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Mike Winkler

Publications and source records attributed to Mike Winkler.

16 recordsLinked to original sources
arXiv2026

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

Let $q\geq3$ be odd and put $m_r=\lfloor r\log_2q\rfloor$. We count parity sequences with exactly $r$ entries equal to $1$ for which the coefficient in the $qx+1$ iteration first drops below $1$ at step $m_r+1$. Their number $a_q(r)$ also counts the corresponding residue classes modulo $2^{m_r+1}$. By classifying increasing integer sequences according to the first index at which they exceed a prescribed bound, we obtain a recurrence which, for $q=3$, establishes the five- or six-term block pattern conjectured in 2015, allowing a shorter final block. We establish the sharp inequalities \[ \frac1r\binom{m_r-1}{r-1}\leq a_q(r)\leq\frac1r\binom{m_r}{r-1} \] Equality in the lower (respectively upper) bound occurs exactly when $\{r\log_2q\}$ is smaller (respectively larger) than all preceding fractional parts $\{j\log_2q\}$, $1\leq j<r$. These indices are characterized by denominators of convergents and semiconvergents from below and above; at these indices $r\geq2$, the values $a_q(r)$ are rational Catalan numbers. We also characterize the dependence on $q$ and prove, for each fixed odd $q\geq3$, that $a_q(r)=Θ_q(r^{-3/2}B_q^r)$, where $B_q=(\log_2q)^{\log_2q}/(\log_2q-1)^{\log_2q-1}$. For $q=3$, this proves Hikawa's conjectured polynomial factor for the number of finite binary words of given weight for which the length of every nonempty prefix is smaller than $\log_2 3$ times the number of its ones.

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arXiv2026

Harborth's theorem p(5)=18 revisited

Let $p(n)$ be the minimum cardinality of a finite planar point set in which each point has exactly $n$ unit-distance neighbours. Harborth proved $p(5)=18$ in work presented in 1985 and published in 1987, using a detailed case analysis. We revisit the proof using convex-hull arguments, incidence counts, and elementary properties of intersections of unit circles. These tools give an alternative treatment of configurations with $14$ and $16$ points and organize the final case distinction by a common vector-sum structure. We also treat a configuration not explicitly covered by the case analysis in the online PDF copy of Harborth's paper available to us, and give an explicit justification of the six-point classification used in the argument. No result subsequent to Harborth's proof is required.

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arXiv2026

Fibonacci Enumeration of Residue Classes via Forbidden Binary Factors

Binary words avoiding adjacent ones are counted by Fibonacci numbers. We identify a natural residue-class enumeration in which this classical word count arises from parity data of the shortcut $3n+1$ map. For an odd starting value, we introduce a local cut time: the first index at which the trajectory enters the residue class $3$ modulo $4$ immediately after an even term. For each finite cut time, the starting values realizing it form a union of complete residue classes modulo a power of two, and the number of these classes is given by an explicit Fibonacci formula. The key step is a recoding of the cut condition as avoidance of the binary factor $11$ in the free part of a parity word. An explicit comparison with the endpoint conventions of the author's 2014 block construction, including the terminal blocks ending at $1$, shows that the resulting residue-class counts prove the two Fibonacci formulas conjectured there in their complete-class form. The construction is a two-state regular-language enumeration; a related theorem of Reyes Jiménez has the bitwise-complementary forbidden-factor formulation $00$.

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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

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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