# A note on the high power diophantine equations

- 26 Downloads

## Abstract

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 \(x^a+y^b=z^c\).

## Keywords

Simultaneous diophantine equations equal sums of the cubes high power diophantine equations## 2010 Mathematics Subject Classification

Primary: 11D45 Secondary: 11D72 11D25## Notes

### Acknowledgements

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.

## References

- 1.Bremner A, Diophantine equations and nontrivial Racah coefficients,
*J. Math. Phys.***27**(1986) 1181–1184MathSciNetCrossRefGoogle Scholar - 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.Beukers F, The Diophantine equation \(Ax^p+By^q=Cz^r\),
*Duke Math. J.***91(1)**(1988) 61–68CrossRefGoogle Scholar - 4.Choudhry A, Some Diophantine problems concerning equal sums of integers and their cubes,
*Hardy–Ramanujan J.***33**(2010) 59–70MathSciNetzbMATHGoogle Scholar - 5.Dickson L E, History of theory of numbers, vol. 2 (1992) (New York: Chelsea Publishing Company) reprintGoogle Scholar
- 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.Labarthe J J, Parametrization of the linear zeros of \(6j\) coefficients,
*J. Math. Phys.***27**(1986) 2964–2965MathSciNetCrossRefGoogle Scholar