On the Diophantine equation a^x +8^y = z^2

Diophantine equations are so important in solving important real-world problems like network flow problems, pole placement problems, business investment problems, and data privacy problems that researchers are becoming more interested in developing new techniques for analyzing the nature and solutions of the various Diophantine equations. In the present study, authors examined the Diophantine equation〖10〗^X+〖400〗^Y=Z^2, where X,Y,Zare non-negative integers, for the non-negative integer solution of this equation. A result of the present study shows that it has no non-negative integer solution.

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 this paper, we study the Diophantine equation n x + 5 y = z 2 , where n is a positive integer and x, y, z are non-negative integers. We found that if n ≡ 1 (mod 4), then the Diophantine equation has no non-negative integer solution. If n ≡ 3 (mod 20) or n ≡ 7 (mod 20), then the Diophantine equation has all non-negative integer solutions, which are (n, x, y, z) = (n, 1, 0, (n+1)0.5), where (n+1)0.5 is a positive integer.

In this paper, we consider the Diophantine equation (P+ 12) X+ (P+2K)=Z2where p > 3, p are primes and k is natural number, when x, y and z are non-negative integers. It is found that the Diophantine equation has no nonnegative integer solution.

In this paper, we show that (n, x, y, z) = (2, 3, 0, 3) is the unique non-negative integer solution of the Diophantine equation n^x + 10^y = z^2 , where n is a positive integer with n ≡ 2 (mod 30) and x, y, z are non-negative integers. If n = 5, then the Diophantine equation has exactly one non-negative integer solution (x, y, z) = (3, 2, 15). We also give some conditions for non-existence of solutions of the Diophantine equation.

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

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.

In this paper, we study the Diophantine equation (p+n)^x+p^y=z^2, where p, p+n are prime numbers and n is a positive integer such that n equiv mod 4. In case p=3 and n=4, Rao{7} showed that the non-negative integer solutions are (x,y,z)=(0,1,2) and (1,2,4) In case p>3 and pequiv 3pmod4, if n-1 is a prime number and 2n-1 is not prime number, then the non-negative integer solution (x, y, z) is (0, 1,\sqrt {p+1}) or ( 1, 0, \sqrt{p+n+1}). In case pequiv 1pmod4, the non-negative integer solution (x,y,z) is also (0, 1,\sqrt {p+1}) or ( 1,0, \sqrt{p+n+1}).

Diophantische Approximationen

We compute upper and lower bounds for the approxi- mation of certain values ξ of hyperbolic and trigonometric functions by rationals x=y such that x; y satisfy Diophantine equations. We show that there are infinitely many coprime integers x; y such that |yξ − x| ≪ log log y log y and a Diophantine equation holds simultaneously relating x; y and some integer z. Conversely, all positive integers x; y with y ≥ c0 solving the Diophantine equation satisfy |yξ − x| ≫ log log y log y : Moreover, we approximate sin(πα) and cos(πα) by rationals in connection with solutions of a quadratic Diophantine equation when tan(πα=2) is a Liouville number.

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.

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 .

<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>

In this paper, we study the Diophantine equation $(p+4n)^x+p^y=z^2,$ where $n$ is a non-negative integer and $p, p+4n$ are prime numbers such that $p\equiv 7\pmod{12}$. We show that the non-negative integer solutions of such equation are $(x, y, z)\in \{(0, 1, \sqrt {p+1})\} \cup \{ (1, 0, 2\sqrt{n+\frac{p+1}{4}})\}$, where $\sqrt {p+1}$ and $\sqrt{n+\frac{p+1}{4}}$ are integers.

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smory\'nski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smory\'nski's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q.