Glasnik Matematicki, Vol. 53, No. 1 (2018), 33-42.

STRONG EULERIAN TRIPLES

Andrej Dujella, Ivica Gusić, Vinko Petričević and Petra Tadić

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: duje@math.hr

Faculty of Chemical Engineering and Technology , University of Zagreb , Marulićev trg 19, 10000 Zagreb, Croatia
e-mail: igusic@fkit.hr

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: vpetrice@math.hr

Department of Economics and Tourism , Juraj Dobrila University of Pula , 52100 Pula, Croatia
e-mail: ptadic@unipu.hr

Abstract.   We prove that there exist infinitely many rationals a, b and c with the property that a2-1, b2-1, c2-1, ab-1, ac-1 and bc-1 are all perfect squares. This provides a solution to a variant of the problem studied by Diophantus and Euler.

2010 Mathematics Subject Classification.   11D09, 11G05.

Key words and phrases.   Eulerian triples, elliptic curves.

Full text (PDF) (free access)

DOI: 10.3336/gm.53.1.03

References:

  1. N. Adžaga, A. Dujella, D. Kreso and P. Tadić, Triples which are D(n)-sets for several n's, J. Number Theory 184 (2018), 330-341.
    MathSciNet     CrossRef

  2. J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997.
    MathSciNet    

  3. L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1966, pp. 513-520.
    MathSciNet    

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

  5. A. Dujella, A note on Diophantine quintuples, in: Algebraic number theory and Diophantine analysis (F. Halter-Koch, R. F. Tichy, eds.), Walter de Gruyter, Berlin, 2000, pp. 123-127.
    MathSciNet    

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

  7. A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples, Glas. Mat. Ser. III 42(62) (2007), 3-18.
    MathSciNet     CrossRef

  8. A. Dujella, What is ... a Diophantine m-tuple?, Notices Amer. Math. Soc. 63 (2016), 772-774.
    MathSciNet     CrossRef

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

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

  11. A. Dujella and C. Fuchs, On a problem of Diophantus for rationals, J. Number Theory 132 (2012), 2075-2083.
    MathSciNet     CrossRef

  12. A Dujella, I. Gusić and P. Tadić, The rank and generators of Kihara's elliptic curve with torsion ℤ/4ℤ over ℚ(t), Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 105-109.
    MathSciNet     CrossRef

  13. A. Dujella and M. Kazalicki, More on Diophantine sextuples, in: Number theory - Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy's 60th birthday (C. Elsholtz, P. Grabner, Eds.), Springer-Verlag, Berlin, 2017, pp. 227-235.
    MathSciNet    

  14. A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai, There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. IMRN 2017 (2) (2017), 490-508.
    MathSciNet     CrossRef

  15. A. Dujella and J. C. Peral, Elliptic curves with torsion group ℤ/8ℤ or ℤ/2ℤ × ℤ/6ℤ, in: Trends in number theory, Contemp. Math. 649 (2015), 47-62.
    MathSciNet    

  16. A Dujella, J. C. Peral and P. Tadić, Elliptic curves with torsion group ℤ/6ℤ, Glas. Mat. Ser. III 51(71) (2016), 321-333.
    MathSciNet     CrossRef

  17. A. Dujella and V. Petričević, Strong Diophantine triples, Experiment. Math. 17 (2008), 83-89.
    MathSciNet     CrossRef

  18. P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41(61) (2006), 195-203.
    MathSciNet     CrossRef

  19. P. Gibbs, A survey of rational Diophantine sextuples of low height, preprint, 2016.

  20. I. Gusić and P. Tadić, A remark on the injectivity of the specialization homomorphism, Glas. Mat. Ser. III 47(67) (2012), 265-275.
    MathSciNet     CrossRef

  21. I. Gusić and P. Tadić, Injectivity of the specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137-152.
    MathSciNet     CrossRef

  22. T. L. Heath, Diophantus of Alexandria. A study of the history of Greek algebra, With a supplement containing an account of Fermat's theorems and problems connected with Diophantine analysis and some solutions of Diophantine problems by Euler (Cambridge, 1910), Powell's Bookstore, Chicago; Martino Publishing, Mansfield Center, 2003, pp. 177-181.
    MathSciNet    

  23. A. J. MacLeod, Square Eulerian quadruples, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 20 (2016), 1-7.
    MathSciNet    

  24. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
    MathSciNet     CrossRef

  25. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1994.
    MathSciNet     CrossRef

  26. M. Stoll, Diagonal genus 5 curves, elliptic curves over ℚ(t), and rational diophantine quintuples, preprint, 2017.
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