A note on the high power diophantine equations

  • Mehdi Baghalaghdam
  • Farzali Izadi


In this paper, we solve the simultaneous diophantine equations \( x_{1}^\mu + x_{2}^\mu +\cdots + x_{n}^\mu =k \cdot (y_{1}^\mu + y_{2}^\mu +\cdots + y_{\frac{n}{k}}^\mu )\), \(\mu =1,3\), where \( n \ge 3\) and \(k \ne n\) is a divisor of n (\(\frac{n}{k}\ge 2\)), and we obtain a nontrivial parametric solution for them. Furthermore, we present a method for producing another solution for the above diophantine equation (DE) for the case \(\mu =3\), when a solution is given. We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. Also we prove that the other DE \(\sum _{i=1}^n p_{i} \cdot x_{i}^{a_i}=\sum _{j=1}^m q_{j} \cdot y_{j}^{b_j}\), has parametric solution and infinitely many solutions in nonzero integers with the condition that there is an i such that \(p_{i}=1\) and (\(a_{i},a_{1} \cdot a_{2} \cdots a_{i-1} \cdot a_{i+1} \cdots a_{n} \cdot b_{1} \cdot b_{2} \cdots b_{m})=1\), or there is a j such that \(q_{j}=1\) and \((b_{j},a_{1} \cdots a_{n} \cdot b_{1} \cdots b_{j-1} \cdot b_{j+1} \cdots b_{m})=1\). Finally, we study the DE \(x^a+y^b=z^c\).


Simultaneous diophantine equations equal sums of the cubes high power diophantine equations 

2010 Mathematics Subject Classification

Primary: 11D45 Secondary: 11D72 11D25 



The authors would like to express their hearty thanks to the anonymous referee for a careful reading of the paper and for many useful comments and remarks which improved the quality of the paper.


  1. 1.
    Bremner A, Diophantine equations and nontrivial Racah coefficients, J. Math. Phys. 27 (1986) 1181–1184MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bremner A and Brudno S, A complete determination of the zeros of weight-1 \(6j\) coefficients, J. Math. Phys. 27 (1986) 2613–2615MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beukers F, The Diophantine equation \(Ax^p+By^q=Cz^r\), Duke Math. J. 91(1) (1988) 61–68CrossRefGoogle Scholar
  4. 4.
    Choudhry A, Some Diophantine problems concerning equal sums of integers and their cubes, Hardy–Ramanujan J. 33 (2010) 59–70MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dickson L E, History of theory of numbers, vol. 2 (1992) (New York: Chelsea Publishing Company) reprintGoogle Scholar
  6. 6.
    Darmon H and Granville A, On the equation \(z^m=f(x,y)\) and \(Ax^p+By^q=Cz^r\), Bull. London Math. Soc. 27 (1995) 513–543MathSciNetCrossRefGoogle Scholar
  7. 7.
    Labarthe J J, Parametrization of the linear zeros of \(6j\) coefficients, J. Math. Phys. 27 (1986) 2964–2965MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceAzarbaijan Shahid Madani UniversityTabrizIran
  2. 2.Department of Mathematics, Faculty of ScienceUrmia UniversityUrmiaIran

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