Related papers: How to Solve Diophantine Equations
In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…
In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
The paper proposes artificial intelligence technique called hill climbing to find numerical solutions of Diophantine Equations. Such equations are important as they have many applications in fields like public key cryptography, integer…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $\frac{(m-1)(m+n-2)}{2} $ or $\frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd.…
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…
This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient…
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer solutions and check for…
To give a parametrization of the Diophantine equation $A^{3}+B^{3}=C^{3}+D^{3}$ in terms of integral binary quadratic forms in a constructive way.
We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…