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An Effective Method for Solving a Class of Transcendental Diophantine Equations
Zeyu Cai
https://arxiv.org/abs/2510.11753 https://arxiv.org/pdf/2510.11753

An Effective Method for Solving a Class of Transcendental Diophantine Equations
This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental diophantine equation." A general solution to this problem remains an open question; however, the ABC conjecture implies that the number of solutions for any such equation is finite. This work introduces and implements an effective algorithm designed to solve the…

On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
Bibhu Prasad Tripathy, Asutosh Satapathy, Utkal Keshari Dutta, Bijan Kumar Patel
https://arxiv.org/abs/2510.12145

On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b \geq 2$. In this paper, we study the Diophantine equations \[ \mathcal{P}_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] \[ E_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] and \[ N_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] in non-negative integers $n, b$ and positive integer $l$. As a result, we determine the Padovan, Perrin…

Crosslisted article(s) found for cs.MS. https://arxiv.org/list/cs.MS/new
[1/1]:
- An Effective Method for Solving a Class of Transcendental Diophantine Equations
Zeyu Cai

The Fractal Logic of $\Phi$-adic Recursion
Milan Rosko
https://arxiv.org/abs/2510.08934 https://arxiv.org/pdf/2510.08934

The Fractal Logic of $Φ$-adic Recursion
We establish that valid $Σ_1$ propositional inference admits reduction to Fibonacci-indexed witness equations. Specifically, modus ponens verification reduces to solving a linear Diophantine equation in $O(M(\log n))$ time, where $M$ denotes integer multiplication complexity. This reduction is transitive: tautology verification proceeds via Fibonacci index arithmetic, bypassing semantic evaluation entirely. The core discovery is a transitive closure principle in $Φ$-scaled space (Hausdorff di…

Replaced article(s) found for math.NT. https://arxiv.org/list/math.NT/new
[1/1]:
- Diophantine stability for elliptic curves on average
Anwesh Ray, Tom Weston
h…

On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$
Valery Asiryan
https://arxiv.org/abs/2510.07643 https://arxiv.org/pdf/2510.07643

On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$
In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after r…

Cubic incompleteness: Hilbert's tenth problem begins at degree three
Milan Rosko
https://arxiv.org/abs/2510.00759 https://arxiv.org/pdf/2510.00759

Cubic incompleteness: Hilbert's tenth problem begins at degree three
We present a preliminary result addressing a long-standing open question: Are cubic Diophantine equations undecidable?} We answer in the affirmative. By encoding Gödel's first incompleteness theorem via a Fibonacc-based Göodel numbering and the Zeckendorf representation, we reduce the arithmetic complexity sufficiently to construct an explicit \emph{cubic Diophantine equation} whose solvability is independent of a model provided by \emph{Peano axioms}. We present three results. First, a commo…

The inter-universal Teichm\"uller theory and new Diophantine results over rational numbers. II
Zhong-Peng Zhou
https://arxiv.org/abs/2510.05448 https://

The inter-universal Teichmüller theory and new Diophantine results over rational numbers. II
[This is an older version of the paper, which will be updated soon.] In the present paper, we continue our research on the generalized Fermat equation $x^r + y^s = z^t$ with signature $(r, s, t)$, where $r, s, t \ge 2$ are positive integers such that $\frac{1}{r} + \frac{1}{s} + \frac{1}{t} < 1$. All known positive primitive solutions for the generalized Fermat equation when $\frac{1}{r} + \frac{1}{s} + \frac{1}{t} < 1$ are related to the Catalan solutions $1^n + 2^3 = 3^2$ and nine non-Catal…

On convergence of normal form transformations
Valery G. Romanovski, Sebastian Walcher
https://arxiv.org/abs/2510.00925 https://arxiv.org/pdf/2510.00925

On convergence of normal form transformations
We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincare-Dulac normal form, and the convergence of such transformations. We first review A.D. Bruno's approach to formal normalization, as well as convergence results in presence of certain (simplified) versions of Bruno's ``Condition A'', and along the way we also identify a large class of systems that satisfy Bruno's diophantine ``Condition omega''. We retrace the proof steps in Bruno's work, …

A note on general isolation result in Diophantine Approximation
Sergei Pitcyn, Nikolay Moshchevitin
https://arxiv.org/abs/2509.25628 https://arxiv.org/pdf/…

A note on general isolation result in Diophantine Approximation
In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $.

Diophantine analysis and Arthur's trace formula
Yuchan Lee
https://arxiv.org/abs/2509.22314 https://arxiv.org/pdf/2509.22314

Diophantine analysis and Arthur's trace formula
Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_γ\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $γ\in G(k)$ whose centralizer $G_γ$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $κ$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincid…

Diophantine equations involving powers of factorials
Sa\v{s}a Novakovi\'c
https://arxiv.org/abs/2509.18860 https://arxiv.org/pdf/2509.18860

Diophantine equations involving powers of factorials
We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations $(k!)^n-k^n=(n!)^k-n^k$ and $(k!)^n+k^n=(n!)^k+n^k$. We consider the equations $(k!)^{n!}\pm k^n=(n!)^{k!}\pm n^k$ and $(k!)^n\pm k^{n!}=(n!)^k\pm n^{k!}$ and prove a similar statement.