Hostname: page-component-77c78cf97d-cfh4f Total loading time: 0 Render date: 2026-04-27T03:39:38.038Z Has data issue: false hasContentIssue false
  • English
  • Français

Number of solutions to a special type of unit equations in two unknowns, III

Part of: Diophantine approximation, transcendental number theory Diophantine equations

Published online by Cambridge University Press:  07 August 2025

TAKAFUMI MIYAZAKI  and
ISTVÁN PINK
TAKAFUMI MIYAZAKI
Affiliation:
Gunma University, Division of Pure and Applied Science, Graduate School of Science and Technology, Tenjin-cho 1-5-1, Kiryu 376-8515, Japan e-mail: tmiyazaki@gunma-u.ac.jp
ISTVÁN PINK
Affiliation:
University of Debrecen, Institute of Mathematics, H-4002 Debrecen, P.O. Box 400, Hungary e-mail: pinki@science.unideb.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

It is conjectured 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. We develop the methods in our previous work which rely on a variety from Baker’s theory and thoroughly study the conjecture for cases where c is small relative to a or b. Using restrictions derived from the hypothesis that there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture. In particular, we find some, presumably infinitely many, new values of c with the property that for each such c the conjecture holds true except for only finitely many pairs of a and b. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or (10,3) each of which gives exactly two solutions. Further, our study with the help of the Schmidt Subspace Theorem among others more, brings strong contributions to the study of Pillai’s type Diophantine equations, notably a general and satisfactory result on a well-known conjecture of M. Bennett on the equation $a^x-b^y=c$ for any fixed positive integers a,b and c with both a and b greater than 1. Some conditional results are presented under the abc-conjecture as well.

MSC classification

Primary: 11D61: Exponential equations 11D41: Higher degree equations; Fermat's equation 11D45: Counting solutions of Diophantine equations 11J86: Linear forms in logarithms; Baker's method
Secondary: 11J61: Approximation in non-Archimedean valuations 11J87: Schmidt Subspace Theorem and applications

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 179 , Issue 3 , November 2025 , pp. 737 - 784
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Dedicated to Professor Masaaki Amou on the occasion of his retirement from Gunma University.

†

Supported by JSPS KAKENHI (No. 20K03553).

‡

Supported by the NKFIH grants 150284 and ANN130909. The research was supported in part by the University of Debrecen Program for Scientific Publication.

References

Bauer, M. and Bennett, M.A.. Applications of the hypergeometric method to the generalised Ramanujan-Nagell equation. Ramanujan J. 6(2) (2002), 209–270.Google Scholar
Bennett, M.A.. On some exponential equations of S. S. Pillai, Canad. J. Math. 53(5) (2001), 897–922.Google Scholar
Bennett, M.A.. Rational approximation to algebraic numbers of small height: The Diophantine equation $|ax^n-by^n|=1$ . J. Reine. Angew. Math. 535 (2001), 1–49.Google Scholar
Bennett, M.A.. J.S. Ellenberg and N.C. Ng, The Diophantine equation. $A^4+2^\delta B^2=C^n$ , Int. J. Number Theory 6(2) (2010), 311–338.Google Scholar
Bennett, M.A. and S. Siksek. Differences between perfect powers: prime power gaps. Algebra Number Theory. 17(10) (2023), 1789–1846.Google Scholar
Beukers, F.. On the generalised Ramanujan-Nagell equation I. Acta Arith. 38 (1981), 389–410.Google Scholar
Bilu, Y.F., Y. Bugeaud and M. Mignotte. The Problem of Catalan. (Springer, 2014).CrossRefGoogle Scholar
Bosma, W., J. Cannon and C. Playoust. The Magma Algebra System I: The User Language. J. Symb. Comp. 24(3-4) (1997), 235–265.Google Scholar
Bruin, N.. The Diophantine equations $x^2\pm y^4=\pm z^6$ and $x^2+y^8=z^3$ . Compositio Math. 118(3) (199), 305–321.CrossRefGoogle Scholar
Bugeaud, Y.. On the greatest prime factor of $ax^m+by^n$ , II. Bull. London Math. Soc. 32(6) (2000), 673–678.Google Scholar
Bugeaud, Y.. Linear forms in two m-adic logarithms and applications to Diophantine problems. Compositio Math. 132(2) (2002), 137–158.Google Scholar
Bugeaud, Y.. Linear forms in logarithms and applications. IRMA Lect. Math. Theor. Phys. vol. 28, (European Mathematical Society (EMS), Zürich, 2018).CrossRefGoogle Scholar
Bugeaud, Y. and Laurent, M.. Minoration effective de la distance p-adique entre puissances de nombres algébriques. J. Number Theory 61(2) (1996), 311–342.CrossRefGoogle Scholar
Bugeaud, Y. and F. Luca. On Pillai’s Diophantine equation. New York J. Math. 12 (2006), 193–217.Google Scholar
Cohen, H.. Number Theory. Vol. II. Analytic and Modern Tools. Grad. Texts in Math. vol. 240 (Springer, 2007).Google Scholar
Ellenberg, J.S.. Galois representations attached to $\mathbb Q$ -curves and the generalised Fermat equation $A^4+B^2=C^p$ . Amer. J. Math. 126(4) (2004), 763–787.Google Scholar
Evertse, J.H. and K. Györy. Unit Equations in Diophantine Number Theory (Cambridge University Press Cambridge, 2015).CrossRefGoogle Scholar
Guy, R.K.. Unsolved Problems in Number Theory (Springer, 2004).CrossRefGoogle Scholar
Hu, Y.-Z. and M.-H. Le. A note on ternary purely exponential diophantine equations, Acta Arith. 17(2) (2015), 173–182.Google Scholar
Hu, Y.-Z. and M.-H. Le. An upper bound for the number of solutions of ternary purely exponential diophantine equations. J. Number Theory. 183 (2018), 62–73.Google Scholar
Hu, Y.-Z. and M.-H. Le. An upper bound for the number of solutions of ternary purely exponential diophantine equations II. Publ. Math. Debrecen 95(3-4) (2019), 335–354.Google Scholar
Le, M.-H.. A note on the diophantine equation $ax^m-by^n=k$ . Indag. Math. 3 (1992), 185–191.Google Scholar
Le, M.-H.. A conjecture concerning the exponential diophantine equation $a^x+b^y=c^z$ . Acta Arith. 106(4) (2003), 345–353.Google Scholar
Le, M.-H. and T. Miyazaki. General sharp bounds for the number of solutions to purely exponential equations with three terms. Preprint: ArXiv:2407.07407 (2025).Google Scholar
LeVeque, W.J.. On the equation $a^x-b^y=1$ . Amer. J. Math. 74 (1952), 325–331.Google Scholar
Luca, F.. On the diophantine equation $p^{x_1}-p^{x_2}=q^{y_1}-q^{y_2}$ . Indag. Math. (N.S.) 14(2) (2003), 207–222.CrossRefGoogle Scholar
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(4) (2012), 373–392.Google Scholar
Mihăilescu, P.. Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine. Angew. Math. 572 (2004), 167–195.Google Scholar
Miyazaki, T. and I. Pink. Number of solutions to a special type of unit equations in two unknowns. Amer. J. Math. 146(2) (2024), 295–369.Google Scholar
Miyazaki, T. and I. Pink. Number of solutions to a special type of unit equations in two unknowns, II. Res. Number Theory 10(2) (2024), 41 pp.Google Scholar
Pillai, S.S.. On the inequality $0 \lt a^x-b^y \leq n$ . J. Indian Math. Soc. 19 (1931), 1–11.Google Scholar
S.S. Pillai, On $a^x-b^y=c$ . J. Indian Math. Soc. (N.S.) 2 (1936), 119–122 and 215.Google Scholar
Ribenboim, P.. Catalan’s Conjecture: are 8 and 9 the only Consecutive Powers?. (MA: Academic Press, Boston, 1994).Google Scholar
Ribenboim, P.. The Little Book of Bigger Primes (Springer, 1991).CrossRefGoogle Scholar
Ridout, D.. Rational approximations to algebraic numbers. Mathematika 4 (1957), 125–131.CrossRefGoogle Scholar
Schmidt, W.M.. Diophantine approximation. Lecture Notes in Math. vol. 785 (Springer, 1980).Google Scholar
Schmidt, W.M.. Diophantine approximations and Diophantine equations. Lecture Notes in Math. vol. 1467 (Springer, 1991).CrossRefGoogle Scholar
Scott, R.. On the equations $p^x-b^y=c$ and $a^x+b^y=c^z$ . J. Number Theory 44(2) (1993), 153–165.Google Scholar
Scott, R. and R. Styer. On $p^x-q^y=c$ and related three term exponential Diophantine equations with prime bases, J. Number Theory. 105(2) (2004), 212–234.Google Scholar
Scott, R. and R. Styer. On the generalised Pillai equation $\pm a^x \pm b^y=c$ . J. Number Theory. 118(2) (2006), 236–265.Google Scholar
Scott, R. and R. Styer. The number of solutions to the generalised Pillai equation $\pm r a^x \pm s b^y=c$ . J. Theor. Nombres Bordeaux. 25(1) (2013), 179–210.Google Scholar
Scott, R. and R. Styer. Bennett’s Pillai theorem with fractional bases and negative exponents allowed. J. Theor. Nombres Bordeaux. 27(1) (2015), 289–307.Google Scholar
Scott, R. and R. Styer. Number of solutions to $a^x+b^y=c^z$ . Publ. Math. Debrecen. 88(1-2) (2016), no. 1-2, 131–138.Google Scholar
Scott, R. and R. Styer. Number of solutions to $a^x+b^y=c^z$ with $\gcd(a,b) \gt 1$ . Preprint: ArXiv:2401.04197 (2024).Google Scholar
Shorey, T.N. and R. Tijdeman. Exponential Diophantine Equations. Cambridge Tracts in Math. vol. 87, (Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
Stroeker, R.J. and R. Tijdeman. Diophantine Equations. Computational Methods in Number Theory, M.C. Tract 155, (Centre for Mathematics and Computer Science, 1982), pp. 321–369.Google Scholar
Styer, R.. At most one solution to $a^x+b^y=c^z$ . for some ranges of a, b, c. Glas. Mat. Ser. III. 59(2) (2024), 277–298.Google Scholar
Terai, N.. Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations. Acta Arith. 90(1) (1999), 17–35.Google Scholar
Waldschmidt, M.. Perfect Powers: Pillai’s works and their developments. Preprint, ArXiv:0908.4031 (2009).Google Scholar