Glasnik Matematicki, Vol. 59, No. 2 (2024), 299-312. \( \)

ON A CONJECTURE CONCERNING THE NUMBER OF SOLUTIONS TO \(a^x+b^y=c^z\), II

Maohua Le, Reese Scott and Robert Styer

Institute of Mathematics, Lingnan Normal College, Zhanjiang 524048, Guangdong, China
e-mail:lemaohua2008@163.com

Somerville, MA, USA

Department of Mathematics, Villanova University, Villanova, PA, USA
e-mail:robert.styer@villanova.edu

Abstract.   Let \(a\), \(b\), \(c\) be distinct primes with \(a\lt b\). Let \(N(a,b,c)\) denote the number of positive integer solutions \((x,y,z)\) of the equation \(a^x + b^y = c^z\). In a previous paper [16] it was shown that if \((a,b,c)\) is a triple of distinct primes for which \(N(a,b,c)\gt 1\) and \((a,b,c)\) is not one of the six known such triples then \((a,b,c)\) must be one of three cases. In the present paper, we eliminate two of these cases (using the special properties of certain continued fractions for one of these cases, and using a result of Dirichlet on quartic residues for the other). Then we show that the single remaining case requires severe restrictions, including the following: \(a=2\), \(b \equiv 1 \bmod 48\), \(c \equiv 17 \mod 48\), \(b \gt 10^9\), \(c \gt 10^{18}\); at least one of the multiplicative orders \(u_c(b)\) or \(u_b(c)\) must be odd (where \(u_p(n)\) is the least integer \(t\) such that \(n^t \equiv 1 \bmod p\)); 2 must be an octic residue modulo \(c\) except for one specific case; \(2 \mid v_2(b-1) \le v_2(c-1)\) (where \(v_2(n)\) satisfies \(2^{v_2(n)} \parallel n\)); there must be exactly two solutions \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) with \(1 = z_1 \lt z_2\) and either \(x_1 \ge 28\) or \(x_2 \ge 88\). These results support a conjecture put forward in [28] and improve results in [16].

2020 Mathematics Subject Classification.   11D61

Key words and phrases.   Ternary purely exponential Diophantine equation, upper bound for number of solutions

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https://doi.org/10.3336/gm.59.2.03

References:

  1. L. J. Alex, Diophantine equations related to finite groups, Comm. Algebra 4 (1976), 77–100.
    MathSciNet    CrossRef

  2. M. Bauer and M. A. Bennett, Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation, Ramanujan J. 6 (2002), 209–270.
    MathSciNet    CrossRef

  3. M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897–922.
    MathSciNet    CrossRef

  4. M. A. Bennett, Differences between perfect powers, Canad. Math. Bull. 51 (2008), 337–347.
    MathSciNet    CrossRef

  5. F. Beukers and H. P. Schlickewei, The equation \(x+y=1\) in finitely generated groups, Acta Arith. 78 (1996), 189–199.
    MathSciNet    CrossRef

  6. J. L. Brenner and L. L. Foster, Exponential Diophantine equations, Pacific J. Math. 101 (1982), 263–301.
    MathSciNet

  7. G. Lejeune Dirichlet, Ueber den biquadratischen Character der Zahl "Zwei", J. Reine Angew. Math. 57 (1860), 187–188.
    MathSciNet    CrossRef

  8. A. Gelfond, Sur la divisibilité de la différence des puissances de deux nombres entiers par une puissance d'un idéal premier, Rec. Math. [Mat. Sbornik] N.S. 7(49) (1940), 7–25.
    MathSciNet

  9. R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 2004.
    MathSciNet    CrossRef

  10. T. Hadano, On the Diophantine equation \(a^{x}=b^{y}+c^{z}\), Math. J. Okayama Univ. 19 (1976/77), 25–29.
    MathSciNet

  11. A. Herschfeld, The equation \(2^x-3^y=d\), Bull. Amer. Math. Soc. 42 (1936), 231–234.
    MathSciNet    CrossRef

  12. Y. Hu and M. Le, An upper bound for the number of solutions of ternary purely exponential diophantine equations, J. Number Theory 183 (2018), 62–73.
    MathSciNet    CrossRef

  13. Y. Hu and M. Le, An upper bound for the number of solutions of ternary purely exponential Diophantine equations II, Publ. Math. Debrecen 95 (2019), 335–354.
    MathSciNet    CrossRef

  14. M. H. Le, On the Diophantine equation \(a^x + b^y = c^z\), J. Changchun Teachers College Ser. Nat. Sci. 2 (1985), 50–62 (in Chinese).

  15. M. Le, A conjecture concerning the exponential Diophantine equation \(a^x+b^y=c^z\), Acta Arith. 106 (2003), 345–353.
    MathSciNet    CrossRef

  16. M. Le and R. Styer, On a conjecture concerning the number of solutions to \(a^x+b^y=c^z\), Bull. Aust. Math. Soc. 108 (2023), 40–49.
    MathSciNet    CrossRef

  17. K. Mahler, Zur Approximation algebraischer Zahlen. I, Math. Ann. 107 (1933), 691–730.
    MathSciNet    CrossRef

  18. T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two variables, Amer. J. Math. 146 (2024), 295–369.
    MathSciNet

  19. T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two variables II, Res. Number Theory 10 (2024), Paper No. 36.
    MathSciNet

  20. D. Z. Mo and R. Tijdeman, Exponential Diophantine equations with four terms, Indag. Math. (N.S.) 3 (1992), 47–57.
    MathSciNet    CrossRef

  21. T. Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951.
    MathSciNet

  22. T. Nagell, Sur une classe d'équations exponentielles, Ark. Mat. 3 (1958), 569–582.
    MathSciNet    CrossRef

  23. O. Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, 1950.
    MathSciNet

  24. R. Scott, On the equations \(p^x-b^y=c\) and \(a^x+b^y=c^z\), J. Number Theory 44 (1993), 153–165.
    MathSciNet    CrossRef

  25. R. Scott, Elementary treatment of \(p^a \pm p^b + 1=x^2\), (2006), arxiv:math.0608796.
    Link

  26. R. Scott and R. Styer, On \(p^x-q^y=c\) and related three term exponential Diophantine equations with prime bases, J. Number Theory 105 (2004), 212–234.
    MathSciNet    CrossRef

  27. R. Scott and R. Styer, On the generalized Pillai equation \(\pm a^x\pm b^y=c\), J. Number Theory 118 (2006), 236–265.
    MathSciNet    CrossRef

  28. R. Scott and R. Styer, Number of solutions to \(a^x+b^y=c^z\), Publ. Math. Debrecen 88 (2016), 131–138.
    MathSciNet    CrossRef

  29. C. G. Reuschle, Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabellen, Programm zum Schlusse des Schuljahrs 1855–56 am Königlichen Gymnasium zu Stuttgart, (1856), 61 pp.

  30. S. Uchiyama, On the Diophantine equation \(2^{x}=3^{y}+13^{z}\), Math. J. Okayama Univ. 19 (1976/77), 31–38.
    MathSciNet

  31. B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), 325–367.
    MathSciNet    CrossRef

  32. A. E. Western, Some criteria for the residues of eighth and other powers, Proc. London Math. Soc. (2) 9 (1911), 244–272.
    MathSciNet    CrossRef

  33. A. L. Whiteman, The sixteenth power residue character of \(2\), Canad. J. Math. 6 (1954), 364–373.
    MathSciNet    CrossRef
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