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We consider the problem of subharmonic solutions and unbounded solutions for a class of indefinite sublinear equations of the form x ″ + f (t , x) = p (t). We overcome the difficulty due to the indefiniteness of the nonlinearity and establish a new approach to study the spiral property of solutions through precise estimations of increments and geometric analysis on the phase plane. Focusing on a typical model equation x ″ + a (t) x 1 / 3 = 0 with a (t) being sign-changing and 2 π -periodic, we can show that it has infinitely many subharmonic solutions when ∫ 0 2 π a (t) d t > 0 , with the amplitude becoming larger and larger, and its large amplitude solutions are unbounded when ∫ 0 2 π a (t) d t < 0. [ABSTRACT FROM AUTHOR]

We give an algorithm that produces all solutions of the equation ∑ i = 1 n 1 / x i = 1 in integers of the form 2 a k b , where k is a fixed positive integer that is not a power of 2, a is an element of { 0 , 1 , 2 } that can vary from term to term, and b is a nonnegative integer that can vary from term to term. We also completely characterize the pairs (k , n) for which this equation has a nontrivial solution in integers of this form. [ABSTRACT FROM AUTHOR]

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected. [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the Diophantine equation from the title. [ABSTRACT FROM AUTHOR]

Here we are interested in the familiar identity but only in as far as it provides the solution (1, 2, ..., n) of the Diophantine equation (1) where n and the aj are positive integers. In fact, not only is (1, 2, ..., n) a solution of (1), it has the remarkable property that, up to a permutation of its entries, it is the only solution of (1) in which the a j are distinct (see Section 4). Thus any other solution will necessarily contain repetitions among the a j and, for this reason, a solution must be defined to be a sequence (which allows repetitions) rather than a set. However, we shall not distinguish between two solutions whose entries are permutations of each other. [ABSTRACT FROM AUTHOR]

Let F n n ≥ 0 and M n n ≥ 0 be the Fibonacci and Mulatu sequences. Let b be a positive integer such that b ≥ 2. In this paper, we prove that the following Diophantine equation F n + ϵ M m = ρ (b ± 1) b ℓ ± 1 , where (ϵ , ρ) ∈ { (1 , 1) , (- 1 , 1) , (- 1 , - 1) } have only finitely many solutions in non-negative integers n, m, b and positive integer ℓ. Additionally, we present a method to determine all solutions within the range 2 ≤ b ≤ 12. All this is done using linear forms in logarithms of algebraic numbers and Dujella–Pethő’s reduction method. [ABSTRACT FROM AUTHOR]

This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method's effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach. [ABSTRACT FROM AUTHOR]

We provide the sufficient and necessary conditions for the existence of a solution to the Diophantine equation 4x' +(m² - 1)x = y². More- over, for an even positive integer m, we show that this equation can only be solved if and only if m is a prime power of 2. Furthermore, we provide all of its solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Diophantine equation ax + (a+5b)y = z² when α z² 1 (mod5) and b is a positive integer. We establish that the equation has no solutions in positive integers x, y and z. We start with the Diophantine equation px + (p+5a)y = z² where p and p+5a are both primes and p z² 1 (mod5) and a is a positive integer. [ABSTRACT FROM AUTHOR]

This paper investigates the exponential Diophantine equation 8x + ny = z², where n > 1 is an odd positive integer. We characterize solutions for the base cases (x = 0 or y = 0) and describe, based on implications of Bennett and Skinner's theorem, that no solutions exist for y > 2 in certain cases. For y = 1 and y = 2, we employ elliptic curve methods, focusing on the equations zz² = tz²+n and zz² = t³+nz², where t = 2x. This work generalizes known results for specific cases and provides insights into this class of Diophantine equations and their associated elliptic curves. [ABSTRACT FROM AUTHOR]

The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan's conjecture and Congruence theory. The Diophantine equation 3x + 15y = x² is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 4) in non-negative integers. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Diophantine equation (pn)x + (4mp + 1)y = z² in N for p = 3 and 1+4mp are prime integers. Concretely, using the congruent method, we prove that this equation has no non-negative solutions if p > 3. For the case p = 3, we will show that this equation has no solutions if m > 1. Furthermore, in this case, when m = 1 using the elliptic curves, we will show that this equation has only solution (x, y, z) = (3, 2, 14) if n = 1 and (x, y, z) = (1, 2, 14) if n = 3. [ABSTRACT FROM AUTHOR]

This work is concerned with the planar indefinite Schrödinger equation with nonlinearity $ f(x, t) $ has supercritical exponential growth on $ t $, which behaviors as $ \exp\left[\left(\beta+|x|^{\alpha}\right) t^{2}\right] $ with $ \beta $, $ \alpha>0 $. Previous results in this framework required $ f $ to be subcritical or critical exponential growth, thus missing the paramount case of supercritical exponential growth. The purpose of this paper is to fill this gap and obtain a ground state solution. [ABSTRACT FROM AUTHOR]

This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.

In this article, the author shows that the equilibrium equations of any one-dimensional theory can be obtained by direct manipulation of the indefinite 3D elasticity equations: σ i j , i + p j = 0. It is found that there is no need to refer to Newton-type or Lagrange-type equations to write the equilibrium in terms of stress resultants. Such 'direct manipulation' is illustrated by considering various problems, from the simplest to the most complex. First, the bar case is analyzed and the well-known equilibrium normal stress resultant equation is obtained: N z z , z + q z = 0 ; second, the Timoshenko beam theory is tackled; finally, the case of arbitrary higher order beam theories based on the Carrera Unified Formulation is treated. The equations of a new, unprecedented higher-order beam theory are derived directly from the indefinite equilibrium equations as an example of the capabilities of the proposed method. The manipulation introduced consists of the following points: (i) the 3D elasticity is first written for each degree of freedom (dof) of the beam theories; (ii) the respective stress resultants are used to replace stress components in the latter equations; (iii) the derivatives on the beam axis z do not affect the nature of the indefinite equilibrium equations and these derivatives are applied directly to the respective stress resultants; (iv) the derivatives on the section coordinate x – y should 1 – change the sign of the respective stress resultants, 2 – be applied to the functions of x, y used to define the stress resultants. These points have been proved by referring to the principle of virtual work and splitting the differential operators of the strain-displacement relations into two parts: over the section derivatives along the beam axis derivatives. Some possible further generalizations are discussed in the appendix; in particular, the extension to is briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this paper are observations regarding a previously known upper bound on the Frobenius number where, in particular, we observe that a previously presented argument features a subtle error, which alters the value of the upper bound. Despite this, we demonstrate that the subtle error does not impact upon on the validity of the upper bound, although it does impact on the upper bounds tightness. Notably, we formally state the corrected result and additionally compare the relative tightness of the corrected upper bound with the original. In particular, we show that the updated bound is tighter in all but only a relatively "small" number of cases using both formal techniques and via Monte Carlo simulation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we solve three Diophantine equations: 2 x ± (2 k p) y = z 2 and − 2 x + (2 k 3) y = z 2 with k ≥ 0 and prime p ≡ ± 3 (mod 8) . We obtain all the non-negative integer solutions by using elementary methods and the database of elliptic curves in "The L-functions and modular forms database" (LMFDB). [ABSTRACT FROM AUTHOR]

The article "Heron Triangles and the Hunt for Unicorns" delves into the search for Pythagorean triples and Heron triangles with rational sides and medians. It explores the concept of perfect triangles with integer sides, medians, and area, as well as the mathematical beauty of Somos-5 sequences and their connection to integrable maps. The text discusses the work of Buchholz and Rathbun in finding Heron triangles with two rational medians using specific algorithms, presenting tables and equations to illustrate the relationships between the parameters of these triangles. It also mentions the existence of an infinite family of such triangles and sporadic cases that do not conform to this family, highlighting the intricate mathematical patterns and challenges in studying these triangles. [Extracted from the article]

In this paper, we establish a new linking theorem for local Lipschitz functionals without the τ -upper semi-continuity assumption. As an application, we study the following equation with strongly indefinite structure and sign-changing discontinuous nonlinearity - Δ u + V (x) u = | u | p - 2 u - μ H e (a - | u |) | u | q - 2 u , a.e. in R N , u ∈ H 1 (R N) , N ≥ 2 , where V is 1-periodic and 0 lies in a gap of the spectrum of - Δ + V , 2 < q < p < 2 ∗ and μ , a > 0 are parameters. We prove the existence of a nontrivial solution of the equation above when μ > 0 is small enough. [ABSTRACT FROM AUTHOR]

Equation discovery, also known as symbolic regression, is the field of machine learning that studies algorithms for discovering quantitative laws, expressed as closed-form equations or formulas, in collections of observed data. The latter is expected to come from measurements of physical systems and, therefore, noisy, moving the focus of equation discovery algorithms towards discovering approximate equations. These loosely match the noisy observed data, rendering them inappropriate for applications in mathematics. In this article, we introduce Diofantos, an algorithm for discovering equations in the ring of integers that exactly match the training data. Diofantos is based on a reformulation of the equation discovery task into the task of solving linear Diophantine equations. We empirically evaluate the performance of Diofantos on reconstructing known equations for more than 27,000 sequences from the online encyclopedia of integer sequences, OEIS. Diofantos successfully reconstructs more than 90% of these equations and clearly outperforms SINDy, a state-of-the-art method for discovering approximate equations, that achieves a reconstruction rate of less than 70%. [ABSTRACT FROM AUTHOR]

This paper proposes a fractal modification of tropical algebra for noise removal and optimal control. Fractal addition, fractal multiplication, and fractal dot product are defined and explained in a fractal space. A fractal tropical polynomials function can model a coastline at any scale, providing a new tool for fractal analysis of any irregular curve. An example is given to solve an optimal control subject to a fractal tropical polynomials function. This paper provides a new window for tropical algebra, combining fractal geometry and algebra. [ABSTRACT FROM AUTHOR]

If X is a nonempty subset of Qk, the cone generated by X is C(X)= {q1x1 + ... + qnxn / n ∈ N\{0}, {q1,...,qn} ⊂ Q0+ and {x1,...,xn} ⊂ X}. In this work we present an algorithm which calculates from {(a1, b1), (a2,b2)} ⊂ N², the minimal system of generators of the affine semigroup C({(a1,b1), (a2,b2)}) ∩N². This algorithm is based on the study of proportionally modular Diophantine inequalities carried out in [1]. Also, we present an upper bound for the embedding dimension of this semigroup. [ABSTRACT FROM AUTHOR]

In this paper we consider some fragments of IOpen (Robinson arithmetic Q with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that I (lit) is equivalent to IOpen and is not finitely axiomatizable over Q , establish some inclusion relations between I (=) , I (≠) , I (⩽) and I (⩽̸) . We also prove that the set of diophantine equations solvable in models of I (=) is (algorithmically) decidable. [ABSTRACT FROM AUTHOR]

The steering tool combined bearing is a mechanical component that can support forces, guide movement, and bear vibrations at the same time. In the process of analyzing the contact and load distribution inside the combined bearing, there are several contact points, and the static equilibrium equation is statically indeterminate. Based on dynamics theory and Hertz elastic contact theory, the concept of a load integral and the displacement estimation formula for roller bearings are introduced, and a numerical solution to the statically indeterminate problem of the steering tool combined bearing is given. According to the statically indeterminate equation matrix of the combined bearing, the relationship between the load and displacement of the combined bearing is studied to improve the statically indeterminate equation matrix. Moreover, the simulation results of the maximum stress and displacement distribution results of the inner and outer ring eccentricity from 1 to 6mm are compared with the algorithm prediction results. The error between the numerical solution of the statically indeterminate problem of the steering tool combined bearing and the simulation result is small, improving the calculation efficiency. This provides theoretical support for the design and reasonable application of the steering tool combined bearing. [ABSTRACT FROM AUTHOR]

We consider stationary Schrödinger equations with indefinite potential and nonlinearity sublinear at u = 0. Using linking theorem and symmetric mountain pass theorem, existence and multiplicity results are obtained. [ABSTRACT FROM AUTHOR]

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation a 1 x 1 + ⋯ + a k x k = n ( a 1 , ... , a k are given positive integers with gcd (a 1 , ... , a k) = 1) does not have a non-negative integer solution (x 1 , ... , x k). The generalized Frobenius number (called the p -Frobenius number) is the largest integer such that this linear equation has at most p solutions. That is, when p = 0 , the 0 -Frobenius number is the original Frobenius number. In this paper, we introduce and discuss p -numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer p , p -gaps, p -symmetric semigroups, p -pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When p = 0 , they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively. [ABSTRACT FROM AUTHOR]

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if char D ≠ 2 , the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) on D ∖ { 0 } with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if char D = 2. [ABSTRACT FROM AUTHOR]

AbstractConsider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a2 + b2 = c2 + d2 = e2 in integers. To this system we associate a family of elliptic curves with defining equation y2 = (x−a2)(x−b2)(x−c2)+a2b2c2. We show that there exists a subfamily of rank ≥ 3 over ℚ(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over ℚ(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a2 + b2 = c2 + d2.) [ABSTRACT FROM AUTHOR]

In this paper, based on the Diophantine equation technique, we first prove the circulant graph C (Z n , { k 1 , k 2 }) is isomorphic to the disjoint union of gcd (n , k 1 , k 2) identical graphs, which belong to one of the three types of graphs: (i) C (Z n { 1 , k }) ; (i i) the Cartesian graph bundle C s □ φ C t over cycles; (i i i) the Cartesian product of the complete graph K 2 and a cycle. Naturally, to study the dispersability of C (Z n , { k 1 , k 2 }) can be reduced to study that of the above three types. Next, we solve the dispersability of the circulant graph C (Z n , { 1 , k }). Specifically, C (Z n , { 1 , k }) is dispersable if it is a bipartite; otherwise, it is nearly dispersable. In addition, we prove that bipartite circulant graphs are dispersable. [ABSTRACT FROM AUTHOR]

Let d(n) and φ (n) denote the number of positive divisors of n and the Euler's phi function of n, respectively. In this paper, we prove various (conditional and unconditional) results about the solvability of the Diophantine equation d(n) = d(φ(n)) and a related inequality. For further research, we present some open problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study balancing hybrinomials, i.e., polynomials being a generalization of balancing hybrid numbers. We provide some properties of the balancing hybrinomials, including Catalan, Cassini, d'Ocagne, and Vajda identities, among others. Moreover, we present a matrix representation of the hybrinomials. [ABSTRACT FROM AUTHOR]

A generalisation of the well-known Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$ , $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $n\ge 0$ is the k -generalised Pell sequence $\{P^{(k)}_n\}_{n\ge -(k-2)}$ whose first k terms are $0,\ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+\cdots +P^{(k)}_{n-k}$. For the Pell sequence, the formula $P^2_n+P^2_{n+1}=P_{2n+1}$ holds for all $n\ge 0$. In this paper, we prove that the Diophantine equation $$ \begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m \end{align*} $$ has no solution in positive integers $k, m$ and n with $n>1$ and $k\ge 3$. [ABSTRACT FROM AUTHOR]

The Surányi–Hickerson conjecture is a long-standing unsolved problem of Diophantine equations. This conjecture states that all the solutions to ℓ 1 ! ⋯ ℓ m ! = k ! with k - ℓ m ≥ 2 are (ℓ 1 , ... , ℓ m ; k) = (6 , 7 ; 10) , (3 , 5 , 7 ; 10) , (2 , 5 , 14 ; 16) and (2, 3, 3, 7; 9). In this paper, we generalize the Surányi–Hickerson conjecture to ℓ 1 ! ⋯ ℓ m ! = k 1 ! ⋯ k n ! . We say that a solution (ℓ 1 , ... , ℓ m ; k 1 , ... , k n) is trivial if there exists a pair (i, j) such that | ℓ i - k j | = 1 . As in the Surányi–Hickerson conjecture, we give theoretical and computational results. In particular, we suggest that all non-trivial solutions to the equation ℓ 1 ! ℓ 2 = k 1 ! k 2 ! are (ℓ 1 , ℓ 2 ; k 1 , k 2) = (7 , 13 ; 4 , 15) , (14, 62; 7, 66) and (22, 54; 18, 57). [ABSTRACT FROM AUTHOR]

We say that a simple graph G is Seidel integral if its Seidel spectrum consists entirely of integers. In this work we establish a characterization of Seidel integral graphs which belong to the class αKa ∪ βKb,b, where mG denotes the m-fold union of the graph G. [ABSTRACT FROM AUTHOR]

This paper considers two models of queuing with a varying structure based on the introduction of additional transient intensities into known models or their combinations, which create stationary distributions convenient for calculation. In the first model, it is a probabilistic mixture of known stationary distributions with given weights. In the second model, this uniform distribution is repeatedly used in physical statistics. Both models are based on the selection of states, between which additional transient intensities are introduced. The algorithms used in this paper for introducing new transient intensities are closely related to the concept of flow in a deterministic transport network. The introduced controls are selected so that the marginal distribution of the combined system is a mixture of the marginal distributions of the combined systems with different weights determined by the introduced transient intensities. As a result, the process of functioning of the combined system is obtained by switching processes corresponding to different combined systems at certain points in time. [ABSTRACT FROM AUTHOR]

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Rivest-Shamir-Adleman (RSA) algorithm is a widely utilized technique in asymmetric cryptography, primarily for verifying digital signatures and encrypting messages. Its security relies on the integer factorization problem's difficulty, which is computationally infeasible with large security parameters. However, this study revealed scenarios where an attacker can concurrently factorize multiple RSA moduli N i = p i q i under specific conditions. The attack is feasible when the attacker possesses a set of RSA key pairs with certain flaws, allowing each N i to be factored in polynomial time. We identified vulnerabilities in RSA keys that satisfy particular equations by applying Diophantine approximation and Coppersmith's lattice-based technique. For instance, the study demonstrates that if RSA public exponents e i and moduli N i adhere to e i r − (N i − p i − q i + u i) s i = t i , where r , s i , u i , and t i are small integers, then all N i can be factorized simultaneously. Additionally, another vulnerability arises when RSA parameters satisfy e i r i − s (N i − p i − q i + u i) = t i , enabling concurrent factorization with small integers s , r i , u i , and t i . This research expands the understanding of RSA security by identifying specific conditions under which RSA public-key pairs can be compromised. These findings are relevant to the broader field of cryptography and the ongoing efforts to secure communication systems against sophisticated adversaries. [ABSTRACT FROM AUTHOR]

Copyright of Palestine Journal of Mathematics is the property of Palestine Polytechnic University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Palestine Journal of Mathematics is the property of Palestine Polytechnic University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we study the integer solutions of a family of Fermat‐type equations of signature (2,2n,n)$(2, 2n, n)$, Cx2+qky2n=zn$Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant BC,q$B_{C, q}$ such that if n>BC,q$n > B_{C,q}$, there are no solutions (x,y,z,n)$(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory. [ABSTRACT FROM AUTHOR]

In this note, we revisit the exponential Diophantine equation 8x + 161y = z², initially studied by Manikandan and Venkatraman. Their work established that the equation has the two non-negative integer solutions: (1, 0, 3) and (1, 1, 13). Our findings reveal an additional solution, (2, 1, 15), and we show that these three solutions constitute the complete list of non-negative integer solutions for this equation. This extends and completes the main result presented in their paper. [ABSTRACT FROM AUTHOR]

In this article, we establish that there is a unique non-negative solution to the Diophantine equation 675x + 896y = z². The solution is (x, y, z) = (1, 0, 26). [ABSTRACT FROM AUTHOR]

In this article, we prove that (1, 0, 8) and (0, 1, 18) are the only solutions for the Diophantine equation 63x + 323y = z² where x, y and z are non-negative integers. [ABSTRACT FROM AUTHOR]

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of non-affine type. As an application, a class of Diophantine equations encoded with cluster algebras are studied. [ABSTRACT FROM AUTHOR]