Bidirected graphs, integral quadratic forms and some Diophantine equations

Abstract We establish upper bounds on the lengths of minimal conjugators in 2‐step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds are sharp. This enables us to construct a family of finitely generated 2‐step nilpotent groups such that the conjugator length function of grows like a polynomial of degree .

This mathematical research investigates whether the sum of two powers of Mersenne numbers can be expressed as a square. It also determines whether the sum of powers of these special numbers can be a power of two. Diophantine analysis using elementary methods and well-established results in number theory served as the basis for the work. The results show that there are infinitely many Mersenne numbers of the form 22α −1 whose sum of powers can be written as a perfect square and that the sum of the zeroth power and the first power of Mersenne numbers can be written as powers of 2. Moreover, the sum of two zeroth powers of Mersenne numbers always yields 2. Similarly, the sum of any two positive powers of 1, the first positive Mersenne number, is always equal to 2.

By means of a fully actuated system (FAS) approach, this article is concerned with an anti-disturbance tracking control problem toward a class of lumped disturbances containing the model uncertainties and external disturbances. A FAS predictive control with a generalized proportional-integral observer (GPIO) is presented to address this problem. Concretely, a FAS model of discrete-time nonlinear systems with the lumped disturbances is firstly given as a control-oriented one. Then, a GPIO is developed to achieve an accurate estimation for the lumped disturbances by adopting a less conservative disturbance assumption, which provides a better foundation to construct a disturbance preview. Furthermore, an incremental FAS (IFAS) prediction model with a disturbance preview is constructed by utilizing a new type of Diophantine Equation. Dependent on this IFAS prediction model, the multistep ahead predictions can be obtained to minimize an objective function to yield an optimal anti-disturbance controller, such that the desired tracking performance can be guaranteed. The depth analysis derives a sufficient condition for the bounded stability and tracking performance of the closed-loop FASs. The proposed GPIO-based FAS predictive control provides a solution to the spacecraft attitude control for verifying the feasibility.

Purely exponential Diophantine equations with four terms of consecutive bases: Contribution to Skolem’s conjecture

This paper establishes profound connections between metallic ratios, Diophantine equations, and their underlying symmetry groups. Building on recent work on Diophantine equations and their solutions, we develop a comprehensive theory revealing the algebraic and geometric structures governing families of Diophantine equations associated with metallic ratios. Through detailed investigations of automorphism groups, continued fractions, and arithmetic geometry, we provide complete proofs and explicit examples that illuminate the deep arithmetic properties of metallic ratios. Our work offers a unified framework bridging number theory, group theory, and algebraic geometry, with applications to class field theory and computational number theory.

In this survey, we outline the role of G -functions in arithmetic geometry, notably their link with Picard–Fuchs differential equations and periods. We explain how polynomial relations between special values of G -functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more “motivic” symmetries; and how Bombieri’s principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André–Oort conjecture). At the end, we sketch the recent revival of the G -function method in the context of unlikely intersections and the Zilber–Pink conjecture.

We prove that the average size of a mixed character sum \begin{equation*}\sum_{1\leqslant n \leqslant x} \chi(n) e(n\theta) w(n/x)\end{equation*} (for a suitable smooth function $w$ ) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine condition, where $\chi$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\leqslant r$ . In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $\theta$ . Certain quadratic Diophantine equations play a key role in the present paper.

Recent versions of large language models have become increasingly reliable in providing mathematical arguments, especially in classical topics such as Diophantine problems. We exemplify this development here by using ChatGPT to solve an open Diophantine problem from Majstorović Ergotić and Došlić [MATCH Commun. Math. Comput. Chem. 95 (2026) 265-283].

In this note we consider the title Diophantine equation from both a theoretical as well as experimental point of view. In particular, we prove that for k = 4 , 6 k=4, 6 and each choice of the signs our equation has infinitely many coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) such that no partial sum in the expression x 3 ± y 3 − ( a k ± b k ) x^3 \pm y^3-(a^k \pm b^k) vanishes. The same is true for each k ≢ 0 ( mod 4 ) k\not \equiv 0\pmod {4} and the equation x 3 ± y 3 = a k − b k x^3\pm y^3=a^k-b^k . For k = 5 , 7 k=5, 7 and all choices of the signs we computed all coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) of x 3 ± y 3 = a k + b k x^3\pm y^3=a^k+b^k satisfying the condition b > a ≤ 50000 b>a\leq 50000 .

When studying pure number fields, the discriminant and integral basis are cornerstone ideas that shed light on their structure and mathematical characteristics. The algebraic invariants and arithmetic behavior of a number field are affected by the discriminant, which encodes crucial information regarding the field’s ramification and the geometry of its ring of integers. To examine the field’s features, like the structure of its ideal class group and the solutions to Diophantine equations, an integral basis is a set of elements in the ring of integers that forms a basis over the integers. Class number analysis, ideal class group determination, and Hilbert symbol computing are only a few of the many important areas of number theory that benefit greatly from these ideas. Computerising discriminants for fields of high degree and discovering minimal integral bases for fields with complex ramification remain challenging tasks. We still require a deeper understanding of algebraic number theory and more advanced computational approaches to address these challenges, despite the significant progress we’ve made. In mathematical physics, coding theory, and cryptography, where the algebraic properties of number fields influence the efficacy and security of various algorithms, the study of integral bases and discriminants is fundamental for both theoretical and practical reasons.

To overcome the constraint of communication delay on tracking accuracy, a time-delay model predictive control method is proposed. Integrating trajectory coordinate dynamics and time-delay state equations to construct an enhanced composite model with time-varying delay characteristics. Designed the optimal weighting method to predict communication delay, transformed the uncertain delay equation into a time series model with external inputs, and constructed an augmented prediction equation based on the Diophantine equation. The controller solves the objective function online through rolling optimization and introduces feedback correction to compensate for time delay errors. The key content is to abandon the traditional ideal time series assumption and achieve effective modeling of time-delay uncertainty, as well as collaborative optimization of time-delay prediction and feedback correction. The results show that TDMPC significantly reduces the lateral tracking error compared to standard MPC under communication delay, and the amplitude of lateral angular velocity fluctuation is significantly reduced, significantly improving the tracking stability of complex time-delay scenarios.

Integral solutions of certain Diophantine equations in quadratic fields

Microtubules are cylindrical protein polymers that organize the cytoskeleton and play essential roles in intracellular transport, cell division, and possibly cognition. Their highly ordered, quasi-crystalline lattice of tubulin dimers, notably tryptophan residues, endows them with a rich topological and arithmetic structure, making them natural candidates for supporting coherent excitations at optical and terahertz frequencies. The Penrose–Hameroff Orch OR theory proposes that such coherences could couple to gravitationally induced state reduction, forming the quantum substrate of conscious events. Although controversial, recent analyses of dipolar coupling, stochastic resonance, and structured noise in biological media suggest that microtubular assemblies may indeed host transient quantum correlations that persist over biologically relevant timescales. In this work, we build upon two complementary approaches: the parametric resonance model of Nishiyama et al. and our arithmetic–geometric framework, both recently developed in Quantum Reports. We unify these perspectives by describing microtubules as rectangular lattices governed by the imaginary quadratic field Q(i), within which nonlinear dipolar oscillations undergo stochastic parametric amplification. Quantization of the resonant modes follows Gaussian norms N=p2+q2, linking the optical and geometric properties of microtubules to the arithmetic structure of Q(i). We further connect these discrete resonances to the derivative of the elliptic L-function, L′(E,1), which acts as an arithmetic free energy and defines the scaling between modular invariants and measurable biological ratios. In the appended adelic extension, this framework is shown to merge naturally with the Bost–Connes and Connes–Marcolli systems, where the norm character on the ideles couples to the Hecke character of an elliptic curve to form a unified adelic partition function. The resulting arithmetic–elliptic resonance model provides a coherent bridge between number theory, topological quantum phases, and biological structure, suggesting that consciousness, as envisioned in the Orch OR theory, may emerge from resonant processes organized by deep arithmetic symmetries of space, time, and matter.

A bstract We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $${\mathfrak{u}}_{1}$$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $${\mathfrak{u}}_{1}$$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Kollár) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $${\mathfrak{s}\mathfrak{u}}_{3}\oplus {\mathfrak{s}\mathfrak{u}}_{2}$$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $${\mathfrak{u}}_{1}$$ hypercharges. We also discuss whether it is possible to find all solutions in this way.

Explicit colourings for partition irregularity of certain nonlinear Diophantine equations

The aim of this paper is to show that the Diophantine equation 145^x+147^y=z^2 has no non-negative integer solutions x, y and z.

In this paper, we show that for all solutions of the Diophantine equation where x,y,z are nonnegative integers are present a be a positive integer with a≡8(mod27). It has exactly two non-negative integer infinite solutions (x,y,z)=(1,0,√(a+1)) where a=(9t±3)^2-1 and (x,y,z)=(1,1,√(3a+12)) where a=243t^2±108t+8. Moreover, we prove that (x,y,z)=(3,2,36) is the unique non-negative integer solution and t is an integer.

Let (x,y,z) be a primitive Pythagorean triples such that x^2+y^2=z^2 with gcd⁡(x,y,z)=1. In this paper, we search all primitive Pythagorean triples such that the difference of its right sides is a perfect square. We show that there are infinitely many such Pythagorean triples. In fact, we give the parametric equation of all sides of such Pythagorean triples depending on two parameters. To get the results, we use some elementary methods as well as generalized Pell equations and some classical Diophantine Equations from number theory.

We consider an iterative branching process in which an abstract object can subdivide into other objects. The multiplication process may be varied by the occurrence of random "fatal" events in which some of the subsequent objects or states may fail. The process is also constrained to terminate upon reaching a given number of events or alternatively upon reaching a fixed number of iteration steps. A system of diophantine integer-variable equations capable of describing the aforementioned process is proposed. These equations can be applied prospectively to many branching phenomena of physical, biological and demographic nature. The equations, which we call systems of equations S, Q, U can be reformulated into three main classes based on the behavior of the sum of variables with respect to a fixed principal numerical parameter (TC= 'Total Cases'). These systems always admit solutions and these are sought for the three classes. The mathematical properties of the three systems are presented both analytically and graphically, and the software script for calculating numerical solutions is attached. In the case of high TC values, where direct calculation is not possible, special solutions are also sought for the steady state case and the "most probable" case, the latter using statistical mechanics methods. Solutions examples are given for a wide range of TC parameters. We also refer to real-world examples of applications ranging from prey/predator population dynamics to population mortality modeling and 2d lattice space tiling and also tree leaves branching alternatives. The main purpose of the study here proposed is to implement a mathematical frame that can provide tools to be used in the study of real-world applications.