Volume 15, Issue 1 (4-2020)                   IJMSI 2020, 15(1): 15-21 | Back to browse issues page


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Shabani-Solt. H, Yusefnejad N, Janfada A S. On the Diophantine Equation x^6+ky^3=z^6+kw^3. IJMSI 2020; 15 (1) :15-21
URL: http://ijmsi.ir/article-1-1004-en.html
Abstract:  

Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is not greater than 500. We exhibit a collection of (probably infinitely many) rational numbers k for which this Diophantine equation is satisfied. Finally, appealing these observations, we conjecture that the above result is true for all rational numbers k.

Type of Study: Research paper | Subject: Special

References
1. 1. A. Choudhry, Symmetric Diophantine equations, Rocky Mountain J. Math., 34(4), (2004), 1281-1248. [DOI:10.1216/rmjm/1181069800]
2. H. Inose, On certain Kummer surface which can be realized as non-singular quartic surfaces in P^3, J. Fac. Sci. Univ. Tokyo, 23, (1476), 545-560.
3. A. S. Janfada and A. Abbaspoor, On Diophantine equations X^6+6Z^3=Y^6 ± 6W^3, Int. J. Pure and App. Math., 105(4), (2015), 709-713. [DOI:10.12732/ijpam.v105i4.10]
4. M. Kuwata, Elliptic fibrations on quartic K3 surfaces with large Picard numbers, Pacific J. Math., 171(1), (1995) 231-243. [DOI:10.2140/pjm.1995.171.231]
5. T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge University Press, 1986. [DOI:10.1017/CBO9780511566042]
6. L. C. Washington, Elliptic curves: Number theory and cryptography, Second edition, Taylor & Francis Group LLC, 2008. [DOI:10.1201/9781420071474]

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