On the Diophantine Equation (p+n)^x+p^y=z^2 where p and p+n are Prime Numbers

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 explain all non-negative integer solutions for the nonlinear Diophantine equation of type 8x + py = z2 when p is an arbitrary odd prime number and incongruent with 1 modulo 8. Then, we apply the result to describe all non-negative integer solutions for the equation (8n)x + py = z2 when n ≥ 2. The results presented in this paper generalize and extend many results announced by other authors. HIGHLIGHTS Studying a new series of the equation 8x + py = z2 when p is prime which is not congruent to 1 modulo 8 Describing all non-negative integer solutions of the equation (8n)x + py = z2 which is a generalization of the equation 8x + py = z2 The equation 8x + py = z2 has at most 3 non-negative integer solutions when p is congruent to 1 modulo 8 and p ≠ 17

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.

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.

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.

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.

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.

Farkas type results are available for solutions to linear systems. These can also include restrictions such as nonnegative solutions or integer solutions. They show that the unsolvability can be reduced to a single constraint that is not solvable and this condition is implied by the original system. Such a result does not exist for integer solution to inequality system because a single inequality is always solvable in integers. But a single equation that does not have nonnegative integer solution exists. We present some cases when polynomial algorithms to find nonnegative integer solutions exist.

In this paper, we study two Diophantine equations of the type px + 9y = z2 , where p is a prime number. We find that the equation 2x + 9y = z2 has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5x + 9y = z2 has no non-negative integer solution.

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.

In this study, we solve the Diophantine equation in the title in nonnegative integers $m,n,$ and $a$. The solutions are given by $F_{1}-F_{0}=F_{2}-F_{0}=F_{3}-F_{2}=F_{3}-F_{1}=F_{4}-F_{3}=5^{0}$ and $F_{5}-F_{0}=F_{6}-F_{4}=F_{7}-F_{6}=5.$ Then we give a conjecture that says that if $a\geq 2$ and $p>7$ is prime, then the equation $F_{n}-F_{m}=p^{a}$ has no solutions in nonnegative integers $m,n.$

In this paper, authors have examined the Diophantine equation 8α + 67β = γ2, where α,β, γ are non-negative integers, for non-negative integer solutions. Authors used Catalan's conjecture for this purpose. Results of the present paper show that the Diophantine equation 8α + 67β = γ2, where α,β,γ are non-negative integers, has a unique solution in non-negative integers and this solution is given by (α, β, γ) = (1,0,3).

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. This class of equations includes as special cases many of the known examples of natural problems apparently occupying intermediate stages in the P − NP hierarchy, i.e., problems not known to be solvable in polynomial time nor to be NP-complete, for example the problem of factoring integers. Let L(F) denote the length of the binary encoding of the binary quadratic Diophantine equation F given by ax 2 +bx1x2+cx 2 +dx1+ex2+f = 0. Suppose F is such an equation having a nonnegative integer solution. This paper shows that there is a proof (i.e., “certificate”) that F has such a solution which can be checked in O(L(F) 5 logL(F)log logL(F)) bit operations. A corollary of this result is that the set � = {F : F has a nonnegative integer solution} is in

A generalization of the integer linear infeasibility problem

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions dn to a set of parametric linear Diophantine equations. We present an algorithm based on generating functions for constructing a formula for these numbers dn. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let dn(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑n≥ 0dntn and construct a formula for dn.