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  2. J. H. van Lint, On a set of diophantine equations, T. H.-Report 68 - WSK-03, Department of Mathematics, Technological University Eindhoven, Eindhoven, 1968.

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  53. Z. Y. Chen, The Diophantine system of equations 5x2 - 3y2 = 2, 16y2 - 5z2 = 11, J. Central China Normal Univ. Natur. Sci. 30 (1996), 381-384 (in Chinese).

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  55. A. Dujella, Generalized Fibonacci numbers and the problem of Diophantus, Fibonacci Quart. 34 (1996), 164-175.

  56. A. Dujella, Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 328 (1996), 25-30.

  57. A. Dujella, Generalization of the Problem of Diophantus and Davenport, Dissertation, University of Zagreb, 1996 (in Croatian).

  58. S. T. Thakar, The role of "T" and "S" of IMTS, Mathematics and Informatics Quarterly 6 (1996), 23-26.

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  60. Z. Y. Chen, Upper bounds for positive integer solutions of the indeterminate equations x2 - 7y2 = 2, z2 - 32y2 = -23, J. Central China Normal Univ. Natur. Sci. 31 (1997), 253-256 (in Chinese).

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  62. A. Dujella, On Diophantine quintuples, Acta Arith. 81 (1997), 69-79.

  63. A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1-10.

  64. A. Dujella, The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen 51 (1997), 311-322.

  65. A. Dujella, The problem of Diophantus and Davenport, Math. Commun. 2 (1997), 153-160.

  66. M. N. Deshpande, One property of triangular numbers, Portugaliae Math. 55 (1998), 381-383.

  67. A. Dujella, On the exceptional set in the problem of Diophantus and Davenport, Application of Fibonacci Numbers, Vol. 7 (G. E. Bergum, A. N. Philippou, A. F. Horadam, eds.), Kluwer, Dordrecht, 1998, pp. 69-76.

  68. A. Dujella, A problem of Diophantus and Pell numbers, Application of Fibonacci Numbers, Vol. 7 (G. E. Bergum, A. N. Philippou, A. F. Horadam, eds.), Kluwer, Dordrecht, 1998, pp. 61-68.

  69. A. Dujella, A problem of Diophantus and Dickson's conjecture, Number Theory, Diophantine, Computational and Algebraic Aspects (K. Gyory, A. Petho, V. T. Sos, eds.), Walter de Gruyter, Berlin, 1998, pp. 147-156.

  70. A. Dujella, Some estimates of the number of Diophantine quadruples, Publ. Math. Debrecen 53 (1998), 177-189.

  71. A. Dujella, Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), 59-67.

  72. A. Dujella, Diophantine quadruples and quintuples modulo 4, Notes Number Theory Discrete Math. 4 (1998), 160-164.

  73. A. Dujella and A. Pethoe, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49 (1998), 291-306.

  74. K. S. Kedlaya, When is (xy+1)(yz+1)(zx+1) a square?, Math. Mag. 71 (1998), 61-63.

  75. K. S. Kedlaya, Solving constrained Pell equations, Math. Comp. 67 (1998), 833-842.

  76. Z. Y. Chen, The system of Diophantine equations (m + 2)x2 - my2 = 2, (4m + 4)y2 - (m + 2)z2 = 3m + 2, J. Central China Normal Univ. Natur. Sci. 33 (1999), 1-5.

  77. A. Dujella, A proof of the Hoggatt-Bergum conjecture, Proc. Amer. Math. Soc. 127 (1999), 1999-2005.

  78. A. Dujella, An extension of an old problem of Diophantus and Euler, Fibonacci Quart. 37 (1999), 312-314.

  79. Z. Franco, Solution of Problem 10622, Amer. Math. Monthly 106 (1999), 868.

  80. P. Gibbs, A generalised Stern-Brocot tree from regular Diophantine quadruples, XXX Mathematics Archive math.NT/9903035.

  81. E. Herrmann, A. Pethoe and H. G. Zimmer, On Fermat's quadruple equations, Abh. Math. Sem. Univ. Hamburg 69 (1999), 283-291.

  82. L. Jones, A polynomial approach to a Diophantine problem, Math. Mag. 72 (1999), 52-55.

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  86. M. N. Deshpande, An interesting conjecture, The Mathematical Gazette 84 (2000), 296-298.

  87. A. Dujella, Diophantine triples and construction of high-rank elliptic curves over Q with three non-trivial 2-torsion points, Rocky Mountain J. Math. 30 (2000), 157-164.

  88. A. Dujella, A note on Diophantine quintuples, Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch, R. F. Tichy, eds.), Walter de Gruyter, Berlin, 2000, pp. 123-127.

  89. A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), 87-101.

  90. A. Dujella, Irregular Diophantine m-tuples and high-rank elliptic curves, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 66-67.

  91. A. Dujella and A. Pethoe, Integer points on a family of elliptic curves, Publ. Math. Debrecen 56 (2000), 321-335.

  92. O. Kihel, On the extendibility of the P-1-set {1,2,5}, Fibonacci Quart. 38 (2000), 464-466.

  93. Gh. Udrea, A problem of Diophantus-Fermat and Chebyshev polynomials of the first kind, Rev. Roumaine Math. Pures Appl. 45 (2000), 531-535.

  94. E. Assaf and S. Gueron, Characterization of regular Diophantine quadruples, Elem. Math. 56 (2001), 71-81.

  95. M. N. Deshpande and E. Brown, Diophantine triplets and the Pell sequence, Fibonacci Quart. 39 (2001), 242-249.

  96. A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150.

  97. A. Dujella, Diophantine m-tuples and elliptic curves, J. Theor. Nombres Bordeaux 13 (2001), 111-124.

  98. P. Gibbs, Diophantine quadruples and Cayley's hyperdeterminant, XXX Mathematics Archive math.NT/0107203

  99. K. Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), 53-65.

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  102. A. Kihel and O. Kihel, On the intersection and the extendability of Pt sets, Far East J. Math. Sci. 3 (2001), 637-643.

  103. A. F. Beardon and M. N. Deshpande, Diophantine triples, The Mathematical Gazette 86 (2002), 258-260.

  104. M. N. Deshpande, One interesting family of diophantine triplets, Internat. J. Math. Ed. Sci. Tech. 33 (2002), 253-256.

  105. M. N. Deshpande, A problem in number theory, Resonance 7(7) (2002), 89-91.

  106. M. N. Deshpande and A. Dujella, An interesting property of a reccurence related to the Fibonacci sequence, Fibonacci Quart. 40 (2002), 157-160.

  107. A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), 23-33.

  108. A. Dujella, An extension of an old problem of Diophantus and Euler. II, Fibonacci Quart. 40 (2002), 118-123.

  109. A. Dujella, C. Fuchs and R. F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002), 21-33.

  110. C. Fuchs, Quantitative finiteness results for Diophantine equations, Dissertation, TU Graz, 2002.

  111. K. Gyarmati, A. Sarkozy and C. L. Stewart, On shifted products which are powers, Mathematika 49 (2002), 227-230.

  112. M. J. Jacobson, Jr. and H. C. Williams, Modular arithmetic on elements of small norm in quadratic fields, Des. Codes and Cryptogr. 27 (2002), 93-110.

  113. Y. Bugeaud and A. Dujella, On a problem of Diophantus for higher powers, Math. Proc. Cambridge Philos. Soc. 135 (2003), 1-10.

  114. M. N. Deshpande, Families of Diophantine triplets, Bulletin of the Marathwada Mathematical Society, 4 (2003), 19-21.

  115. A. Dujella and C. Fuchs, A polynomial variant of a problem of Diophantus and Euler, Rocky Mountain J. Math. 33 (2003), 797-811.

  116. F. S. Abu Muriefah and A. Al- Rashed, On the exendibility of the Diophantine triple {1,5,c}, Internat. J. Math. Math. Sci. 33 (2004), 1737-1746.

  117. F. S. Abu Muriefah and A. Al- Rashed, Some Diophantine quadruples in the ring Z[√-2], Math. Commun. 9 (2004), 1-8.

  118. J. Almeida and A. Machiavelo, Jose Morgado, Bulletin of International Center for Mathematics 17 (2004), 24-27.

  119. J. Almeida and A. Machiavelo, Jose Morgado: in memoriam, Boletim da SPM 50 (2004), 1-18.

  120. Y. Bugeaud, On the Diophantine equation (xk-1)(yk-1) = (zk-1), Indag. Math. 15 (2004), 21-28.

  121. Y. Bugeaud and K. Gyarmati, On generalizations of a problem of Diophantus, Illinois J. Math. 48 (2004), 1105-1115.

  122. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.

  123. A. Dujella, Bounds for the size of sets with the property D(n), Glas. Mat. Ser. III 39 (2004), 199-205.

  124. A. Dujella, Diophantine quadruples and Fibonacci numbers, Bulletin of Kerala Mathematical Association 1 (2004), 133-147.

  125. A. Dujella and C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus, J. Number Theory 106 (2004), 326-344.

  126. Z. Franusic, Diophantine quadruples in the ring Z[√2], Math. Commun. 9 (2004), 141-148.

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  128. D. Saraswathy, A Study on Sets of Numbers with Diophantine Property, M.Phil. Dissertation, Department of Mathematics, Pondicherry University, Pondicherry, 2004.

  129. M. Waldschmidt, Open Diophantine problems, Moscow Math. J. 4 (2004), 245-305.

  130. R. Dietmann, C. Elsholtz, K. Gyarmati, M. Simonovits, Shifted products that are coprime pure powers, J. Combin. Theory Ser. A 111 (2005), 24-36.

  131. A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52.

  132. A. Dujella and F. Luca, Diophantine m-tuples for primes, Intern. Math. Research Notices 47 (2005), 2913-2940.

  133. A. Dujella and A. M. S. Ramasamy, Fibonacci numbers and sets with the property D(4), Bull. Belg. Math. Soc. Simon Stevin 12 (2005), 401-412.

  134. A. Filipin, Non-extendibility of D(-1)-triples of the form {1,10,c}, Internat. J. Math. Math. Sci. 35 (2005) 2217-2226.

  135. Z. Franusic, Diophantine Quadruples in Quadratic Fields, Dissertation, University of Zagreb, 2005 (in Croatian).

  136. C. Fuchs, Upper Bounds for the Solutions of Diophantine Problems, Habilitation Thesis, TU Graz, 2005.

  137. K. Gyarmati, A polynomial extension of a problem of Diophantus, Publ. Math. Debrecen 66 (2005), 389-405.

  138. F. Luca, On shifted products which are powers, Glas. Mat. Ser. III 40 (2005), 13-20.

  139. F. S. Abu Muriefah and A. Al- Rashed, On the simultaneous Diophantine equations y2 - 5x2 = 4 and z2 - 442x2 = 441, Arab. J. Sci. Eng. Sect. A Sci. 31 (2006) 207-211.

  140. A. Dujella, C. Fuchs and P. G. Walsh, Diophantine m-tuples for linear polynomials. II. Equal degrees, J. Number Theory 120 (2006), 213-228.

  141. A. Filipin, Systems of Pellian Equations and the Problem of Extension of Some Diophantine Triples, Dissertation, University of Zagreb, 2006 (in Croatian).

  142. Y. Fujita, The unique representation d = 4k(k2 - 1) in D(4)-quadruples {k-2, k+2, 4k, d}, Math. Commun. 11 (2006), 69-81.

  143. Y. Fujita, The non-extensibility of D(4k)-triples {1, 4k(k-1), 4k2+1} with |k| prime, Glas. Mat. Ser. III 41 (2006), 205-216.

  144. P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41 (2006), 195-203.

  145. P.-C. Hu, C.-C. Yang, Value Distribution Theory Related to Number Theory, Birkhäuser, Basel, 2006, pp. 335-336.

  146. Y. Li, An upper bound for the positive integer solutions of the system of Diophantine equations 7x2-5y2=2, 24y2-7z2=17, J. Chongqing Norm. Univ. Nat. Sci. Ed. 23 (2006), 33-35. (in Chinese).

  147. A. Silvester, Fast and Unconditional Principal Ideal Testing, Master's thesis, University of Calgary, 2006.

  148. Y. Bugeaud, A. Dujella and M. Mignotte, On the family of Diophantine triples {k - 1, k + 1, 16k^3 - 4k}, Glasgow Math. J. 49 (2007), 333-344.

  149. A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples, Glas. Mat. Ser. III 42 (2007), 3-18.

  150. A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(-1)-quadruple conjecture, Acta Arith. 128 (2007), 319-338.

  151. A. Dujella and F. Luca, On a problem of Diophantus with polynomials, Rocky Mountain J. Math. 37 (2007), 131-157.

  152. A. Filipin, Extensions of some parametric families of D(16)-triples, Internat. J. Math. Math. Sci. 2007 (2007), Article ID 63739, 12 pages

  153. Y. Fujita, The extensibility of D(-1)-triples {1,b,c}, Publ. Math. Debrecen 70 (2007), 103-117.

  154. Y. Fujita, The D(1)-extensions of D(-1)-triples {1, 2, c} and integer points on the attached elliptic curves, Acta Arith. 128 (2007), 349-375.

  155. K. Gyarmati and C. L. Stewart, On powers in shifted products, Glas. Mat. Ser. III 42 (2007), 273-279.

  156. T. Liqun, On the property P-1, Integers 7 (2007), #A47

  157. K. Kaygisiz, H. Senay, Constructions of some new nonextandable Pk sets, International Mathematical Forum 2 (2007), 2869-2874.

  158. A. M. S. Ramasamy, Sets and sequences linked with a question of Diophantus, Bulletin of Kerala Mathematics Association 4 (2007), 109-125.

  159. M. N. Deshpande, Diophantine triples from recurrence relations, Internat. J. Math. Ed. Sci. Tech., to appear.

  160. A. Dujella, On the number of Diophantine m-tuples, Ramanujan J. 15 (2008), 37-46.

  161. A. Dujella, C. Fuchs and F. Luca, A polynomial variant of a problem of Diophantus for pure powers, Int. J. Number Theory 4 (2008), 57-71.

  162. A. Dujella and V. Petricevic, Strong Diophantine triples, Experiment. Math. 17 (2008), 83-89.

  163. A. Filipin, On the size of sets in which xy + 4 is always a square, Rocky Mountain J. Math. 39 (2009), 1195-1224.

  164. A. Filipin, There does not exist a D(4)-sextuple, J. Number Theory 128 (2008), 1555-1565.

  165. A. Filipin and Y. Fujita, Any polynomial D(4)-quadruple is regular, Math. Commun. 13 (2008), 45-55.

  166. Z. Franusic, Diophantine quadruples in Z[√(4k+3)], Ramanujan J. 17 (2008), 77-88.

  167. Z. Franusic, A Diophantine problem in Z[(1+√d)/2], Studia Sci. Math. Hungar. 46 (2009), 103-112.

  168. Y. Fujita, The extensibility of Diophantine pairs {k-1, k+1}, J. Number Theory 128 (2008), 322-353.

  169. Y. Fujita, The Hoggatt-Bergum conjecture on D(-1)-triples {F2k+1, F2k+3, F2k+5} and integer points on the attached elliptic curves, Rocky Mountain J. Math. 39 (2009), 1907-1932.

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  171. G. Campbell and E. H. Goins, Heron triangles, Diophantine problems and elliptic curves, preprint.

  172. Y. Fujita, Any Diophantine quintuple contains a regular Diophantine quadruple, J. Number Theory 129 (2009), 1678-1697.

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  176. R. Tamura, Non-extendibility of D(-1)-triples {1,b,c}, preprint.

  177. C. Fuchs, F. Luca and L. Szalay, Diophantine triples with values in binary recurrences, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 579-608.

  178. F. Luca and L. Szalay, Fibonacci Diophantine triples, Glas. Mat. Ser. III 43 (2008), 253-264.

  179. K. Kaygisiz, H. Senay and N. Bircan, Construction of some new nonextended Pk sets, preprint.

  180. A. Dujella, Conjectures and results on the size and number of Diophantine tuples, Diophantine Analysis and Related Fields (DARF 2007/2008), AIP Conf. Proc. 976 (T. Komatsu, ed.), Amer. Inst. Phys., Melville, NY, 2008, pp. 58-61.

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  346. B. He, F. Luca, A. Togbé, Diophantine triples of Fibonacci numbers, Acta Arith. 175 (2016) , 57-70.

  347. A. Dujella, M. Kazalicki, Diophantine m-tuples in finite fields and modular forms, preprint.

  348. P. E. Gibbs, Regular rational Diophantine sextuples, preprint.

  349. P. E. Gibbs, A survey of rational Diophantine sextuples of low height, preprint.

  350. B. He, A. Togbé, V. Ziegler, There is no Diophantine quintuple, preprint.

  351. N. Adžaga, A. Filipin, On the extension of D(-8k2)-pair {8k2, 8k2 + 1}, Mosc. Math. J., to appear.

  352. D. M. Burton, Elementary Number Theory, 7th edition, McGraw-Hill, New York, 2011, p. 33.

  353. T. Cai, The Book of Numbers, World Scientific, Singapore, 2017, pp. 206-209.

  354. N. Irmak, M. Alp, Pellans sequence and its Diophantine triples, Publ. Inst. Math. 100 (2016), 259-269.

  355. M. Cipu, Y. Fujita, M. Mignotte, Two-parameter families of uniquely extendable Diophantine triples, Sci. China Math., to appear.

  356. S. Bujačić, A. Filipin, Linear forms in logarithms, in Diophantine Analysis: Course Notes from a Summer School (J. Steuding, Ed.), Birkhäuser, Basel, 2016, pp. 1-59.

  357. I. Chajda, Diofantovské n-tice, Matematika - fyzika - informatika 26 (2017), 1-6.

  358. Ö. Özer, On the some particular sets, Kirklareli University Journal of Engineering and Science 2 (2016) 99-108.


Classical references (Diophantus, Fermat, Euler)

Links to related sites

Papers of Andrej Dujella


1. Introduction
2. Diophantine quintuple conjecture
3. Sets with the property D(n)
4. Connections with Fibonacci numbers
5. Rational Diophantine m-tuples
6. Connections with elliptic curves
7. Various generalizations


Diophantine m-tuples page Andrej Dujella home page