On convergents formed from Diophantine equations

Let $\xi=\sqrt{v/u}\tanh(uv)^{-1/2}$, where $u$ and $v$ are positive integers, and let $\eta=|h(\xi)|$, where $h(t)$ is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values $\eta$ of hyperbolic functions by rationals $x/y$ such that $x$ and $y$ satisfy Diophantine equations. We show that there are infinitely many coprime integers $x$ and $y$ such that $|y\eta-x|\ll\log\log y/\log y$ and a Diophantine equation holds simultaneously relating $x$ and $y$ and some integer $z$. Conversely, all positive integers $x$ and $y$ with $y\ge c_0$ solving the Diophantine equation satisfy $|y\eta-x|\gg\log\log y/\log y$.

We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms x i · x j = x k and x i + 1 = x i has only finitely many solutions in non-negative integers x 1 , . . . , x i , then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

A method for direct construction of continued fraction elements, presenting a square root from a natural number, is proposed. This technique resembles the usual division process. Derivation of the computing procedures is based on the Euclidean algorithm, Fermat's and Euler's theorems on a sum of two squares, and formulation of the basic equation for the continued fraction elements as Diophantine equation. The solution of Pell equation is presented explicitly through the same functions, which describe the Euclidean algorithm and solutions to linear Diophantine equations. It is supposed that similar procedures could be used by Fermat when he challenged to solve the quadratic Diophantine equation for special difficult cases found by him.

The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five solutions in non-negative integers. %For example the equation $y^2=130\cdot 3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also find new examples of equations $x^2=A2^{n}+B$ having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions: \begin{equation*} \begin{array}{ll} x^2= 57\cdot 2^{n}+117440512 & n=0, 14, 16, 20, 24, 25, x^2= 165\cdot 2^{n}+26404 & n=0, 5, 7, 8, 10, 12. \end{array} \end{equation*} Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.

In Chapters 3 and 4 we were concerned with quadratic equations in two variables, but were only allowing ourselves integer solutions. An equation involving polynomials together with the constraint that we are only interested in integer solutions is called a Diophantine equation. In this sense we have been considering ‘quadratic Diophantine equations’.

In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next we turn our attention to the familiar sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and express the squares $S_k^2$, $S_m^2$, and the product $S_k S_m$ as a linear combination of power sums. These expressions, along with the above quadratic-form solution for the cubic equation, allows one to generate an infinite number of relations of the form $Q_1^3 + Q_2^3 + Q_3^3 = Q_4^3$, with each $Q_i$ being a linear combination of power sums. Also, we briefly consider the quadratic Diophantine equations $a^2 + b^2 + c^2 = d^2$ and $a^2 + b^2 = c^2$, and give a family of corresponding solutions $Q_1^2 + Q_2^2 + Q_3^2 = Q_4^2$ and $Q_1^2 + Q_2^2 = Q_3^2$ in terms of sums of powers of integers.

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .

Numerous researches have been devoted in finding the solutions , in the set of non-negative integers, of Diophantine equations of type (1), where the values p and q are fixed. In this paper, we also deal with a more generalized form, that is, equations of type (2), where n is a positive integer. We will present results that will guarantee the non-existence of solutions of such Diophantine equations in the set of positive integers. We will use the concepts of the Legendre symbol and Jacobi symbol, which were also used in the study of other types of Diophantine equations. Here, we assume that one of the exponents is odd. With these results, the problem of solving Diophantine equations of this type will become relatively easier as compared to the previous works of several authors. Moreover, we can extend the results by considering the Diophantine equations (3) in the set of positive integers.

One of the areas of Number theory that has attracted many mathematicians since antiquity is the subject of diophantine equations. A diophantine equation is a polynomial equation in two or more unknowns such that only the integer solutions are determined. No doubt that diophantine equation possess supreme beauty and it is the most powerful creation of the human spirit. A pell equation is a type of non-linear diophantine equation in the form where and square-free. The above equation is also called the Pell-Fermat equation. In Cartesian co-ordinates, this equation has the form of a hyperbola. The binary quadratic diophantine equation having the form

We know that Diophantine equations are polynomial equations with integer coefficients and they are having integer solutions. In this paper we are revisits one of the Diophantine Equation xn + yn=znin different perspective, to study some of its inherent properties. In this paper we are proven transcendental representation of above Diophantine equations is zyn2=1+2x2-1. By substituting n = 2, the quadratic Diophantine equation is satisfies Pythagorean theorem, which is having transcendental representation zy=1+2x2-1. Also we are finding all primitive and non primitive Pythagorean triples by choosing of x value from following four disjoint Sets (whose union is becomes to Set of all positive integers). A = x,y,z:zy=1+2x2-1ifxisoddprimenumberoritspowersB = x,y,z:zy=1+2x2p-122-1ifxisoddcompositeanditspowers,forsomep=1,2,3..C = x,y,z:zy=1+2x22-1ifxisgeometricpowerof2 D=x,y,z:zy=1+2x2p22-1ifxisevencompositebutnotgeometricpowerof2,forsomep=1,2,3⋯. And with using of programming coding of ‘c’ language for above transcendental representation of Diophantine equation,we are proven Fermat’s Last Theorem for n > 2.

Talking about the Diophantine analysis’ history, namely, the problem of rational solutions of Diophantine equations, we should note the longevity of the algebraic approach, which goes back to Diophantus’ “Arithmetica”. Indeed, after the European mathematicians of the second half of the XVI century became acquainted with Diophantus’ oeuvre, algebraic apparatus of variable changes, substitutions and transformations turned into the main tool of finding rational solutions of Diophantine equations. Despite the limitations of this apparatus, there were obtained important results on rational solutions of quadratic, cubic and quartic indeterminate equations in two unknowns. Detailed historico-mathematical analysis of these results was done, inter alia, by I. G. Bashmakova and her pupils. The paper examines the departure from this algebraic treatment of Diophantine equations, typical for most of the research up to the end of XIX century, towards a more general viewpoint on this subject, characterized also by radical expansion of the tools used in the Diophantine equations’ investigations. The works of A. L. Cauchy, C. G. J. Jacobi and ´E. Lucas, where this more general approach was developed, are analyzed. Special attention is paid to the works of J. J. Sylvester on Diophantine equations and the paper “On the Theory of Rational Derivation on a Cubic Curve” by W. Story, which were not in the focus of the research on history of the Diophantine analysis and where apparatus of algebraic curves was used in a pioneering way.

Diophantine equations are gradually drawing attention in the study of hydrogen spectrum, eco- nomics, Biology, quantum Hall effect, chemistry, cryptography etc. Different types of schemes are employed to find solution of Diophantine equations. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation (3x+63y=z2) is addressed in this paper to find the solution(s) in non-negative integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,8) in non-negative integers.

Number theory is the second large field of mathematics that comes to us from the Pythagoreans via Euclid. The Pythagorean theorem led mathematicians to the study of squares and sums of squares; Euclid drew attention to the primes by proving that there are infinitely many of them. Euclid’s investigations were based on the so-called Euclidean algorithm, a method for finding the greatest common divisor of two natural numbers. Common divisors are the key to basic results about prime numbers, in particular unique prime factorization, which says that each natural number factors into primes in exactly one way. Another discovery of the Pythagoreans, the irrationality of \(\sqrt{2}\), has repercussions in the world of natural numbers. Since\(\sqrt{2}\neq m/n\) for any natural numbers m, n, there is no solution of the equation \(x^2 - 2y^2 = 0\) in the natural numbers. But, surprisingly, there are natural number solutions of \(x^2 - \rm{2}y^2 = 1\), and in fact infinitely many of them. The same is true of the equation \(x^2 - Ny^2 = 1\) for any nonsquare natural number N. The latter equation, called Pell’s equation, is perhaps second in fame only to the Pythagorean equation \(x^2 + y^2 = z^2\), among equations for which integer solutions are sought. Methods for solving the Pell equation for general N were first discovered by Indian mathematicians, whose work we study in Chapter 5. Equations for which integer or rational solutions are sought are called Diophantine, after Diophantus. The methods he used to solve quadratic and cubic Diophantine equations are still of interest. We study his method for cubics in this chapter, and take it up again in Chapters 11 and 16.

<p>Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich's conjecture on finite-fold Diophantine representations are jointly inconsistent.</p>