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Mike Winkler

Publications and source records attributed to Mike Winkler.

15 recordsLinked to original sources
arXiv2026

Admissible $qx+1$ Sequences, Semiconvergents, and Rational Catalan Numbers

For an odd integer \(q\geq3\), let \(a_q(r)\) count finite words in \(\{q/2,1/2\}\) containing exactly \(r\) factors \(q/2\), with every proper prefix product greater than \(1\) and total product less than \(1\). For \(q=3,5,7\), these are OEIS \OEIS{A100982}, \OEIS{A174795}, and \OEIS{A174796}. Put \[ α_q=\log_2q,\qquad m_r(q)=\lfloor rα_q\rfloor. \] Using the classical cycle lemma in this \(q\)-specific setting, we recover the two natural Beatty passage bounds \[ \frac1r\binom{m_r(q)-1}{r-1} \leq a_q(r)\leq \frac1r\binom{m_r(q)}{r-1}, \] and prove their equality criteria directly. The lower and upper equality orders are the denominators of the strict lower and upper one-sided best approximations to \(\log_2q\); together they are the denominators of all convergents and semiconvergents. Equivalently, they are the strict record minima and maxima of the binary mantissas \(q^r/2^{m_r(q)}\). At every nontrivial equality order, the admissible words are in explicit bijection with rational Dyck paths, so that \(a_q(r)\) is a rational Catalan number. We also prove monotonicity in \(q\), compare the three OEIS sequences, and derive their growth constants and exact normalized oscillations.

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arXiv2026

An Identity of Fillipi Related to Fermat-Type Equations

We give a structural proof of an algebraic identity proposed by Fillipi in 2014. For every integer $n\ge2$, the identity represents an explicit multiple of $x^n+y^n-z^n$ as $\mathcal A^2+\mathcal B^2-\mathcal C^2$. We identify this expression with a determinant arising from a weighted Gram matrix and show that Fillipi's particular fourth-power terms are obtained from a simpler identity by one polynomial column transformation. The same transformation yields an explicit polynomial rank-one factorisation modulo $x^n+y^n-z^n$. Thus the associated Pythagorean relation is a universal polynomial consequence of the Fermat-type equation and supplies no additional algebraic solvability condition.

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arXiv2026

Fibonacci Enumeration of Parity Blocks in Collatz Trajectories

Let $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. For an odd starting number $s$, let $\ell(s)$ be the first positive index $k$ for which $T^k(s)\equiv3\pmod4$ and $T^{k-1}(s)$ is even. The use of this single local cut time is the main structural step: it replaces the two block parameters of an earlier formulation and converts the endpoint condition into a local condition on consecutive parity symbols. We prove that, for every $r\geq2$, the starting numbers $s\equiv1\pmod4$ with $\ell(s)=r$ form exactly $F_{r-1}$ residue classes modulo $2^{r+2}$, while for every $r\geq3$ the starting numbers $s\equiv3\pmod4$ with $\ell(s)=r$ form exactly $F_r-1$ such classes. Fixing any odd residue class modulo $12$ leaves these counts unchanged and yields the two Fibonacci formulas conjectured in a 2014 preprint by the author as special cases. The proof uses the classical finite parity correspondence of Terras and Everett. A term is congruent to $3$ modulo $4$ exactly when two consecutive parities are $1$, so the cut conditions reduce to binary words without the factor $11$. More generally, every finite-state condition on a parity word gives a residue-class counting sequence governed by a finite transfer matrix. The present Fibonacci formulas are the two-state instance of this principle. We also show that the exceptional trajectories for which no cut occurs reach $1$, and that these exceptions have density zero in every odd residue class modulo $12$.

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arXiv2026

Thinned Wallis-type prime products in residue classes modulo $2^m$

For odd primes $p$ we consider the factors $A(p)=(p-χ_4(p))/(p+χ_4(p))$, where $χ_4$ is the quadratic Dirichlet character modulo $4$, and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, express the limiting constant in terms of Mertens-type constants in arithmetic progressions and Dirichlet $L$-values, and give reproducible high-precision computations of the resulting constants.

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arXiv2026

Deterministic Structures in the Stopping Time Dynamics of the 3x+1 Problem

The $3x+1$ problem concerns the iteration of the map $T:\mathbb{Z}\to\mathbb{Z}$ defined by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. We study the coefficient stopping time in the sense of Terras. For each order $n$, we characterize the corresponding residue classes modulo $2^{σ_n}$ by the admissible positions of the odd iterates. These position vectors form a directed rooted tree under deletion of the final odd position. This description yields a Pascal-type recursion for the number of classes and proves that the recursive generation produces exactly the admissible vectors, each exactly once. The affine iterate formula gives explicit congruence relations for the classes and arithmetic transition rules between related parity vectors. For every fixed $N$, the union of the classes generated up to order $N$ is periodic with period $2^{σ_N}$. Its density is given by an exact finite sum, and its largest initial interval of coverage can be computed from one period. These results concern finite coefficient stopping-time structures. They neither prove that every starting value has finite coefficient stopping time nor establish equality between the coefficient stopping time and the classical stopping time.

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arXiv2026

Realized Rank Certificates for Matchstick Frameworks and Insertion Edges

We give an exact, checkable rank-certificate method for realized planar unit-distance frameworks. The method is motivated by Vogel's computations for matchstick graphs and by the insertion-edge tests used in the Matchstick Graphs Calculator. Its algebraic core is independent of geometry. A singular square matrix $M$ is replaced by a sparse perturbation $B=M+UCV^T$. If $B$ is nonsingular and the inverse satisfies $V^TB^{-1}U=C^{-1}$, then the columns of $B^{-1}U$ and the rows of $V^TB^{-1}$ form bases of the right and left kernels of $M$, and the rank defect of $M$ is certified. Applied to the equilibrium matrix of a planar framework, this gives finite exact certificates for self-stresses, infinitesimal motions, redundant edges, and candidate edges whose constraints are already forced by the realized framework. The certificate data can be checked independently from the search that produced it, using exact matrix identities. We emphasize that matchstick frameworks are not generic: unit distances, triangles, rhombi, and symmetries can change the realized rank. The method therefore concerns the coordinate-dependent representation of a given drawing, not only the generic rigidity matroid of the abstract graph.

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arXiv2026

On the existence of 4-regular matchstick graphs

A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 vertices are known only for $n \in \{52, 54, 57, 60\}$, we prove the existence of such graphs for every integer $n \geq 63$.

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arXiv2021

A catalog of 4-regular and (2;4)-regular matchstick graphs

The first part (page 1 - 7) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (page 8 - 15) presents the currently known examples of $(2;4)$-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2.

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arXiv2021

New results on the stopping time behaviour of the Collatz 3x + 1 function

Let $σ_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{σ_n})$ of starting numbers $s$ with finite stopping time $σ(s)=σ_n$. Let $z_n$ be the number of these residue classes for each $n\geq0$ as listed in the OEIS as A100982. It is conjectured that for each $n\geq4$ the value of $z_n$ is given by the formula \begin{align*} z_n=\frac{(m+n-2)!}{m!\cdot(n-2)!}-\sum_{i=2}^{n-1}\binom{\big\lfloor\frac{3(n-i)+δ}{2}\big\rfloor}{n-i}\cdot z_i, \end{align*} where $m=\big\lfloor(n-1)\cdot\log_23\big\rfloor-(n-1)$ and $δ\in\mathbb{Z}$ assumes different values within the sum at intervals of 5 or 6 terms. This allows us to create an iterative algorithm which generates $z_n$ for each $n>6$.

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arXiv2021

Aperiodic Sets of Prototiles Extracted From the Penrose Rhomb Tiling

We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any matching rules. We also give an approximate solution to an aperiodic monotile that tessellates the plane (including five types of gaps) only in a nonperiodic way.

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arXiv2020

Approximate Solutions of 4-regular Matchstick Graphs with 50-62 Vertices

A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertices $\geq$ 52 except for 53, 55, 56, 58, 59, 61, and 62. In this article we present 38 different examples with 50 - 62 vertices which contain two, three, or four distances which differ slightly from the unit length. These graphs should show why this subject is so extraordinarily difficult to deal with and should also be an incentive for the interested reader to find solutions for the missing numbers of vertices.

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arXiv2019

A 3-regular matchstick graph of girth 5 consisting of 54 vertices

In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.

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arXiv2018

New minimal (4; n)-regular matchstick graphs

A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph ($m;n)$-regular if every vertex has only degree $m$ or $n$. In this article the authors present the latest known $(4;n)$-regular matchstick graphs for $4\leq n\leq11$ with a minimum number of vertices.

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arXiv2018

Minimal completely asymmetric (4; n)-regular matchstick graphs

A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph $(m;n)$-regular if every vertex has only degree $m$ or $n$. In this article we present the latest known $(4;n)$-regular matchstick graphs for $4\leq n\leq11$ with a minimum number of vertices and a completely asymmetric structure. We call a matchstick graph completely asymmetric, if the following conditions are complied. 1) The graph is rigid. 2) The graph has no point, rotational or mirror symmetry. 3) The graph has an asymmetric outer shape. 4) The graph can not be decomposed into rigid subgraphs and rearrange to a similar graph which contradicts to any of the other conditions.

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