arXiv subjects
Publications and source records attributed to Ajai Choudhry.
In this paper we obtain several parametric solutions of the quartic diophantine equation $(x_1^4+x_2^4)(y_1^4+y_2^4)=z_1^4+z_2^4$. We also show how infinitely many parametric solutions of this equation may be obtained by using elliptic curves.
In this paper we use matrices, whose entries satisfy certain linear conditions, to obtain composition identities $f(x_i)f(y_i)=f(z_i)$, where $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables ($n$ being $3,\,4,\,6$ or $8$), and $x_i,\,y_i,\;i=1,\,2,\,\ldots,\,n$, are independent variables while the values of $z_i,\;i=1,\,2,\,\ldots,\,n$, are given by bilinear forms in the variables $x_i,\,y_i$. When $n=2,\,4$ or $8$, we also obtain composition identities $f(x_i)f(y_i)f(z_i)=f(w_i)$ where, as before, $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables while $x_i,\,y_i,z_i,\;i=1,\,2,\,\ldots,\,n$, are independent variables and the values of $w_i,\;i=1,\,2,\,\ldots,\,n$, are given by trilinear forms in the variables $x_i,\,y_i,\,z_i$, and such that the identities cannot be derived from any identities of the type $f(x_i)f(y_i)=f(z_i)$. Further, we describe a method of obtaining both these types of composition identities for forms of higher degrees. The composition identities given in this paper have not been obtained earlier. We also obtain infinitely many solutions in positive integers of certain quartic and octic diophantine equations $f(x_1,\,\ldots,\,x_n)=1$ where $f(x_1,\,\ldots,\,x_n)$ is a form that admits a composition identity and $n=4$ or $8$.
In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $ ϕ(x_1,\,y_1,\,z_1)=ϕ(x_2,\,y_2,\,z_2)=ϕ(x_3,\,y_3,\,z_3)=k$ where $ϕ(x,\,y,\,z)$ is a sextic form defined by $ϕ(x,\,y,\,z)$ $=x^6+y^6+z^6-2x^3y^3-2x^3z^3-2y^3z^3$ and $k$ is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve $y^2=x^3+k/4$. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank $\geq 6$ using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is $\geq 6$.
In 1851 Prouhet showed that when $N=j^{k+1}$ where $j$ and $k$ are positive integers, $j \geq 2$, the first $N$ consecutive positive integers can be separated into $j$ sets, each set containing $j^k$ integers, such that the sum of the $r$-th powers of the members of each set is the same for $r=1,\,2,\,\ldots,\,k$. In this paper we show that even when $N$ has the much smaller value $2j^k$, the first $N$ consecutive positive integers can be separated into $j$ sets, each set containing $2j^{k-1}$ integers, such that the integers of each set have equal sums of $r$-th powers for $r=1,\,2,\,\ldots,\,k$. Moreover, we show that this can be done in at least $\{(j-1)!\}^{k-1}$ ways. We also show that there are infinitely many other positive integers $N=js$ such that the first $N$ consecutive positive integers can similarly be separated into $j$ sets of integers, each set containing $s$ integers, with equal sums of $r$-th powers for $r=1,\,2,\,\ldots,\,k$, with the value of $k$ depending on the integer $N$.
In this paper we find a parametric solution to the hitherto unsolved problem of finding three positive integers such that their sum, the sum of their squares and the sum of their cubes are simultaneously perfect squares.
In this paper we obtain cyclic pentagons and hexagons with rational sides, diagonals and area all of which are expressed in terms of rational functions of several arbitrary rational parameters. On suitable scaling, we obtain cyclic pentagons and hexagons whose sides, diagonals and area are all given by integers.
A cyclic quadrilateral is called a Brahmagupta quadrilateral if its four sides, the two diagonals and the area are all given by integers. In this paper we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with equal perimeters and equal areas. We obtain two parametric solutions of the problem -- the first solution generates examples in which each quadrilateral has two equal sides while the second solution gives quadrilaterals all of whose sides are unequal. We also show how more parametric solutions of the problem may be obtained.
While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb{Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct several parametrized families of elliptic curves with their generic ranks ranging from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve $\leq 13$. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves with much higher rank.
In this paper we obtain a new parametric solution of the problem of finding two triads of biquadrates with equal sums and equal products.
This paper is concerned with the diophantine equation $\sum_{i=1}^na_ix_i^4= \sum_{i=1}^na_iy_i^4$ where $n \geq 3$ and $a_i,\,i=1,\,2,\,\ldots,\,n$, are arbitrary integers. While a method of obtaining numerical solutions of such an equation has recently been given, it seems that an explicit parametric of this diophantine equation has not yet been published. We obtain a multi-parameter solution of this equation for arbitrary values of $a_i$ and for any positive integer $n \geq 3$, and deduce specific solutions when $n=3$ and $n=4$. The numerical solutions thus obtained are much smaller than the integer solutions of such equations obtained earlier.
In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine equations, in five or more variables, with one of the equations being of degree $\geq 4$. We show that, under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, some examples being a sextic equation in four variables, a tenth degree equation in six variables, and two simultaneous equations of degrees four and six in six variables. The method of solving these homogeneous equations also simultaneously yields arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. In contrast to existing methods, we obtain the arbitrarily large number of solutions without finding a parametric solution of the equations under consideration and without relating the solutions to rational points on an elliptic curve of positive rank. It appears from the examples given in the paper that there may exist projective varieties on which there are an arbitrarily large number of integer points and on which a curve of genus 0 or 1 does not exist.
In this paper we describe a new method of obtaining ideal solutions of the well-known Tarry-Escott problem, that is, the problem of finding two distinct sets of integers $x_i,\;i=1,\,2,\,\dots,\,k+1$ and $y_i,\;i=1,\,2,\,\dots,\,k+1$ such that $ \sum_{i=1}^{k+1} x_i^r = \sum_{i=1}^{k+1} y_i^r,\;\;\;r = 1,\,2,\,\dots,\,k$, where $k$ is a given positive integer. When $k > 3$, only a limited number of parametric/ numerical ideal solutions of the Tarry-Escott problem are known. In this paper, by applying the new method mentioned above, we find several new parametric ideal solutions of the problem when $k \leq 7$. The ideal solutions obtained by this new approach are more general and very frequently, simpler than the ideal solutions obtained by the earlier methods. We also obtain new parametric solutions of certain diophantine systems that are closely related to the Tarry-Escott problem. These solutions are also more general and simpler than the solutions of these diophantine systems published earlier.
Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not known, Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees 5 and 2 respectively. In this paper, we present several new solutions of this equation when $h$ is given by polynomials of degrees $2,\;3$ and 4.
An elliptic curve defined by an equation of the type $y^2=x^3+d$ is called a Mordell curve. We obtain a parametrised family of Mordell curves whose rank, in general, is at least three, and whose torsion group is $\mathbb{Z}/3\mathbb{Z}$.
In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\, 3$ or 4 to produce numerical curios involving square roots, cube roots and fourth roots, and as another application of these solutions, we show how to construct examples of geometric series with an interesting property.
This paper is concerned with the diophantine system, $\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\, r=1,\,2,\,\ldots,\,k, $ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as small as possible. We define $β(k)$ to be the minimum value of $s_1+s_2$ for which there exists a nontrivial solution of this diophantine system. We find nontrivial integer solutions of this diophantine system when $k < 6$, and thereby show that $β(2) =4,\;\, β(3) = 6,\;\, 7 \leq β(4) \leq 8$ and $8 \leq β(5) \leq 10$.
In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this result to establish a necessary, though not sufficient, condition for the solvability of the simultaneous quadratic diophantine equations $Q_j(x_1,\,x_2,\,x_3,\,x_4)=0,\;j=1,\,2,$ and give a method of obtaining their complete solution. In general, when these two equations have a rational solution, they represent an elliptic curve but we show that there are several cases in which their complete solution may be expressed by a finite number of parametric solutions and/ or a finite number of primitive integer solutions. Finally we relate the solutions of the quartic equation $y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4$ to the solutions of a pair of quadratic diophantine equations, and thereby obtain new formulae for deriving rational solutions of the aforementioned quartic equation starting from one or two known solutions.
In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over $\Q$ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.