arXiv subjects

Mehdi Baghalagdam

Publications and source records attributed to Mehdi Baghalagdam.

4 recordsLinked to original sources

arXiv2017

Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$?

The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer values of $h \le 5000$ and $A, B, C, D \le 100000$, except for some numbers, while a solution of this Diophantine equation is not known for arbitrary positive integer values of $h$. Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees $5$ and $2$ respectively. Also Choudhry presented some new solutions of this equation when $h$ is given by polynomials of degrees $2$, $3$, and $4$. In this paper, by using the elliptic curves theory, we study this Diophantine equation, where $h$ is a fixed arbitrary rational number. We work out some solutions of the Diophantine equation for certain values of $h$, in particular for the values which has not already been found a solution in the range where $A, B, C, D \le 100000$ by computer search. Also we present some new parametric solutions for the Diophantine equation when $h$ is given by polynomials of degrees $3$, $4$. Finally We present two conjectures such that if one of them is correct, then we may solve the above Diophantine equation for arbitrary rational number $h$.

math.NT↗

arXiv2017

On the sums of many biquadrates in two different ways

The beautiful quartic Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been studied by some mathematicians for many years. Although Choudhry, Gerardin and Piezas presented solutions of this equation for many values of $h$, the solutions were not known for arbitrary positive integer values of $h$. In a separate paper (see the arxiv), the authors completely solved the equation for arbitrary values of $h$, and worked out many examples for different values of $h$, in particular for the values which has not already been given a solution. Our method, give rise to infinitely many solutions and also infinitely many parametric solutions for the equation for arbitrary rational values of $h$. In the present paper, we use the above solutions as well as a simple idea to show that how some numbers can be written as the sums of two, three, four, five, or more biquadrates in two different ways. In particular we give examples for the sums of $2$, $3$, $\cdots$, and $10$, biquadrates expressed in two different ways.

math.NT↗

arXiv2017

On the Diophantine equations $ \sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3 $

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are fixed arbitrary nonzero integers. By our method, we may find infinitely many nontrivial positive solutions and also obtain infinitely many nontrivial parametric solutions for the Diophantine equations for every arbitrary integers $n$, $m$, $a_i$ and $b_i$.

math.NT↗

arXiv2017

On the Diophantine equation $ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 $

In this paper, by using the elliptic curves theory, we study the fourth power Diophantine equation ${ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }$, where $a_i$ and $n\geq3$ are fixed arbitrary integers. We solve the equation for some values of $a_i$ and $n=3,4$, and find nontrivial solutions for each case in natural numbers. By our method, we may find infinitely many nontrivial solutions for the above Diophantine equation and show, among the other things, that how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for every $a_i$ and $n\geq 3$, this paper will be restricted to the examples where $n=3,4$. In the end, we explain how to solve it in general cases without giving concrete examples.

math.NT↗