2 citations · 6 across the 28 of their papers we have counts for
35 papers · 1 filter
Construction of parametrized families of general pseudo-bimagic and bimagic squares
Ajai Choudhry
A magic square of order is a square array of distinct integers such that the integers of every row, every column and the two diagonals have the same common sum. If a magi…
Two sets of integers such that all elements of the sumset of the two sets are perfect squares
Ajai Choudhry
This paper is concerned with the problem of finding two sets of integers, , and , such that all the sums $a_i+b_j, i=1,…
Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares
Ajai Choudhry
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…
An arbitrary number of squares whose sum, on excluding any one of them, is also a square
Ajai Choudhry
This paper is concerned with the problem of finding distinct squares such that, on excluding any one of them, the sum of the remaining squares is a square. While parametr…
Expressing three consecutive integers as sums of three cubes
Ajai Choudhry
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…
Expressing an integer as a sum of cubes of polynomials
Ajai Choudhry
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…
Circles with four rational points in geometric progression
Ajai Choudhry
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 po…
Finite sequences of integers expressible as sums of two squares
Ajai Choudhry, Bibekananda Maji
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 such that $n, n…
Two pairs of biquadrates with equal sums
Ajai Choudhry
In this paper we present a new method of solving the classical diophantine equation . Two methods of solving this equation, given by Euler, yield parametric soluti…
Rational quadrilaterals
Ajai Choudhry
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 quadrilateral…
Ideal solutions of the Tarry-Escott Problem of degree seven
Ajai Choudhry
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=…
An octic diophantine equation and related families of elliptic curves
Ajai Choudhry, Arman Shamsi Zargar
We obtain two parametric solutions of the diophantine equation where is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_…
A diophantine problem concerning third order matrices
Ajai Choudhry
In this paper we find a third order unimodular matrix, none of whose entries is or , such that when each entry of the matrix is replaced by its cube, the resulting matrix i…
New solutions of the Tarry-Escott problem of degrees 2, 3 and 5
Ajai Choudhry
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…
Some diophantine problems concerning a pair of rational triangles with a common circumradius
Ajai Choudhry
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…
A new diophantine equation involving fifth powers
Ajai Choudhry, Oliver Couto
In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation . Further, we show, using ellipti…
Pairs of equiperimeter and equiareal triangles whose sides are perfect squares
Ajai Choudhry, Arman Shamsi Zargar
In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such…
A quartic diophantine equation inspired by Brahmagupta's identity
Ajai Choudhry, Iliya Bluskov, Alexander James
In this paper we obtain several parametric solutions of the quartic diophantine equation . We also show how infinitely many parametric solut…
Matrix morphology and composition of higher degree forms with applications to diophantine equations
Ajai Choudhry
In this paper we use matrices, whose entries satisfy certain linear conditions, to obtain composition identities , where is an irreducible form, with…
A sextic diophantine chain and a related Mordell curve
Ajai Choudhry, Arman Shamsi Zargar
In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain where …
An improvement of Prouhet's 1851 result on multigrade chains
Ajai Choudhry
In 1851 Prouhet showed that when where and are positive integers, , the first consecutive positive integers can be separated into sets, each s…
A diophantine system
Ajai Choudhry
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 c…
Cyclic pentagons and hexagons with integer sides, diagonals and areas
Ajai Choudhry
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 rationa…
Brahmagupta quadrilaterals with equal perimeters and equal areas
Ajai Choudhry
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 uns…