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Ajai Choudhry

Publications and source records attributed to Ajai Choudhry.

At least 19 recordsLinked to original sources
arXiv2025

Two sets of integers such that all elements of the sumset of the two sets are perfect squares

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is known for generating numerical examples of such sets when $m=2$ or 3 and $n$ is arbitrary. When both $m$ and $n$ exceed 2, only one two-parameter solution with $(m, n)=(4, 4)$ has been published. In this paper we obtain several multi-parameter solutions of the problem in three cases when $(m, n)$ is $(3, 3)$ or $(5, 3)$ or $(4, 4)$, and we indicate how more such solutions may be obtained.

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arXiv2025

Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares

Euler had considered the problem of finding three integers whose sum, product, and also the sum of the products of the integers, taken two at a time, are all perfect squares. Euler's methods of solving the problem lead to parametric solutions in terms of polynomials of high degrees and his numerical solutions consisted of very large integers. We obtain, by a new method, several parametric solutions given by polynomials of much smaller degrees and thus we get a number of numerically small solutions of the problem.

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arXiv2025

An arbitrary number of squares whose sum, on excluding any one of them, is also a square

This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n > 4$, only a finite number of numerical solutions, found by computer trials, are known. In fact, efforts to find parametric solutions for $n > 4$ have so far been futile. In this paper we describe two methods of obtaining parametric solutions of the problem, and we apply these methods to get several parametric solutions when $n=5, 6, 7$ or $8$. We also indicate how parametric solutions may be obtained for larger values of $n$.

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arXiv2024

Finite sequences of integers expressible as sums of two squares

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$ are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain $n$ in parametric terms such that all the four integers $n, n+1, n+2, n+4$ are sums of two squares. We also find infinitely many integers $n$ such that all the five integers $n, n+1, n+2, n+4, n+5$ are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference $4$, of five integers all of which are sums of two squares.

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arXiv2024

The diophantine equation $x^4+y^4=z^4+w^4$

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$ respectively.

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arXiv2023

Expressing three consecutive integers as sums of three cubes

This paper is concerned with the problem of expressing three consecutive integers as sums of three cubes. We give several parametric solutions of the problem. We also give somewhat trivial solutions of five or seven consecutive integers that can expressed as sums of three cubes. We conclude the paper with an open problem regarding four or more consecutive integers expressible as sums of three cubes.

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arXiv2023

Expressing an integer as a sum of cubes of polynomials

In this paper we prove that there exist infinitely many integers which can be expressed as a sum of four cubes of polynomials with integer coefficients. We give several identities that express the integers 1 and 2 as a sum of four cubes of polynomials. We also show that every integer can be expressed as a sum of five cubes of polynomials with integer coefficients.

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arXiv2023

Circles with four rational points in geometric progression

A set of rational points on a curve is said to be in geometric progression if either the abscissae or the ordinates of the points are in geometric progression. Examples of three points in geometric progression on a circle are already known. In this paper we obtain infinitely many examples of four points in geometric progression on a circle with rational radius.

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arXiv2023

Two pairs of biquadrates with equal sums

In this paper we present a new method of solving the classical diophantine equation $A^4+B^4=C^4+D^4$. Two methods of solving this equation, given by Euler, yield parametric solutions given by polynomials of degrees 7 and 13. Several other parametric solutions are now known, and with the exception of one solution of degree 11, all the published solutions are of degrees $6n+1$ for some integer $n$. The method described in this paper yields new parametric solutions of degrees 21, 39 and 75, that is, degrees that are expressible as $6n+3$.

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arXiv2022

Rational quadrilaterals

A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an elegant solution of the problem. In 1848 Kummer gave a method of generating all rational quadrilaterals. In this paper we present an alternative method of generating all rational quadrilaterals. For rational cyclic quadrilaterals, we obtain a complete parametrization and for noncyclic rational quadrilaterals, we give several parametrizations in terms of quadratic and quartic polynomials. The parametrizations obtained in this paper are simpler than the known parametrizations of rational quadrilaterals. We also describe how further parametrizations of rational quadrilaterals may be obtained.

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arXiv2022

Ideal solutions of the Tarry-Escott Problem of degree seven

In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While all the known parametric solutions of the problem, with one exception, are given by polynomials of degrees $ \geq 5$, the solutions obtained in this paper are given by quartic polynomials, and are thus simpler than almost all of the known solutions.

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arXiv2022

An octic diophantine equation and related families of elliptic curves

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.

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arXiv2021

A diophantine problem concerning third order matrices

In this paper we find a third order unimodular matrix, none of whose entries is $1$ or $-1$, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square integer matrices $(a_{ij})$, none of the integers $a_{ij}$ being $1$ or $-1$, such that $\det{(a_{ij})}=k$ and $\det{(a_{ij}^3)}=k^3$, where $k$ is a nonzero integer.

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arXiv2021

New solutions of the Tarry-Escott problem of degrees 2, 3 and 5

In this paper we obtain new parametric ideal solutions of the Tarry-Escott problem of degrees 2, 3 and 5, that is, of the diophantine systems $\sum_{i=1}^{k+1}x_i^j=\sum_{i=1}^{k+1}y_i^j,\;j=1,\,2,\,\dots,\,k$, when $k$ is 2, 3 or 5. When $k=2$, we obtain the complete ideal solution in terms of polynomials in six parameters $p, q, r, a, b$ and $c $ such that the common sums $σ_j=\sum_{i=1}^3x_i^j=\sum_{i=1}^3y_i^j$ for both $j=1$ and $j=2$ are symmetric functions of the parameters $p, q, r$ and also symmetric functions of the parameters $a, b, c$. When $k=3$, we obtain a solution in terms of polynomials in four parameters $p, q, r$ and $s$ such that the three common sums $σ_j= \sum_{i=1}^4x_i^j=\sum_{i=1}^4y_i^j, j=1, 2, 3$, are symmetric functions of all the four parameters $p, q, r$ and $s$. When $k=5$, our solution is derived from the solution already obtained when $k=2$, and the common sums, defined as in the cases when $k=2$ or 3, are either 0 or have properties similar to the case when $k=2$.

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arXiv2021

Some diophantine problems concerning a pair of rational triangles with a common circumradius

A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a common inradius or a common area. While several similar problems concerning pairs of rational triangles have been considered in the existing literature, these three problems have not been studied till now. For each of these problems, we give a parametric solution and we also indicate how additional parametric solutions of these problems may be obtained.

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arXiv2021

A new diophantine equation involving fifth powers

In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation $(x_1^5+x_2^5)(x_3^5+x_4^5)=(y_1^5+y_2^5)(y_3^5+y_4^5)$. Further, we show, using elliptic curves, that there exist infinitely many parametric solutions of the aforementioned diophantine equation, and they can be effectively computed.

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