Related papers: How to Solve Diophantine Equations
In [1] it is shown that the Diophantine equation $(k!)^n+k^n=(n!)^k+n^k$ only has the trivial solution $n=k$, and $(k!)^n-k^n=(n!)^k-n^k$ only has the solutions $n=k$, $(n, k)=(1, 2),$ and $(2, 1)$. In this article we find all solutions of…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
It is important in drawing techniques to find combinations of two straight lines and their angle bisectors whose slopes are all rational numbers. This problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$…
In this paper we present a new approach to prove effective results in Diophantine approximation. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with…
In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…
In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for…
In this article we study a system of eikonal equations. Our aim is to isolate the solutions which minimise the discontinuity set of the gradient.
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We discuss how a class of difficult kinematic problems can play an important role in an introductory course in stimulating students' reasoning on more complex physical situations. The problems presented here have an elementary analysis once…
The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many…
The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert's tenth problem. The proposed methodology uses a three layer feed forward neural network…
Let $\{x_{n}\}_{n \geq 0}$ be the balancing-like sequence defined by $x_{n+1} = A x_{n} - x_{n-1}$, for $A>2$, where $x_0 = 0$ and $x_1 = 1$. In this paper, we demonstrate how to find all the solutions of the Diophantine equation,…
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally…