On Terai’s exponential equation with two finite integer parameters

Takafumi Miyazaki
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 84–107
DOI: 10.7546/nntdm.2019.25.1.84-107
Full paper (PDF, 247 Kb)

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Authors and affiliations

Takafumi Miyazaki
Division of Pure and Applied Science
Faculty of Science and Technology
Gunma University
1-5-1 Tenjin-cho, Kiryu, Gunma, Japan

Abstract

Let r be an integer with r > 1, and m be an even positive integer. Define integers A and B by the equation A + B √−1 = (m + √−1)^r. It is proven by F. Luca in 2012 that the equation |A|^x + |B|^y = (m² + 1)^z does not hold for any triple (x, y, z) of positive integers not equal to (2, 2, r), whenever r or m exceeds some effectively computable absolute constant. In our previous work, we estimated this constant explicitly. Here that estimate is substantially improved.

Keywords

• Exponential Diophantine equation
• Linear forms in logarithms of algebraic numbers

2010 Mathematics Subject Classification

• 11D61
• 11J86

References

1. Bugeaud, Y. (1999). Linear forms in p-adic logarithms and the Diophantine equation (xⁿ – 1)/(x – 1) = y^q. Math. Proc. Cambridge Phil. Soc., 127 (3), 373–381.
2. Cao, Z. & Dong, X. (2002). On the Terai-Jeśmanowicz conjecture. Publ. Math. Debrecen, 61 (3–4), 253–265.
3. Cao, Z. & Dong, X. (2003). An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture. Acta Arith., 110 (2), 153–164.
4. Cipu, M. & Mignotte, M. (2009). On a conjecture on exponential Diophantine equations. Acta Arith., 140 (3), 251–269.
5. Jeśmanowicz, L. (1955/56). Several remarks on Pythagorean numbers. Wiadom. Mat., 1 (2), 196–202 (in Polish).
6. Laurent, M. (2008). Linear forms in two logarithms and interpolation determinants II. Acta Arith., 133 (4), 325–348.
7. Laurent, M., Mignotte, M. & Nesterenko, Y. (1995). Formes linéaires en deux logarithmes et déterminants dínterpolation. J. Number Theory, 55 (2), 285–321.
8. Le, M. (2003). A conjecture concerning the exponential Diophantine equation a^x + b^y = c^z. Acta Arith., 106 (4), 345–353.
9. Le, M., Togbe, A. & Zhu, H. (2014). On a pure ternary exponential Diophantine equation. Publ. Math. Debrecen, 85 (3–4), 395–411.
10. Lu, W. (1959). On the Pythagorean numbers 4n² – 1, 4n and 4n² + 1. Acta Sci. Natur. Univ. Szechuan, 2, 39–42 (in Chinese).
11. Luca, F. (2012). On the system of Diophantine equations a² + b² = (m² + 1)^r and a^x + b^y = (m² + 1)^z. Acta Arith., 153 (4), 373–392.
12. Miyazaki, T. (2011). Terai’s conjecture on exponential Diophantine equations. Int. J. Number Theory, 7 (4), 981–999.
13. Miyazaki, T. (2014). A note on the article by F. Luca “On the system of Diophantine equations a² + b² = (m² + 1)^r and a^x + b^y = (m² + 1)^z” (Acta Arith. 153 (2012), 373–392). Acta Arith., 164 (1), 31–42.
14. Scott, R. & Styer, R. (2016). Number of solutions to a^x + b^y = c^z. Publ. Math. Debrecen, 88 (1–2), 131–138.
15. Terai, N. (1994). The Diophantine equation a^x + b^y = c^z. Proc. Japan Acad. Ser. A Math. Sci., 70 (1), 22–26.
16. Terai, N. (1995). The Diophantine equation a^x + b^y = c^z II. Proc. Japan Acad. Ser. A Math. Sci., 71 (6), 109–110.
17. Terai, N. (1996). The Diophantine equation a^x + b^y = c^z III. Proc. Japan Acad. Ser. A Math. Sci., 72 (1), 20–22.
18. Terai, N. (1999). Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations. Acta Arith., 90 (1), 17–35.