Pacific J. Math 14 (1964) 585–596; MR 28 #5035.Google Scholar
  55. W. Ljunggren, Some remarks on the diophantine equation x2Dy 4 = 1 and x 4 — Dy 2 = 1, J. London Math. Soc 41(1966) 542–544; MR 33 #5555.Google Scholar
  56. L. J. Mordell, The diophantine equation y 2 = Dx 4 + 1, J. London Math. Soc 39 (1964) 161–164; MR 29 #65.Google Scholar
  57. T. Nagell, Contributions to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta Soc. Sci. Upsal. (4) 16 (1955) #2; 38 pp; MR 17, 13.Google Scholar
  58. M R. Best and H. J. J. te Riele, On a conjecture of Erdös concerning sums of powers of integers, Report NW 23/76,Mathematisch Centrum Amsterdam, 1976.Google Scholar
  59. P. Erdös, Advanced problem 4347, Amer. Math. Monthly 56 (1949) 343.Google Scholar
  60. K. Györy, R. Tijdeman and M. Voorhoeve, On the equation lk + 2“ + • • • + xl` = y`, Acta Arith. 37 (1980) 233–240.MathSciNetMATHGoogle Scholar
  61. J. van de Lune, On a conjecture of Erdös (I), Report ZW 54/75, Mathematisch Centrum, Amsterdam, 1975.Google Scholar
  62. J. van de Lune and H. J. J. te Riele, On a conjecture of Erdös (II), Report ZW 56/75, Mathematisch Centrum, Amsterdam, 1975.Google Scholar
  63. L. Moser, On the diophantine Equation 1“ + 2” + • • • + (m — 1)“ = m”, Scripta Math. 19 (1953) 84–88; MR 14–950.Google Scholar
  64. J. J. Schäffer, The equation U“ + 2” + • • + n“ = m9, Acta Math. 95 (1956) 155–189; MR 17, 1187.Google Scholar
  65. M.Voorhoeve, K. Györy and R. Tijdeman, On the diophantine equation 1 k + 2k + • • • + x’` + R(x) = yZ, Acta Math. 143 (1979) 1–8; MR 80e:10020 Google Scholar
  66. Leon Bernstein, Explicit solutions of pyramidal Diophantine equations, Canad. Math. Bull. 15 (1972) 177–184; MR 46 #3442.Google Scholar
  67. Hugh Maxwell Edgar, Some remarks on the Diophantine equation x3 + y 3 + z 3 =x + y + z Proc. Amer. Math. Soc. 16 (1965) 148–153; MR 30 #1094.Google Scholar
  68. A. S. Fraenkel, Diophantine equations involving generalized triangular and tetrahedral numbers, in Computers in Number Theory, Proc. Atlas Symp. No. 2, Oxford 1969Google Scholar
  69. Academic Press, London and New York (1971) 99–114.Google Scholar
  70. A. Oppenheim, On the Diophantine equation x3 + y3 + z3 = x + y + z, Proc. Amer. Math. Soc. 17 (1966) 493–496; MR 32 #5590.Google Scholar
  71. A. Oppenheim, On the diophantine equation x3 + y3 — z3 = px + py — qz, Univ. Beograd Publ. Elektrotehn. Fak. Ser. #235 (1968); MR 39 #126.Google Scholar
  72. S. L. Segal, A note on pyramidal numbers, Amer. Math. Monthly 69 (1962) 637–638; Zbl. 105, 36.Google Scholar
  73. W. Sierpinski, Sur une propriété des nombres tétraédraux, Elem. Math. 17 (1962) 29–30; MR 24 #A 3118.Google Scholar
  74. W. Sierpinski, Trois nombres tétraédraux en progression arithmetique, Elem. Math. 18 (1963) 54–55; MR 26 #4957.Google Scholar
  75. M. Wunderlich, Certain properties of pyramidal and figurate numbers, Math. Comput. 16 (1962) 482–486; MR 26 #6115.Google Scholar
  76. R. Tijdeman, On the equation of Catalan, Acta Arith 29 (1976)197–209; MR 53 #7941.Google Scholar
  77. Leo J. Alex, Problem E 2880, Amer. Math. Monthly, 88 (1981) 291.MathSciNetGoogle Scholar
  78. J. L. Brenner and Lorraine L. Foster, Exponential diophantine equations, Pacific J. Math (to appear).Google Scholar
  79. A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart. 11 (1973) 429–438; MR 497’/209 Google Scholar
  80. A. Aigner, Brüche als Summe von Stammbrüchen, J. reine angew. Math. 214/215 (1964) 174–179.Google Scholar
  81. P. J. van Albada and J. H. van Lint, Reciprocal bases for the integers, Amer. Math. Monthly 70 (1963) 170–174.Google Scholar
  82. E. J. Barbeau, Computer Challenge corner: Problem 477: A brute force program, J.Recreational Math 9 (1976/77) 30.Google Scholar
  83. E. J. Barbeau, Expressing one as a sum of distinct reciprocals: comments and a bibliography, Eureka (Ottawa) 3 (1977) 178–181.Google Scholar
  84. Leon Bernstein, Zur Lösung der diophantischen Gleichung m/n = 1 /x + 1/y + 1/z insbesondere im Falle m = 4, J. reine angew. Math 211 (1962) 1–10; MR 26 #77 Google Scholar
  85. M. N. Bleicher, A new algorithm for the expansion of Egyptian fractions, J. Number Theory 4 (1972) 342–382; MR 48 #2052.Google Scholar
  86. M. N. Bleicher and P Erdös, The number of distinct subsums of ~i 1 /i, Math. Comput 29 (1975) 29–42 (and see Notices Amer. Math. Soc 20 (1973) A-516).Google Scholar
  87. M. N. Bleicher and P. Erdös, Denominators of Egyptian fractions, J. Number Theory 8 (1976) 157–168; MR 53 #7925; II, Illinois J. Math 20 (1976) 598–613; MR 54#7359.Google Scholar
  88. Robert Breusch, A special case of Egyptian fractions, Solution to Advanced Problem 4512, Amer. Math. Monthly 61 (1954) 200–201.MathSciNetGoogle Scholar
  89. W. S. Burnside, Theory of Groups of Finite Order, 2nd ed. Cambridge University Press, London, 1911, reprinted Dover, New York, 1955, Note A, 461–462.Google Scholar
  90. N. Burshtein, On distinct unit fractions whose sum equals 1, Discrete Math. 5 (1973) 201–206.MathSciNetMATHGoogle Scholar
  91. Paul J. Campbell, Bibliography of algorithms for Egyptian fractions (preprint) Beloit Coll. Beloit WI 53511, U.S.A.Google Scholar
  92. J. W. S. Cassels, On the representation of integers as the sum of distinct summands taken from a fixed set, Acta Sci. Math. Szeged 21 (1960) 111–124.MathSciNetMATHGoogle Scholar
  93. A. B. Chace, The Rhind Mathematical Papyrus, M.A.A., Oberlin, 1927.Google Scholar
  94. Robert Cohen, Egyptian fraction expansions, Math. Mag 46 (1973) 76–80; MR 47 #3300.Google Scholar
  95. D. Culpin and D. Griffiths, Egyptian fractions, Math. Gaz. 63 (1979) 49–51; MR 80d: 10014.Google Scholar
  96. D. R. Curtiss, On Kellogg’s Diophantine problem, Amer. Math. Monthly, 29 (1922) 380–387.MathSciNetGoogle Scholar
  97. L. E. Dickson, History of the Theory of Numbers, Vol. 2 Diophantine Analysis, Chelsea, New York, 1952, 688–691.Google Scholar
  98. P. Erdös, Egy Kürschàk-féle elemi szàmelméleti tétel hltadanositàsa, Mat. es Phys. Lapok 39 (1932).Google Scholar
  99. P. Erdös, On arithmetical properties of Lambert series, J. Indian Math. Soc 12 (1948) 63–66.MathSciNetMATHGoogle Scholar
  100. P. Erdös, On a diophantine equation (Hungarian. Russian and English summaries), Mat. Lapok 1(1950) 192–210; MR 13, 208.Google Scholar
  101. P. Erdös, On the irrationality of certain series, Nederl. Akad. Wetensch. (Indag. Math.) 60 (1957) 212–219.MATHGoogle Scholar
  102. P. Erdös, Sur certaines séries à valeur irrationnelle, Enseignement Math. 4 (1958) 93–100.MathSciNetMATHGoogle Scholar
  103. P.Erdös, Quelques Problèmes de la Théorie des Nombres,Monographie de l’Enseignement Math. No. 6, Geneva, 1963, problems 72–74.Google Scholar
  104. P. Erdös, Comment on problem E2427, Amer. Math. Monthly, 81 (1974) 780–782.MathSciNetGoogle Scholar
  105. P. Erdös, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci 10 (1975) 1–7.MathSciNetGoogle Scholar
  106. Paul Erdös and Ivan Niven, Some properties of partial sums of the harmonic series, Bull. Amer. Math. Soc. 52 (1946) 248–251; MR 7, 413.Google Scholar
  107. P. Erdös and S. Stein, Sums of distinct unit fractions, Proc. Amer. Math. Soc 14 (1963) 126–131.MathSciNetMATHGoogle Scholar
  108. P. Erdös and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc 27 (1968) 129–133.Google Scholar
  109. P. Erdös and E. G. Straus, Some number theoretic results, Pacific J. Math 36 (1971) 635 —646.Google Scholar
  110. P. Erdös and E. G. Straus, Solution of problem E2232, Amer. Math. Monthly 78 (1971) 302–303.MathSciNetGoogle Scholar
  111. P. Erdös and E. G. Straus, On the irrationality of certain series, Pacific J. Math 55 (1974) 85–92; MR 51 #3069.Google Scholar
  112. P. Erdös and E. G. Straus, Solution to problem 387, Nieuw Arch. Wisk. 23 (1975) 183. Nicola Franceschine, Egyptian Fractions, MA Dissertation, Sonoma State Coll. CA, 1978.Google Scholar
  113. S. W. Golomb, An algebraic algorithm for the representation problems of the Ahmes papyrus, Amer. Math. Monthly 69 (1962) 785–786.MathSciNetGoogle Scholar
  114. S. W. Golomb, On the sums of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math 15 (1963) 475–478.MathSciNetMATHGoogle Scholar
  115. R. L. Graham, A theorem on partitions, J. Austral. Math. Soc 4 (1963) 435–441.Google Scholar
  116. R. L. Graham, On finite sums of unit fractions, Proc. London Math. Soc (3) 14 (1964) 193–207; MR 28 #3968.Google Scholar
  117. R. L. Graham, On finite sums of reciprocals of distinct nth powers, Pacific J. Math 14 (1964) 85–92; MR 28 #3004.Google Scholar
  118. L.-S. Hahn, Problem E2689, Amer. Math. Monthly 85 (1978) 47.MathSciNetGoogle Scholar
  119. J. W. Hille, Decomposing fractions, Math. Gaz 62 (1978) 51–52.Google Scholar
  120. Ludwig Holzer, Zahlentheorie Teil III Ausgewählte Kapitel der Zahlentheorie, Math.-Nat. Bibl. No. 14a, B. G. Teubner-Verlag, Leipzig, 1965, Sect. A, 1–27; MR 34 #4186.Google Scholar
  121. Dag Magne Johannessen, On unit fractions II, Nordisk mat. Tidskr. 25–26 (1978) 85–90; MR 80a:10010; Zbl 384. 10004.Google Scholar
  122. Dag Magne Johannessen and T. V. Sohus, On unit fractions I, ibid 22 (1974) 103–107; MR 55 #252; Zbl. 291. 10010.Google Scholar
  123. Ralph W. Jollensten, A note on the Egyptian problem, Congressus Numerantium XVII,Proc. 7th S. E. Conf. Combin. Graph Theory, Comput. 1976, 351–364; MR 55 #2746.Google Scholar
  124. O. D. Kellogg, On a diophantine problem, Amer. Math. Monthly, 28 (1921) 300–303.MathSciNetMATHGoogle Scholar
  125. E. Kiss, Quelques remarques sur une équation diophantienne (Romanian. French summary) Acad. R. P. Romïne Fil. Cluj, Stud. Cerc. Mat 10 (1959) 59–62.MATHGoogle Scholar
  126. E. Kiss, Remarques relatives à la représentation des fractions subunitaires en somme des fractions ayant le numerateur égal à l’unité (Romanian) Acad. R. P. Romïne Fil. Cluj, Stud. Cerc. Mat 11 (1960) 319–323.MathSciNetMATHGoogle Scholar
  127. Ladis D. Kovach, Ancient algorithms adapted to modern computers, Math. Mag 37 (1964) 159–165.MathSciNetGoogle Scholar
  128. Jôzsef Kürschàk, A harmonikus sorrôl, Mat. es. Phys. Lapok 27 (1918) 299–300.MATHGoogle Scholar
  129. Denis Lawson, Ancient Egypt revisited, Math. Gaz 54 (1970) 293–296; MR 58 #10697 Google Scholar
  130. P. Montgomery, Solution to Problem E2689, Amer. Math. Monthly 86 (1979) 224.Google Scholar
  131. L. J. Mordell, Diophantine Equations, Academic Press, London, 1969, 287–290.MATHGoogle Scholar
  132. T. Nagell, Skr. Norske Vid. Akad. Kristiania I, 1923, no. 13 (1924) 10–15.Google Scholar
  133. M. Nakayama, On the decomposition of a rational number into “Stammbrüche,” Töhoku Math. J. 46 (1939) 1–21.MathSciNetGoogle Scholar
  134. James R. Newman, The Rhind Papyrus, in The World of Mathematics, Allen and Unwin, London, 1960, 169–178.Google Scholar
  135. R. Obldth, Sur l’équation diophantienne 4/n = 1/xi + 1/x2 + 1/x3, Mathesis 59 (1950) 308–316; MR 12, 481.Google Scholar
  136. J. C. Owings, Another proof of the Egyptian fraction theorem, Amer. Math. Monthly 75 (1968) 777–778.MathSciNetMATHGoogle Scholar
  137. G. Palamà, Su di una congettura di Sierpinski relativa alla possibilità in numeri naturali della 5/n = 1/x 1 + 1/x 2 +11x 3, Boll. Un. Mat. Ital (3) 13 (1958) 65–72; MR 20 #3821.Google Scholar
  138. G. Palamà, Su di una congettura di Schinzel, Boll. Un. Mat. Ital (3) 14 (1959) 82–94; MR 22 #7989.Google Scholar
  139. T. E. Peet, The Rhind Mathematical Papyrus, Univ. Press of Liverpool, London, 1923. L. Pisano, Scritti, Vol. 1, B. Boncompagni, Rome, 1857.Google Scholar
  140. Y. Ray, On the representation of a rational number as a sum of a fixed number of unit fractions, J. reine angew. Math 222 (1966) 207–213.MathSciNetGoogle Scholar
  141. L. A. Rosati, Sull’equazione diofantea 4/n = 1/x i + 1/x 2 + 1/x 3, Boll. Un. Mat. Ital. (3) 9 (1954) 59–63; MR 15, 684.Google Scholar
  142. H. D. Ruderman, Problem E2232, Amer. Math. Monthly 77 (1970) 403.MathSciNetGoogle Scholar
  143. Harry Ruderman, Bounds for Egyptian fraction partitions of unity, Problem E2427, Amer. Math. Monthly 80 (1973) 807.MathSciNetGoogle Scholar
  144. H. E. Salzer, The approximation of numbers as sums of reciprocals, Amer. Math. Monthly 54 (1947) 135–142; MR 8, 534.Google Scholar
  145. H. E. Salzer, Further remarks on the approximation of numbers as sums of reciprocals, Amer. Math. Monthly 55 (1948) 350–356; MR 10, 18.Google Scholar
  146. Andrzej Schinzel, Sur quelques propriétés des nombres 3/n et 4/n, où n est un nombre impair, Mathesis 65 (1956) 219–222; MR 18, 284.Google Scholar
  147. Sedlàcek, Über die Stammbrüche, Casopis Pëst. Mat 84 (1959) 188–197; MR 23 #A829.Google Scholar
  148. Ernest S. Selmer, Unit fraction expansions and a multiplicative analog, Nordisk mat. Tidskr. 25–26 (1978) 91–109; Zbl 384. 10005.Google Scholar
  149. W. Sierpinski, Sur les décompositions de nombres rationnels en fractions primaires, Mathesis 65 (1956) 16–32; MR 17, 1185.Google Scholar
  150. W. Sierpinski, On the Decomposition of Rational Numbers into Unit Fractions (Polish), Pànstwowe Wydawnictwo Naukowe, Warsaw, 1957.Google Scholar
  151. W. Sierpiüski, Sur une algorithme pour le développer les nombres réels en séries rapidement convergentes, Bull. Int. Acad. Sci. Cracovie Ser. A Sci. Mat 8 (1911) 113–117.Google Scholar
  152. David Singmaster, The number of representations of one as a sum of unit fractions (mimeographed note) 1972.Google Scholar
  153. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973.MATHGoogle Scholar
  154. B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 (1954) 779–785; MR 16, 336.Google Scholar
  155. B. M. Stewart, Theory of Numbers, Macmillan, N.Y., 1964, 198–207.MATHGoogle Scholar
  156. B.M. Stewart and W. A. Webb, Sums of fractions with bounded numerators Canad. J. Math. 18 (1966) 999–1003; MR 33 #7297.Google Scholar
  157. E. G. Straus and M. V. Subbarao, On the representation of fractions as sum and difference of three simple fractions, Congressus Numeratium XX, Proc. 7th Conf. Numerical Math. Comput. Manitoba 1977, 561–579.Google Scholar
  158. J. J. Sylvester, On a point in the theory of vulgar fractions, Amer. J. Math. 3 (1880) 332–335, 388–389.MathSciNetGoogle Scholar
  159. D. G. Terzi, On a conjecture of Erdös-Straus, BIT 11 (1971) 212–216.MathSciNetMATHGoogle Scholar
  160. L. Theisinger, Bemerkung über die harmonische Reihe, Monat. für Math. u. Physik 26 (1915) 132–134.MathSciNetMATHGoogle Scholar
  161. R. C. Vaughan, On a problem of Erdös, Straus and Schinzel, Mathematika 17 (1970) 193–198.MathSciNetMATHGoogle Scholar
  162. C. Viola, On the diophantine equations IT x, - xi = n and I0 1/xi = a/n, ActaArith 22 (1972/73) 339–352.Google Scholar
  163. W. A. Webb, On 4/n = 1/x + 1/y + 1/z, Proc. Amer. Math. Soc. 25 (1970) 578–584.MathSciNetMATHGoogle Scholar
  164. William A. Webb, Rationals not expressible as a sum of three unit fractions, Elem. Math. 29 (1974) 1–6.Google Scholar
  165. William A. Webb, On a theorem of Rav concerning Egyptian fractions, Canad. Math. Bull. 18 (1975) 155–156.MATHGoogle Scholar
  166. William A. Webb, On the unsolvability of k/n = 1/x + 1/y +11z, Notices Amer. Math. Soc. 22 (1975) A-485.Google Scholar
  167. William A. Webb, On the diophantine equation k/n =ai/xi + a2/x2 + a3/x3 (loose Russian summary), Casopis Pést. Mat 101 (1976) 360–365.MATHGoogle Scholar
  168. H. S. Wilf, Reciprocal bases for the integers, Res. Problem 6, Bull. Amer. Math. Soc. 67 (1961) 456.MathSciNetGoogle Scholar
  169. R. T. Worley, Signed sums of reciprocals I, II, J. Australian Math. Soc. 21 (1976) 410–414, 415–417.Google Scholar
  170. Koichi Yamamoto, On a conjecture of Erdös, Mem. Fac. Sci. Kyushû Univ Ser. A, 18 (1964) 166–167; MR 30 #1968 Google Scholar
  171. K. Yamamoto, On the diophantine equation 4/n = 1/x + 1/y + 1/z, Mem. Fac. Sci. Kyushû Univ. Ser. A, 19 (1965) 37–47.MATHGoogle Scholar
  172. J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge, 1957, 27–44.Google Scholar
  173. H. Cohn, Approach to Markoff’s minimal forms through modular functions, Ann. Math. Princeton (2) 61 (1955) 1–12.MATHGoogle Scholar
  174. T. W. Cusick, The largest gaps in the lower Markoff spectrum, Duke Math. J 41(1974) 453–463; MR 57 #5902.Google Scholar
  175. L. E. Dickson, Studies in the Theory of Numbers,Chicago Univ. Press, 1930, Chap. VII. G. Frobenius, Über die Markoffschen Zahlen, S,-B. Preuss. Akad. Wiss. Berlin (1913) 458 —487.Google Scholar
  176. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann 15 (1879) 381–409.MATHGoogle Scholar
  177. R. Remak, Über indefinite binäre quadratische Minimalformen, Math. Ann 92 (1924) 155–182.MathSciNetMATHGoogle Scholar
  178. R. Remak, Über die geometrische Darstellung der indefiniten binären quadratischen Minimalformen, Jber. Deutsch Mat.-Verein 33 (1925) 288–245.Google Scholar
  179. Gerhard Rosenberger, The uniqueness of the Markoff numbers, Math. Comput 30 (1976) 361–365; but see MR 53 #280.Google Scholar
  180. L. Ja. Vulah, The diophantine equation p 2 + 2q2 + 3r2 = 6pqr and the Markoff spectrum (Russian), Trudy Moskov. Inst. Radiotehn. Èlektron. i Avtomat. Vyp. 67 Mat (1973) 105–112, 152; MR 58 #21957.Google Scholar
  181. Don B.Zagier, Distribution of Markov numbers, Abstract 796—A37, Notices Amer. Math. Soc,26 (1979) A-543.Google Scholar
  182. Chao Ko, Note on the diophantine equation xxyY = zz, J. Chinese Math. Soc. 2 (1940) 205–207; MR 2, 346.Google Scholar
  183. W. H. Mills, An unsolved diophantine equation, in Report Inst. Theory of Numbers, Boulder, Colorado, 1959, 258–268.Google Scholar
  184. T. Baker, The Gentleman’s Diary, or Math. Repository, London, 1839, 33–5, Quest. 1385.Google Scholar
  185. C. Gill, Application of the Angular Analysis to Indeterminate Problems of Degree 2, N.Y. 1848, p. 60.Google Scholar
  186. J. Lagrange, Cinq nombres dont les sommes deux à deux sont des carrés, SéminaireDelange-Pisot-Poitou (Théorie des nombres) 12e année,20, 1970–71, lOpp.Google Scholar
  187. Jean Lagrange, Six entiers dont les sommes deux à deux sont des carrés, Acta Arith (to appear).Google Scholar
  188. Jean-Louis Nicolas, 6 nombres dont les sommes deux à deux sont des carrés, Bull. Soc. Math. France,Mém No 49–50 (1977) 141–143; MR 58 #482.Google Scholar
  189. A. W. Thatcher, A prize problem, Math. Gaz 61 (1977) 64.Google Scholar
  190. P. Erdös, On consecutive integers, Nieuw Arch. Wisk 3 (1955) 124–128.MathSciNetMATHGoogle Scholar
  191. P. Erdös and J. L. Selfridge, the product of consecutive integers is never a power, Illinois J. Math 19 (1975) 292–301.MathSciNetMATHGoogle Scholar
  192. T. Bromhead, On square sums of squares, Math. Gaz. 44 (1960) 219–220; MR 23 #A1594.Google Scholar
  193. W. S. Burnside, Note on the symmetric group, Messenger of Math. 30 (1900–01) 148–153; J’buch 32, 141–142.Google Scholar
  194. E. Z. Chein, On the derived cuboid of an Eulerian triple, Canad. Math. Bull. 20 (1977) 509–510; MR 57 #12375.Google Scholar
  195. J. H. Conway and H. S. M. Coxeter, Triangulated polygons and frieze patterns, Math. Gaz. 57 (1973) 87–94, 175–183 and refs. on pp. 93–94.Google Scholar
  196. H. S. M. Coxeter, Frieze patterns, Acta Arith 18 (1971) 297–310; MR 44 #3980.Google Scholar
  197. L. E. Dickson, History of the Theory of Numbers, Vol. 2, Diophantine Analysis, Washington, 1920: ch. 15, ref. 28, p. 448 and cross-refs. to pp. 446–458; ch. 19 refs 1–30, 40–45, pp. 497–502, 505–507.Google Scholar
  198. Martin Gardner, Mathematical Games, Scientific Amer 223 #1 (Jul. 1970) 118; correction. ibid #3 (Sep. 1970) 218.Google Scholar
  199. W. Howard Joint, Cycles, Note 1767, Math. Gaz 28 (1944) 196–197.Google Scholar
  200. Maurice Kraitchik, On certain rational cuboids, Scripta Math 11 (1945) 317–326; MR8,6.Google Scholar
  201. Maurice Kraitchik, Théorie des Nombres, t. 3, Analyse Diophantine et applications aux cuboides rationnels, Paris, 1947.Google Scholar
  202. Maurice Kraitchik, Sur les cuboides rationnels, in Proc. Internat. Congr. Math. 1954, Vol. 2, Amsterdam, 1954, 33–34.Google Scholar
  203. M. Lal and W. J. Blundon, Solutions of the Diophantine equation x2 + y2 = 12, y 2 + z2 = m2, z2 + x2 = n2, Math. Comput. 20 (1966) 144–147; MR 32 #4082.Google Scholar
  204. J. Lagrange, Sur le dérivé du cuboïde eulerien, Canad. Math. Bull. 22 (1979) 239–241; MR 80h: 10022.Google Scholar
  205. J. Leech, The location of four squares in an arithmetic progression with some applications, in Computers in Number Theory,Academic Press, London, 1971, 83–98; MR 47 #4913.Google Scholar
  206. J. Leech, The rational cuboid revisited, Amer. Math. Monthly 84 (1977) 518–533. Corrections (Jean Lagrange) 85 (1978) 473; MR 58 #16492.Google Scholar
  207. J. Leech, Five tables related to rational cuboids, Math. Comput. 32 (1978) 657–659.Google Scholar
  208. R. C. Lyness, Cycles, Note 1581, Math. Gaz. 26 (1942) 62; Note 1847 ibid 29 (1945) 231–233; Note 2952, ibid 45 (1961) 207–209.Google Scholar
  209. Mahatma“, Problem 78, The A.M.A. (J. Assist. Masters Assoc. London) 44 (1949) 188. Solutions J. Hancock, J. Peacock, N. A. Phillips, ibid. 225.Google Scholar
  210. Eliakim Hastings Moore, The cross-ratio group of n Cremona-transformations of order n — 3 in flat space of n — 3 dimensions, Amer. J. Math. 30 (1900) 279–291; J’buch 31, 655.Google Scholar
  211. L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Cambridge Philos. Soc 21 (1922) 179–192.Google Scholar
  212. H. C. Pocklington, Some diophantine impossibilities, Proc. Cambridge Philos. Soc 17 (1914) 110–121, esp. p. 116.Google Scholar
  213. W. W. Sawyer, Lyness’s periodic sequence, Note 2951, Math. Gaz 45 (1961) 207.Google Scholar
  214. W. Sierpinski, A Selection of Problems in the Theory of Numbers, Pergamon, Oxford, 1964, p. 112.MATHGoogle Scholar
  215. W. G. Spohn, On the integral cuboid, Amer. Math. Monthly,79 (1972) 57–59; MR 46 #7158 Google Scholar
  216. W. G. Spohn, On the derived cuboid, Canad. Math. Bull 17 (1974) 575–577; MR 51 #12693.Google Scholar
  217. R. B. Eggleton, Tiling the plane with triangles, Discrete Math. 7 (1974) 53–65.MathSciNetMATHGoogle Scholar
  218. R. B. Eggleton, Where do all the triangles go? Amer. Math. Monthly 82 (1975) 499–501.MathSciNetMATHGoogle Scholar
  219. Ronald Evans, Problem E2685, Amer. Math. Monthly 84 (1977) 820.MathSciNetGoogle Scholar
  220. N. J. Fine, On rational triangles, Amer. Math. Monthly 83 (1976) 517–521.MathSciNetMATHGoogle Scholar
  221. J. G. Mauldon, An impossible triangle, Amer. Math. Monthly 86 (1979) 785–786.MathSciNetGoogle Scholar
  222. C. Pomerance, On a tiling problem of R. B. Eggleton, Discrete Math. 18 (1977) 63–70.MathSciNetMATHGoogle Scholar
  223. J. H. J. Almering, Rational quadrilaterals, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math. 25 (1963) 192–199; II, ibid. 68 = 27 (1965) 290–304; MR 26 #4963, 31 #3375.Google Scholar
  224. D. D. Ang, D. E. Daykin and T. K. Sheng, On Schoenberg’s rational polygon problem, J. Austral. Math. Soc 9 (1969) 337–344; MR 39 #6816.Google Scholar
  225. A. S. Besicovitch, Rational polygons, Mathematika 6 (1959) 98; MR 22 #1557.Google Scholar
  226. D. E. Daykin, Rational polygons, Mathematika 10 (1963) 125–131; MR 30 #63.Google Scholar
  227. D. E. Daykin, Rational triangles and parallelograms, Math. Mag 38 (1965) 46–47.MathSciNetMATHGoogle Scholar
  228. L. J. Mordell, Rational quadrilaterals, J. London Math. Soc 35 (1960) 277–282.MathSciNetMATHGoogle Scholar
  229. T. K. Sheng, Rational polygons, J. Austral. Math. Soc 6 (1966) 452–459; MR 35 #137 Google Scholar
  230. T. K. Sheng and D. E. Daykin, On approximating polygons by rational polygons, Math. Mag 38 (1966) 299–300; MR 34 #7463.Google Scholar
  231. M. Misiurewicz, Ungelöste Probleme, Elem. Math 21 (1966) 90.Google Scholar
  232. H. Brocard, Question 1532, Nouv. Corresp. Math. 2 (1876) 287; Nouv. Ann. Math. (3) 4 (1885) 391.Google Scholar
  233. P. Erdös and R. Oblâth, Über diophantische Gleichungen der Form n! = xP ± yP und n! ± m! = x“, Acta Szeged 8 (1937) 241–255.Google Scholar
  234. M. Kraitchik, Recherches sur la Théorie des Nombres, t. 1, Gauthier-Villars, Paris, 1924, 38–41.Google Scholar
  235. Richard M. Pollack and Harold N. Shapiro, The next to last case of a factorial diophan- tine equation, Comm. Pure Appl. Math 26 (1973) 313–325; MR 50 #12915.Google Scholar
  236. G. J. Simmons, A factorial conjecture, J. Recreational Math 1 (1968) 38.Google Scholar
  237. H. M. Stark, Problem 23, Summer Institute on Number Theory, Stony Brook, 1969.Google Scholar
  238. Ronald Alter, The congruent number problem, Amer. Math. Monthly 87 (1980) 43–45.MathSciNetMATHGoogle Scholar
  239. R. Alter and T. B. Curtz, A note on congruent numbers, Math. Comput 28 (1974) 303–305; MR 49 #2504.Google Scholar
  240. R. Alter, T. B. Curtz and K. K. Kubota, Remarks and results on congruent numbers, Congressus Numerantium VI, Proc. 3rd S. E. Conf. Combin. Graph Theory Comput. 1972, 27–35; MR 50 #2047.Google Scholar
  241. L. Bastien, Nombres congruents, Intermédiaire des Math. 22 (1915) 231–232.Google Scholar
  242. B. J. Birch, Diophantine analysis and modular functions. Proc. Bombay Colloq. Alg. Geom. 1968.Google Scholar
  243. J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc 41 (1966) 193–291.MathSciNetGoogle Scholar
  244. L. E. Dickson, History of the Theory of Numbers, Vol. 2, Diophantine Analysis, Washington, 1920, 459–472.Google Scholar
  245. A. Genocchi, Note analitiche copra Tre Scritti Annali di Sc. Mat. e Fis. 6 (1855) 273–317.Google Scholar
  246. A. Gérardin, Nombres congruents, Intermédiaire des Math. 22 (1915) 52–53.Google Scholar
  247. H. J.Godwin, A note on congruent numbers, Math. Comput. 32 (1978) 293–295 and 33(1979)847; MR 58 #495;80c:10018 Google Scholar
  248. Jean Lagrange, Thèse d’Etat de l’Université de Reims, 1976.Google Scholar
  249. Jean Lagrange, Construction d’une table de nombres congruents, Bull. Soc. Math. France Mém. No. 49–50 (1977) 125–130; MR 58 #5498.Google Scholar
  250. L. J. Mordell, Diophantine Equations, Academic Press, London, 1969, 71–72.MATHGoogle Scholar
  251. S. Roberts, Note on a problem of Fibonacci’s, Proc. London Math. Soc. 11 (1879–80) 35–44.Google Scholar
  252. N. M. Stephens, Congruence properties of congruent numbers, Bull. London Math. Soc. 7 (1975) 182–184; MR 53 #260.Google Scholar
  253. L. J. Mordell, Research problem 6, Canad. Math. Bull 17 (1974) 149.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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