Glasnik Matematicki, Vol. 55, No. 2 (2020), 195-201.

A NOTE ON THE EXPONENTIAL DIOPHANTINE EQUATION (A2n)x+(B2n)y=((A2+B2)n)z

Maohua Le and Gökhan Soydan

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

Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
e-mail: gsoydan@uludag.edu.tr

Abstract.   Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

2010 Mathematics Subject Classification.   11D61

Key words and phrases.   Ternary purely exponential Diophantine equation

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https://doi.org/10.3336/gm.55.2.03

References:

  1. W.-J. Guan and S. Che, On the Diophantine equation 2yny-x=(b+2)x-bx, J. Northwest Univ. Nat. Sci. 44 (2014), 534-536. (in Chinese)
    MathSciNet    

  2. L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Math. 1 (1955/56), 196-202.
    MathSciNet    

  3. M.-H. Le, R. Scott and R. Styer, A survey on the ternary purely exponential Diophantine equation ax+by=cz, Surv. Math. Appl. 14 (2019), 109-140.
    MathSciNet    

  4. R. Scott and R. Styer, On px-qy=c and related three term exponential Diophantine equations with prime bases, J. Number Theory 105 (2004), 212-234.
    MathSciNet     CrossRef

  5. G. Soydan, M. Demirci, I. N. Cangül and A. Togbé, On the conjecture of Jesmanowicz, Int. J. Appl. Math. Stat. 56 (2017), 46-72.
    MathSciNet    

  6. C.-F. Sun and M. Tang, On the Diophantine equation (an)x + (bn)y = (cn)z, Chinese Ann. Math. Ser. A 39 (2018), 87-94. (in Chinese)
    MathSciNet    

  7. M. Tang and Q.-H. Yang, The Diophantine equation (an)x + (2n)y = ((b+2)n)z, Colloq. Math. 132 (2013), 95-100.
    MathSciNet     CrossRef

  8. Y.-H. Yu and Z.-P. Li, The exceptional solutions of the exponential Diophantine equation (bn)x + (2n)y = ((b+2)n)z, Math. Pract. Theo. 44 (2014), 290-293. (in Chinese)
    MathSciNet    

  9. P.-Z. Yuan and Q. Han, Jeśmanowicz' conjecture and related equations, Acta Arith. 184 (2018), 37-49.
    MathSciNet     CrossRef
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