Number Theory

Diophantine Equations

11Dxx

  1. Fadwa S. Abu Muriefah, Florian Luca, and Alain Togbé, On the Diophantine equation x2+5a13b=yn, Glasg. Math. J. 50 (2008), no. 1, 175–181.[MR]
  2. S. Akhtari, A. Togbé, and P. G. Walsh, On the equation aX4-bY2 = 2, Acta Arith. 131 (2008), no. 2, 145–169.[MR]
  3. Shabnam Akhtari, The diophantine equation aX4 – bY2 = 1, Journal fur die Reine und Angewandte Mathematik, to appear (2009), 20 pages.[arXiv]
  4. Shabnam Akhtari, The method of Thue-Siegel for binary quartic forms, preprint (2009), 35 pages.[arXiv]
  5. M. A. Bennett, N. Bruin, K. Győry, and L. Hajdu, Powers from products of consecutive terms in arithmetic progression, Proc. London Math. Soc. (3) 92 (2006), no. 2, 273–306.[MR]
  6. Michael A. Bennett, The Diophantine equation (xk-1)(yk-1)=(zk-1)t, Indag. Math. (N.S.) 18 (2007), no. 4, 507–525.[MR]
  7. Michael A. Bennett, Kálmán Győry, and Ákos Pintér, On the Diophantine equation 1k+2k+.s+xk=yn, Compos. Math. 140 (2004), no. 6, 1417–1431.[MR]
  8. A. Bérczes, A. Pethő, and V. Ziegler, Parameterized norm form equations with arithmetic progressions, J. Symbolic Comput. 41 (2006), no. 7, 790–810.[MR]
  9. Attila Bérczes and Attila Pethő, Computational experiences on norm form equations with solutions forming arithmetic progressions, Glas. Mat. Ser. III 41(61) (2006), no. 1, 1–8.[MR]
  10. A. Bremner and Jean-Joël Delorme., On equal sums of ninth powers, Math. Comp 79 (2009), 603–612.
  11. A. Bremner and N. Tzanakis, Lucas sequences whose 8th term is a square, preprint (2004), 44 pages.[arXiv]
  12. A. Bremner and N. Tzanakis, On squares in Lucas sequences, J. Number Theory 124 (2007), no. 2, 511–520.[MR]
  13. Andrew Bremner, On the equation Y2=X5 + k, Experiment. Math. 17 (2008), no. 3, 371–374.[MR]
  14. Andrew Bremner, A problem of Ozanam, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 1, 37–44.[MR/doi]
  15. Andrew Bremner and Nikos Tzanakis, On the equation Y2 = X6 + k, Annales des Sciences Mathématiques du Québec, to appear (2010), 23 pages.[arXiv]
  16. R. de la Bret'che and T. D. Browning, Manin's conjecture for quartic del Pezzo surfaces with a conic fibration, preprint (2008).[arXiv]
  17. David Brown, Primitive integral solutions to x2 + y3 = z10, preprint (2009), 11 pages.[arXiv]
  18. N. Bruin, K. Győry, L. Hajdu, and Sz. Tengely, Arithmetic progressions consisting of unlike powers, Indag. Math. (N.S.) 17 (2006), no. 4, 539–555.[MR]
  19. Nils Bruin, The primitive solutions to x3+y9=z2, J. Number Theory 111 (2005), no. 1, 179–189.[MR]
  20. Nils Bruin, Some ternary Diophantine equations of signature (n,n,2), Discovering Mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 63–91.[MR]
  21. Nils Bruin and Michael Stoll, Deciding existence of rational points on curves: an experiment, Experiment. Math. 17 (2008), no. 2, 181–189.[MR/arXiv]
  22. Nils Bruin and Michael Stoll, The Mordell-Weil sieve: Proving non-existence of rational points on curves, LMS J. Comput. Math 13 (2010), 272–306.[arXiv]
  23. Ralph H. Buchholz, Triangles with three rational medians, J. Number Theory 97 (2002), no. 1, 113–131.[MR]
  24. Ralph H. Buchholz and James A. MacDougall, Cyclic polygons with rational sides and area, J. Number Theory 128 (2008), no. 1, 17–48.[MR]
  25. Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On perfect powers in Lucas sequences, Int. J. Number Theory 1 (2005), no. 3, 309–332.[MR]
  26. Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I: Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969–1018.[MR]
  27. Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations II: The Lebesgue-Nagell equation, Compos. Math. 142 (2006), no. 1, 31–62.[MR]
  28. Yann Bugeaud, Maurice Mignotte, and Samir Siksek, A multi-Frey approach to some multi-parameter families of Diophantine equations, Canad. J. Math. 60 (2008), no. 3, 491–519.[MR/link]
  29. Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll, and Szabolcs Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), no. 8, 859–885.[MR/arXiv]
  30. I. N. Cangül, M. Demirci, G. Soydan, and N. Tzanakis., On the diophantine equation x2+5a·11b=yn, Funct. Approx. Comment. Math, to appear (2011), 21 pages.[arXiv]
  31. Imin Chen, A Diophantine equation associated to X0(5), LMS J. Comput. Math. 8 (2005), 116–121 (electronic).[MR]
  32. Imin Chen, On the equation s2 + y2p = α3, Math. Comp. 77 (2008), no. 262, 1223–1227.[MR]
  33. Imin Chen and Samir Siksek, Perfect powers expressible as sums of two cubes, J. Algebra 322 (2009), no. 3, 638–656.[MR/doi]
  34. C. Chisholm and J. A. MacDougall, Rational and Heron tetrahedra, J. Number Theory 121 (2006), no. 1, 153–185.[MR]
  35. C. Chisholm and J. A. MacDougall, Rational tetrahedra with edges in geometric progression, J. Number Theory 128 (2008), no. 2, 251–262.[MR]
  36. Mihai Cipu, Gröbner bases and Diophantine analysis, J. Symbolic Comput. 43 (2008), no. 10, 681–687.[MR]
  37. Mihai Cipu, Florian Luca, and Maurice Mignotte, Solutions of the Diophantine equation xy+yz+zx = n!, Glasg. Math. J. 50 (2008), no. 2, 217–232.[MR]
  38. Henri Cohen, Number theory: volume I: tools and diophantine equations, Springer, Berlin, 2007, pp. xxii+650.
  39. Gunther Cornelissen, Thanases Pheidas, and Karim Zahidi, Division-ample sets and the Diophantine problem for rings of integers, J. Théor. Nombres Bordeaux 17 (2005), no. 3, 727–735.[MR]
  40. Robert S. Coulter, Marie Henderson, and Felix Lazebnik, On certain combinatorial Diophantine equations and their connection to Pythagorean numbers, Acta Arith. 122 (2006), no. 4, 395–406.[MR]
  41. Luis V. Dieulefait, Solving Diophantine equations x4+y4=qzp, Acta Arith. 117 (2005), no. 3, 207–211.[MR]
  42. Shanshan Ding, Smallest irreducible of the form x2-dy2, Int. J. Number Theory (2007), 7 pages.
  43. Konstantinos A. Draziotis, Integer points on the curve Y2=X3±pkX, Math. Comp. 75 (2006), no. 255, 1493–1505 (electronic).[MR]
  44. Konstantinos Draziotis and Dimitrios Poulakis, Practical solution of the Diophantine equation y2=x(x+2apb)(x-2apb), Math. Comp. 75 (2006), no. 255, 1585–1593 (electronic).[MR]
  45. Konstantinos Draziotis and Dimitrios Poulakis, Corrigendum to "Solving the Diophantine equation y2=x(x2-n2)" [J. Number Theory 129 (1) (2009) 102–121] [mr2468473], J. Number Theory 129 (2009), no. 3, 739–740.[MR/doi]
  46. Konstantinos Draziotis and Dimitrios Poulakis, Solving the Diophantine equation y2=x(x2-n2), J. Number Theory 129 (2009), no. 1, 102–121.[MR/doi]
  47. Edray Goins, Florian Luca, and Alain Togbé, On the Diophantine equation x2+2α 5β 13γ=yn, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 430–442.[MR/doi]
  48. Enrique Gonzalez-Jimenez and Xavier Xarles, Five squares in arithmetic progression over quadratic fields, preprint (2009), 24 pages.[arXiv]
  49. K. Győry, L. Hajdu, and Á. Pintér, Perfect powers from products of consecutive terms in arithmetic progression, Compos. Math. 145 (2009), no. 4, 845–864.[MR/doi]
  50. K. Győry and Á. Pintér, Almost perfect powers in products of consecutive integers, Monatsh. Math. 145 (2005), no. 1, 19–33.[MR]
  51. K. Győry and Á. Pintér, Correction to the paper: "Almost perfect powers in products of consecutive integers", Monatsh. Math. 146 (2005), no. 4, 341.[MR]
  52. K. Győry and Á. Pintér, On the resolution of equations Axn-Byn = C in integers x,y and n ≥ 3. I, Publ. Math. Debrecen 70 (2007), no. 3-4, 483–501.[MR]
  53. Lajos Hajdu and Szabolcs Tengely, Arithmetic progressions of squares, cubes and n-th powers, Funct. Approx. Comment. Math. 41 (2009), no. 2, 129–138.[MR/arXiv]
  54. Lajos Hajdu, Szabolcs Tengely, and Robert Tijdeman, Cubes in products of terms in arithmetic progression, Publ. Math. Debrecen 74 (2009), no. 1-2, 215–232.[MR]
  55. Robin Hartshorne and Ronald van Luijk, Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces, Math. Intelligencer 30 (2008), no. 4, 4–10.[MR/arXiv]
  56. Robin Hartshorne and Ronald van Luijk, Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces, Math. Intelligencer 30 (2008), no. 4, 4–10.[MR/arXiv]
  57. Bo He and Alain Togbé, On the number of solutions of Goormaghtigh equation for given x and y, Indag. Math. (N.S.) 19 (2008), no. 1, 65–72.[MR]
  58. E. Herrmann, I. Járási, and A. Pethő, Note on: "The Diophantine equation xn=Dy2 + 1" by J. H. E. Cohn, Acta Arith. 113 (2004), no. 1, 69–76.[MR]
  59. E. Herrmann, F. Luca, and P. G. Walsh, A note on the Ramanujan-Nagell equation, Publ. Math. Debrecen 64 (2004), no. 1-2, 21–30.[MR]
  60. Emanuel Herrmann and Attila Pethő, S-integral points on elliptic curves. Notes on a paper of B. M. M. de Weger, J. Théor. Nombres Bordeaux 13 (2001), no. 2, 443–451.[MR]
  61. Akinari Hoshi, On the simplest quartic fields and related Thue equations, preprint (2010), 17 pages.[arXiv]
  62. Stephen P. Humphries and Kenneth W. Johnson, Fusions of character tables and Schur rings of abelian groups, Comm. Algebra 36 (2008), no. 4, 1437–1460.[MR]
  63. Benjamin Kane, Representing sets with sums of triangular numbers, Int. Math. Res. Not. IMRN (2009), no. 17, 3264–3285.[MR]
  64. Tünde Kovács, Combinatorial Diophantine equations—the genus 1 case, Publ. Math. Debrecen 72 (2008), no. 1-2, 243–255.[MR]
  65. Shanta Laishram, T. N. Shorey, and Szabolcs Tengely, Squares in products in arithmetic progression with at most one term omitted and common difference a prime power, Acta Arith. 135 (2008), no. 2, 143–158.[MR]
  66. A. Laradji, M. Mignotte, and N. Tzanakis, On px2 + q2n= yp and related Diophantine equations, preprint (2010), 22 pages.[arXiv]
  67. Dino Lorenzini and Thomas J. Tucker, Thue equations and the method of Coleman-Chabauty, preprint (2000), 30 pages.[arXiv]
  68. F. Luca, P. Stanica, and A. Togbé, On a Diophantine equation of Stroeker, Bull. Belg. Math. Soc. Simon Stevin (2008), 10.
  69. Florian Luca and Alain Togbé, On the Diophantine equation x2+2α13β=yn, Colloq. Math. 116 (2009), no. 1, 139–146.[MR/doi]
  70. Florian Luca and Peter Gareth Walsh, On a sequence of integers arising from simultaneous Pell equations, Funct. Approx. Comment. Math. 38 (2008), no. , part 2, 221–226.[MR/link]
  71. F. S. Abu Muriefah, F. Luca, S. Siksek, and S. Tengely, On the Diophantine equation x2+C=2yn, Int. J. Number Theory (2008).
  72. Á. Pintér, On a class of Diophantine equations related to the numbers of cells in hyperplane arrangements, J. Number Theory 129 (2009), no. 7, 1664–1668.[MR]
  73. Ákos Pintér, On the power values of power sums, J. Number Theory 125 (2007), no. 2, 412–423.[MR/doi]
  74. Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of X(7) and primitive solutions to x2+y3=z7, Duke Math. J. 137 (2007), no. 1, 103–158.[MR]
  75. Diana Savin, About certain prime numbers, preprint (2009), 9.[arXiv]
  76. Samir Siksek, The modular approach to diophantine equations, Number Theory, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007, pp. 495-527.[doi]
  77. Samir Siksek and John E. Cremona, On the Diophantine equation x2+7=ym, Acta Arith. 109 (2003), no. 2, 143–149.[MR]
  78. Samir Siksek and Michael Stoll, On a problem of Hajdu and Tengely, preprint (2009), 8 pages.[arXiv]
  79. N. P. Smart, Thue and Thue-Mahler equations over rings of integers, J. London Math. Soc. (2) 56 (1997), no. 3, 455–462.[MR]
  80. Nigel P. Smart, The Algorithmic Resolution of Diophantine Equations, London Mathematical Society Student Texts, vol. 41, Cambridge University Press, Cambridge, 1998, pp. xvi+243.[MR]
  81. Sz. Tengely, Note on the paper: "An extension of a theorem of Euler" by N. Hirata-Kohno, S. Laishram, T. N. Shorey and R. Tijdeman, Acta Arith. 134 (2008), no. 4, 329–335.[MR]
  82. Szabolcs Tengely, On the Diophantine equation x2+a2=2yp, Indag. Math. (N.S.) 15 (2004), no. 2, 291–304.[MR]
  83. Szabolcs Tengely, Effective methods for Diophantine equations, PhD Thesis, Leiden University, 2005.
  84. Szabolcs Tengely, Triangles with two integral sides, Ann. Math. Inform. 34 (2007), 89–95.[MR]
  85. P. G. Walsh, On a very particular class of Ramanujan-Nagell type equations, Far East J. Math. Sci. (FJMS) 24 (2007), no. 1, 55–58.[MR]
  86. Huilin Zhu and Jianhua Chen, Integral points on a class of elliptic curve, Wuhan Univ. J. Nat. Sci. 11 (2006), no. 3, 477–480.[MR]