Diophantus of Alexandria was a greek mathematician, around 200 AD, who studied mathematical problems, mostly geometrical ones, which he reduced to equations in rational integers or rational numbers. He was interested in producing at least one solution. Such equations are now called Diophantine equations. An example is y − x = 1, a solution of which is (x = 2, y = 3). More generally, a Diophanti...

In this remark, we use some properties of simple continued fractions of quadratic irrational numbers to prove that the equation x3 − 1 x− 1 = y − 1 y − 1 , x, y, n ∈ N, x > 1, y > 1, n > 3, 2 ∤ n has only the solutions (x, y, n) = (5, 2, 5) and (90, 2, 13). For any positive integer N with N > 2, let s(N) denote the number of solutions (x,m) of the equation (1) N = x − 1 x− 1 , x,m ∈ N, x ≥ 2, m...

In this paper we study the equation x+7 = y, in integers x, y, m with m ≥ 3, using a Frey curve and Ribet’s level lowering theorem. We adapt some ideas of Kraus to show that there are no solutions to the equation with m composite and m > 15, and none with m prime and 11 ≤ m < 10.

We develop an algorithm for solving a diophantine equation with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction, and rst nds short vectors satisfying the diophantine equation. The next step is to branch on linear combinations of these vectors, which either yields a vector that satis es the bound constraints or provides a proof that no such vector exist...

The contribution is focused on a control design and simulation of multi input output (MIMO) linear continuous-time systems. Suitable and efficient tools for description and controller derivation are algebraic notions as rings, polynomial matrices, and Diophantine equations. The generalized MIMO PI controller design is studied for stable and unstable systems. A unified approach through matrix Di...

1. Introduction. Interest in the questions of Diophantine definability and decid-ability goes back to a question that was posed by Hilbert: Given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z? This question, otherwise known as Hilbert's tenth problem, has been answered negatively in the work of...

This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the correspon...

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

Absact This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the co...

Solving Diophantine equation is one of the main problem in number theory for a long time. It is very difficult but wonderful. For example, it took over 300 years to see that Xn + Y n = Zn has no nontrivial integers solution when n ≥ 3. We would like to consider an easier problem: solving the Diophantine equation modulo p, where p is a prime number. We expect that this problem is easy enough to ...