Glasnik Matematicki, Vol. 41, No.2 (2006), 195-203.


Philip Gibbs

6 Welbeck Drive, Langdon Hills, Basildon SS16 6BU, England

Abstract.   A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.

2000 Mathematics Subject Classification.   11D09.

Key words and phrases.   Diophantine sextuple, regular Diophantine quintuple, irregular Diophantine quintuple.

Full text (PDF) (free access)

DOI: 10.3336/gm.41.2.02


  1. J. Arkin, D. C. Arney, F. R. Giordano, R. A. Kolb and G. E. Bergum, An extension of an old classical Diophantine problem, in: Applications of Fibonacci Numbers, Vol. 5, Kluwer Acad. Publ., Dordrecht, 1993, 45-48.

  2. J. Arkin, V. E. Hoggatt and E. G. Straus, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.

  3. A. Baker and H. Davenport, The equations 3x2-2=y2 and 8x2-7=z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.

  4. L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea Publishing Co., New York, 1966, 513-520.

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

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

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

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

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

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

  11. P. Fermat, Observations sur Diophante, Oeuvres de Fermat, Vol. 1 (eds. P. Tannery and C. Henry), 1891.

  12. Y. Fujita, The extensibility of Diophantine pairs {k - 1, k + 1}, preprint.

  13. P. E. Gibbs, Computer Bulletin 17 (1978), 16.

  14. T. L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra, Dover Publications, Inc., New York, 1964.

Glasnik Matematicki Home Page