A diophantine system

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.

A note on the third cuboid conjecture. Part I

The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain Diophantine equation of the order 12. In this paper a structural theorem for the solutions of this Diophantine equation is proved.

A note on the second cuboid conjecture. Part I

The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain polynomial Diophantine equation of the order 10. In this paper a structural theorem for the solutions of this Diophantine equation is proved and some examples of its application are considered.

On an ideal of multisymmetric polynomials associated with perfect cuboids

A perfect Euler cuboid is a rectangular parallelepiped with integer edges, with integer face diagonals, and with integer space diagonal as well. Finding such parallelepipeds or proving their non-existence is an old unsolved mathematical problem. Algebraically the problem is described by a system of Diophantine equations. Symmetry approach to the cuboid problem is based on the natural $S_3$ symmetry of its Diophantine equations. Factorizing these equations with respect to their $S_3$ symmetry, one gets some certain ideal within the ring of multisymmetric polynomials. In the present paper this i…

求解具有四项的无限多个纯指数丢番图方程

Solving an infinite number of purely exponential Diophantine equations with four terms

本文研究了具有连续基数的四项纯指数丢番图方程的无限族，证明了当 $n \equiv 3 \pmod{4}$ 时方程 $n^x+(n+1)^y+(n+2)^z=(n+3)^w$ 的所有正整数解，结合了初等同余论证、Baker 方法和基于 Skolem 猜想变体的算法。

Brahmagupta quadrilaterals with equal perimeters and equal areas

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.