The identity $7^{3n}+7^{3n+1}=(2\cdot 7^n)^3$ and its generalizations

On certain Diophantine equations of the form z2 = f(x)2 ± g(y)2

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.

Objectives: Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation . Methods: Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Findings: Many distinct patterns of integer solutions are obtained. Novelty: The main thrust is to illustrate different ways of obtaining various choices of solutions in integers to second-degree equations with four variables . Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Utilization of substitution strategy reduces the given equation to a ternary quadratic equation for which solutions can be found easily. Mathematics Subject Classification:11D09 Keywords: Homogeneous second degree with four variables, Solutions in integers, Factorization method, Linear transformation, Polynomial diophantine equation

A significant and important subject area of Theory of Numbers is the theory of Diophantine equations which concentrates on attempting to determine solutions in integers for higher degree and many parameters indeterminate equations. Obviously, polynomial Diophantine equations are many due to definition. Especially, the third degree Diophantine equation in two parameters falls into the theory of elliptic curves which is a developed theory. There are numerous motivating cubic equations with multiple variables which have kindled the interest among Mathematicians. For example, the representation of integers by binary cubic forms is known very little. In this context, for simplicity and brevity, refer various forms of equations of degree three having many variables in [Carmichael.,1959, Dickson.,1952, Mordell.,1969, Gopalan et.al., 2015a, Gopalan et.al., 2015b, Premalatha, Gopalan et., 2020, Premalatha et.al., 2021, Shanthi, Gopalan.,2023, Thiruniraiselvi, Gopalan.,2021, Thiruniraiselvi, Gopalan., 2024a, Thiruniraiselvi, Gopalan., 2024b, Thiruniraiselvi et.al., 2024 Vidhyalakshmi, Gopalan., 2022a, Vidhyalakshmi, Gopalan., 2022b, Vidhyalakshmi, Gopalan., 2022c]. The focus in this book is on solving multivariable third degree Diophantine equations. These types of equations are significant since they concentrate on obtaining solutions in integers which satisfy the considered algebraic equations. These solutions play a vital role in different area of mathematics & science and help us in understanding the significance of number patterns. This book contains a reasonable collection of cubic Diophantine equations with three, four, five and six unknowns. The procedure in obtaining varieties of solutions in integers for the polynomial Diophantine equations of degree three with three , four, five and six unknowns considered in this book are illustrated in an elegant manner.

Objectives: This research article focuses on finding non-zero integer solutions to the Transcendental equation with six unknowns represented by . Methods: There is no universal method for finding integer solutions to Diophantine equations involving surds. Various substitutions strategies are employed to obtain non-zero distinct integer solutions to the surd equation under consideration. Findings: Five different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the transcendental equation with six unknowns in title. Novelty: In this analysis, the transcendental equation with six unknowns given by has been reduced to either Space Pythagorean equation or non-homogeneous ternary quadratic equation through suitable transformations and for which integer solutions can be found elegantly. Keywords: Transcendental Equation With Six Unknowns, Surd Equation, Integer Solutions, Space Pythagorean Equation, Ternary Quadratic Equation

In this paper, we solve the simultaneous diophantine equations $$ x_{1}^\mu + x_{2}^\mu +\cdots + x_{n}^\mu =k \cdot (y_{1}^\mu + y_{2}^\mu +\cdots + y_{\frac{n}{k}}^\mu )$$ , $$\mu =1,3$$ , where $$ n \ge 3$$ and $$k \ne n$$ is a divisor of n ( $$\frac{n}{k}\ge 2$$ ), and we obtain a nontrivial parametric solution for them. Furthermore, we present a method for producing another solution for the above diophantine equation (DE) for the case $$\mu =3$$ , when a solution is given. We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. Also we prove that the other DE $$\sum _{i=1}^n p_{i} \cdot x_{i}^{a_i}=\sum _{j=1}^m q_{j} \cdot y_{j}^{b_j}$$ , has parametric solution and infinitely many solutions in nonzero integers with the condition that there is an i such that $$p_{i}=1$$ and ( $$a_{i},a_{1} \cdot a_{2} \cdots a_{i-1} \cdot a_{i+1} \cdots a_{n} \cdot b_{1} \cdot b_{2} \cdots b_{m})=1$$ , or there is a j such that $$q_{j}=1$$ and $$(b_{j},a_{1} \cdots a_{n} \cdot b_{1} \cdots b_{j-1} \cdot b_{j+1} \cdots b_{m})=1$$ . Finally, we study the DE $$x^a+y^b=z^c$$ .

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.

Let $$f\in \mathbb {Q}[x]$$ be a polynomial without multiple roots and $$\deg {f}\ge 2$$. We give conditions for $$f=x^2+bx+c$$ under which the Diophantine equation $$2f(x)f(y)=f(z)(f(x)+f(y))$$ has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for $$f=x^2+bx$$ with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.

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

<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>

The Undecidability of Exponential Diophantine Equations

Objectives: The purpose of this research article is to find non-zero distinct integer solutions to the system of triple equations with five unknowns represented by . Different sets of distinct integer solutions are presented. Methods: Various substitution strategies and factorization techniques are employed for finding non-zero distinct integer solutions. Findings: Four different choices of substitutions are utilized to determine many non-zero distinct integer solutions to the given system of triple equations with five unknowns. Novelty: In this analysis, the given system of triple equations with five unknowns is reduced to a single non-homogeneous cubic equation for which solutions can be found elegantly. Keywords: System of triple equations, Triple equations with five unknowns, Non-homogeneous cubic Equation, Ternary cubic equation, Integer solutions

In this paper, we study the solvability of quadratic Diophantine equations x2 - Dy2 =N, where x and y are unknown integers, and D is a positive integer that is a square free and N is a nonzero integer. We use elementary and quadratic ring methods to find integer solutions of these equations. These methods involve concepts like units, fundamental units, norms, and conjugates in quadratic rings. We propose efficient algorithms to solve the equations for cases where |N| > \sqrt{D} and |N| < \sqrt{D}. The algorithms include the continued fraction algorithm, periodic quadratic algorithm, Lagrange-Matthew-Mollin algorithm, and brute-force search. These algorithms can be implemented in programming languages. Finally, we compare the algorithms and analyze their time complexity.

A b-repdigit is a positive integer that has only one distinct digit in its base b expansion, i.e. a number of the form a(bm – 1)=(b – 1), for some positive integers m ≥ 1, b ≥ 2 and 1 ≤ a ≤ b – 1. Let r; s be non-zero integers with r ≥ 1 and s ∈ {±1}, let {Un } n ≥0 be the Lucas sequence given by Un +2 = rU n+1 + sU n , with U 0 = 0 and U 1 = 1: In this paper, we give effective bounds for the solutions of the Diophantine equation where a; b; n; k and m are positive integers such that 1 ≤ a ≤ b – 1; n; k ≥ 1 and 2 ≤ b ≤ 10. Then, we explicitly solve the above Diophantine equation for the Fibonacci, Pell and balancing sequences.

In a remark on page 80 of his classical book 250 Problems in Elementary Number Theory, Sierpiński stated that it was not known if the equation x∕y+y∕z+z∕x=4 has solutions in positive integers. Bondarenko (Investigation of a class of Diophantine equations, Ukraïn. Mat. Zh. 52:6 (2000), 831–836) gave a negative answer to Sierpiński’s remark by showing that the equation x∕y+y∕z+z∕x=4k2 does not have solutions in positive integers if 3∤k. However, Garaev (Diophantine equations of the third degree, Tr. Mat. Inst. Steklova 218 (1997), 99–108) had already proved that the equation x3+y3+z3=nxyz has no positive integer solutions if n=4k, n=8k−1, or n=22m+1(2k−1)+3, where m,k∈ℤ+, which Bondarenko’s result is a consequence of. In this paper, we shall partially extend Garaev’s result by showing that the equation x∕y+y∕z+m⋅(z∕x)=nm does not have solutions in positive integers if m is odd and 4∣n or 8∣n+1. Our method is different from Garaev’s method and has been successfully applied to several situations.