Abstract
We explicitly solve the equation Ax n − By n = ±1 and, along the way, we obtain new results for a collection of equations Ax n − By n = z m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p α q β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. A. Bennett, “Recipes for ternary diophantine equations of signature (p, p, k),” Proc. RIMS Kokyuroku, 1319, pp. 51–55 (2003).
M. A. Bennett, “Products of consecutive integers,” Bull. London Math. Soc., 36, 683–694 (2004).
M. A. Bennett, K. Győry, M. Mignotte, and Á. Pintér, “Binomial Thue equations and polynomial powers,” Compositio Math., 142, 1103–1121 (2006).
M. A. Bennett, V. Vatsal, and S. Yazdani, “Ternary Diophantine equations of signature (p, p, 3),” Compositio Math., 140, 1399–1416 (2004).
Y. Bugeaud and K. Győry, “On binomial Thue–Mahler equations,” Period. Math. Hungar., 49, 25–34 (2004).
Y. Bugeaud, M. Mignotte, and S. Siksek, “A multi-Frey approach to some multi-parameter families of Diophantine equations,” Can. J. Math., 60, No. 3, 491–519 (2008).
H. Darmon and L. Merel, “Winding quotient and some variants of Fermat’s last theorem,” J. Reine Angew. Math., 490, 81–100 (1997).
J. H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, “S-unit equations and their applications,” in: A. Baker, ed., New Advances in Transcendence Theory, Cambridge Univ. Press (1988), pp. 110–174.
K. Győry, “Über die diophantische Gleichung x p + y p = cz p,” Publ. Math. Debrecen, 13, 301–305 (1966).
K. Győry, “On the number of solutions of linear equations in units of an algebraic number field,” Comment. Math. Helv., 54, 583–600 (1979).
K. Győry, “Some recent applications of S-unit equations,” Astérisque, 209, 17–38 (1992).
K. Győry, “Applications of unit equations,” in: Analytic Number Theory, Kyoto (1996), 62–78.
K. Győry and K. Yu, “Bounds for the solutions of S-unit equations and decomposable form equations,” Acta Arith., 123, 9–41 (2006).
K. Győry, I. Pink, and Á. Pintér, “Power values of polynomials and binomial Thue–Mahler equations,” Publ. Math. Debrecen, 65, 341–362 (2004).
K. Győry and Á. Pintér, “On the resolution of equations Ax n − By n = C in integers x, y and n ≥ 3. I,” Publ. Math. Debrecen, 70, 483–501 (2007).
E. Halberstadt and A. Kraus, “Courbes de Fermat: résultats et problémes,” J. Reine Angew. Math., 548, 167–234 (2002).
A. Kraus, “Majorations effectives pour l’équation de Fermat générelisée,” Can. J. Math., 49, 1139–1161 (1997).
M. Mignotte, “A kit of linear forms of three logarithms,” Publ. Ins. Rech. Math. Av. (Strasbourg), to appear.
K. Ribet, “On the equation a p + 2α b p + c p = 0,” Acta Arith., 79, 7–16 (1997).
J. P. Serre, “Sur les représentations modulaires de degré 2 deGal\( \left( {{{{\bar{\mathbb{Q}}}} \left/ {\mathbb{Q}} \right.}} \right) \),” Duke Math. J., 54, 179–230 (1987).
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Univ. Press, Cambridge (1986).
S. Siksek “The modular approach to Diophantine equations,” in: H. Cohen, Number Theory, Vol. II, Analytic and Modern Tools, Grad. Texts Math., Vol. 240, Springer, Berlin (2007), pp. 1107–1138.
V. G. Sprindzuk, Classical Diophantine Equations, Lect. Notes Math., Vol. 1559, Springer, Berlin (1993).
A. Wiles, “Modular elliptic curves and Fermat’s last theorem,” Ann. Math., 141, 443–551 (1995).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 5, pp. 61–77, 2010.
Rights and permissions
About this article
Cite this article
Győry, K., Pintér, Á. Binomial Thue equations, ternary equations and power values of polynomials. J Math Sci 180, 569–580 (2012). https://doi.org/10.1007/s10958-012-0656-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0656-z