Abstract
This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a, b and c all greater than 1 there is at most one solution to the equation \(a^x+b^y=c^z\) in positive integers x, y and z, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo c on a and b. As applications the conjecture is confirmed to be true if c takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott (J Number Theory 44(2):153-165, 1993) solving the conjecture for \(c=2\) in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of c not being perfect powers. The main novelty is to apply a special type of the p-adic analogue to Baker’s theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker’s theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation.
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References
Bennett, M.A.: On some exponential equations of S. S. Pillai. Can. J. Math. 53, 897–922 (2001)
Bennett, M.A.: Pillai’s conjecture revisited. J. Number Theory 98, 228–235 (2003)
Bennett, M.A., Ellenberg, J.S., Ng, N.C.: The Diophantine equation \(A^4+2^\delta B^2=C^n\). Int. J. Number Theory 6, 311–338 (2010)
Bennett, M.A., Mihăilescu, P., Siksek, S.: The generalized Fermat equation. In: Open Problems in Mathematics, pp. 173–205. Springer (2016)
Bertók, C., Hajdu, L.: A Hasse-type principle for exponential diophantine equations and its applications. Math. Comp. 85, 849–860 (2016)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comp. 85, 849–860 (2016)
Bruin, N.: The Diophantine equations \(x^2\pm y^4=\pm z^6\) and \(x^2+y^8=z^3\). Compos. Math. 118, 305–321 (1999)
Bugeaud, Y.: Linear forms in two \(m\)-adic logarithms and applications to Diophantine problems. Compos. Math. 132, 137–158 (2002)
Bugeaud, Y.: Linear Forms in Logarithms and Applications. IRMA Lectures in Mathematics and Theoretical Physics, vol. 28. European Mathematical Society (2017)
Bugeaud, Y., Laurent, M.: Minoration effective de la distance \(p\)-adique entre puissances de nombres algébriques. J. Number Theory 61, 311–342 (1996)
Cao, Z.F., Dong, X.L.: An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture. Acta Arith. 110, 153–164 (2003)
Cipu, M., Mignotte, M.: On a conjecture on exponential Diophantine equations. Acta Arith. 140, 251–269 (2009)
Cohen, H.: Number Theory. Vol. II: Analytic and Modern Tools, Grad. Texts in Math., vol. 240. Springer, Berlin (2007)
Ellenberg, J.S.: Galois representations attached to \({{\mathbb{Q} }}\)-curves and the generalized Fermat equation \(A^4+B^2=C^p\). Am. J. Math. 126(4), 763–787 (2004)
Evertse, J.H., Győry, K.: Unit Equations in Diophantine Number Theory. Cambridge University Press, Cambridge (2015)
Guy, R.K.: Unsolved Problems in Number Theory. Springer, Berlin (2004)
Hu, Y., Le, M.: A note on ternary purely exponential diophantine equations. Acta Arith. 171, 173–182 (2015)
Hu, Y., Le, M.: An upper bound for the number of solutions of ternary purely exponential diophantine equations. J. Number Theory 183, 62–73 (2018)
Hu, Y., Le, M.: An upper bound for the number of solutions of ternary purely exponential diophantine equations II. Publ. Math. Debrecen 95, 335–354 (2019)
Laurent, M., Mignotte, M., Nesterenko, Y.: Formes linéaires en deux logarithmes et déterminants dínterpolation. J. Number Theory 55, 285–321 (1995)
Luca, F.: On the diophantine equation \(p^{x_1}-p^{x_2}=q^{y_1}-q^{y_2}\). Indag. Math. (N.S.) 14, 207–222 (2003)
Luca, F.: On the system of Diophantine equations \(a^2+b^2=(m^2+1)^r\) and \(a^x+b^y=(m^2+1)^z\). Acta Arith. 153, 373–392 (2012)
Mihăilescu, P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)
Miyazaki, T.: Exceptional cases of Terai’s conjecture on Diophantine equations. Arch. Math. (Basel) 95, 519–527 (2010)
Miyazaki, T.: Contributions to some conjectures on a ternary exponential Diophantine equation. Acta Arith. 186, 1–36 (2018)
Miyazaki, T., Pink, I.: Number of solutions to a special type of unit equations in two unknowns. Am. J. Math. (to appear)
Nagell, T.: Sur une classe d’équations exponentielles. Ark. Mat. 3, 569–582 (1958). ((in French))
Nakamula, K., Pethő, A.: Squares in binary recurrence sequences. In: Number Theory (Eger, 1996), pp. 409–421. de Gruyter (1998)
Pillai, S.S.: On the inequality \(0<a^x-b^y \le n\). J. Indian Math. Soc. 19, 1–11 (1931)
Pillai, S.S.: On \(a^x-b^y=c\). J. Indian Math. Soc. (N.S.) 2, 119–122, 215 (1936)
Scott, R.: On the equations \(p^x-b^y=c\) and \(a^x+b^y=c^z\). J. Number Theory 44, 153–165 (1993)
Scott, R., Styer, R.: On \(p^x-q^y=c\) and related three term exponential Diophantine equations with prime bases. J. Number Theory 105, 212–234 (2004)
Scott, R., Styer, R.: Number of solutions to \(a^x+b^y=c^z\). Publ. Math. Debrecen 88, 131–138 (2016)
Shorey, T.N., Tijdeman, R.: Exponential Diophantine Equations. Cambridge University Press, Cambridge (1986)
Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. 3, 265–284 (1892)
Acknowledgements
We would like to thank the anonymous referee for the helpful remarks and suggestions. We are also grateful to Mihai Cipu, Reese Scott, Robert Styer and Masaki Sudo for their many comments and remarks which improved an earlier draft.
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Dedicated to Professor Nobuhiro Terai on the occasion of his 60th birthday.
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The first author is supported by JSPS KAKENHI (No. 20K03553). The second author was supported by the NKFIH Grants ANN130909 and K128088.
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Miyazaki, T., Pink, I. Number of solutions to a special type of unit equations in two unknowns, II. Res. number theory 10, 36 (2024). https://doi.org/10.1007/s40993-024-00524-7
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DOI: https://doi.org/10.1007/s40993-024-00524-7