Integer (computer science)
View article: Square Root of Natural Numbers as an infinite Staircase of Continued Fractions
Square Root of Natural Numbers as an infinite Staircase of Continued Fractions Open
When one lists the continued fractions of √ n across the natural numbers, a striking regularity appears: the coefficients rise in discrete steps, forming an arithmetical and visual staircase. This article reveals the algebraic and geometri…
View article: Goldbach in Axiom Zero: A Deterministic Additive-Sieve
Goldbach in Axiom Zero: A Deterministic Additive-Sieve Open
This preprint is part of the Axiom Zero (AZ) programme and should be read as a technical case study rather than a standalone introduction. It assumes familiarity with the core AZ framework developed in Axiom Zero: Structural Irreducibility…
View article: A Practical Coverage-Based Framework for the Collatz Conjecture
A Practical Coverage-Based Framework for the Collatz Conjecture Open
This paper introduces a practical, coverage-based framework for analyzing the Collatz conjecture. The framework is built upon two empirical invariants observed up to M = 10^10 1. SPC - Let S(M) denote the number of seeds in the interval [1…
View article: Pindakaas
Pindakaas Open
Pindakaas is a library to transform pseudo-Boolean and integer constraints into conjunctive normal form, and to efficiently interact with Boolean satisfiability solvers.
View article: ExactCN: Predicting Exact Copy Numbers on Whole Exome Sequencing Data
ExactCN: Predicting Exact Copy Numbers on Whole Exome Sequencing Data Open
The quantification of the precise copy number variations (CNVs) is crucial to understanding the effects of gene dosage, disease severity, and therapeutic response. Although whole-exome sequencing(WES) offers a cost-effective solution for C…
View article: Iterated polynomials are dense
Iterated polynomials are dense Open
For any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P $\in$ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.
View article: A Geometric–Spectral Framework for Prime Segments
A Geometric–Spectral Framework for Prime Segments Open
Companion to the CDRH I–III preprint series on composite–zero resonance in the mod 6±1 universe. We introduce a new geometric–spectral framework for analysing the distribution of prime numbers, based on a 5-adic orbital embedding of the in…
View article: Wavefront Reconstruction for Fractional Lateral Shear Measurements using Weighted Integer Shear Averages
Wavefront Reconstruction for Fractional Lateral Shear Measurements using Weighted Integer Shear Averages Open
Wavefront reconstruction in lateral shearing interferometry typically assumes that the shear amount is an integer multiple of the sampling interval. When the shear is fractional, approximating it with the nearest integer value leads to not…
View article: Pindakaas
Pindakaas Open
Pindakaas is a library to transform pseudo-Boolean and integer constraints into conjunctive normal form, and to efficiently interact with Boolean satisfiability solvers.
View article: Trace of the Adjacency Matrix of the Star Graph and Complete Bipartite Graph Raised to a Positive Integer Power
Trace of the Adjacency Matrix of the Star Graph and Complete Bipartite Graph Raised to a Positive Integer Power Open
This research aims to derive the general form of the trace matrix of adjacency from star graphs and complete bipartite graphs with size n × n and raised to a positive integer power. To obtain the general form of the trace matrix of adjacen…
View article: Iterated polynomials are dense
Iterated polynomials are dense Open
For any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P $\in$ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.
View article: A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$
A complete solution of the Erdős-Kleitman matching problem for $n\le 3s$ Open
Given integers $n\ge s\ge 2$, let $e(n,s)$ stand for the maximum size of a family of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. The study of this quantity goes back to the 1960s, when Kleitman determined …
View article: IntAttention: A Fully Integer Attention Pipeline for Efficient Edge Inference
IntAttention: A Fully Integer Attention Pipeline for Efficient Edge Inference Open
Deploying Transformer models on edge devices is limited by latency and energy budgets. While INT8 quantization effectively accelerates the primary matrix multiplications, it exposes the softmax as the dominant bottleneck. This stage incurs…
View article: Geometric Index Sieve (SEMT): A Novel Deterministic Approach to Factoring Large Integers and its High-Efficiency Parallel Implementation
Geometric Index Sieve (SEMT): A Novel Deterministic Approach to Factoring Large Integers and its High-Efficiency Parallel Implementation Open
The security of the RSA cryptosystem relies fundamentally on the presumed intractability of the Integer Factorization Problem (IFP). Current classical solutions, notably the General Number Field Sieve (GNFS), operate with a sub-exponential…
View article: BAMAS: Structuring Budget-Aware Multi-Agent Systems
BAMAS: Structuring Budget-Aware Multi-Agent Systems Open
Large language model (LLM)-based multi-agent systems have emerged as a powerful paradigm for enabling autonomous agents to solve complex tasks. As these systems scale in complexity, cost becomes an important consideration for practical dep…
View article: Pindakaas
Pindakaas Open
Pindakaas is a library to transform pseudo-Boolean and integer constraints into conjunctive normal form, and to efficiently interact with Boolean satisfiability solvers.
View article: Pindakaas
Pindakaas Open
Pindakaas is a library to transform pseudo-Boolean and integer constraints into conjunctive normal form, and to efficiently interact with Boolean satisfiability solvers.
View article: Evaluating the tame Brauer group of open varieties over local fields
Evaluating the tame Brauer group of open varieties over local fields Open
In this document we let $U$ be a smooth variety of pure dimension $d$ over a local field $k_v$ with unit ball $\mathcal{O}_v$ and residue field $\mathbb{F}$ of characteristic $p>0$ and we set $n$ to be a positive integer such that $p\nmid …
View article: On lattices over Fermat function fields
On lattices over Fermat function fields Open
Function field lattices are an interesting example of algebraically constructed lattices. Their minimum distance is bounded below by a function of the gonality of the underlying function field. Known explicit examples--coming mostly from e…
View article: The i = c Identity and the Structural Necessity of the Goldbach Conjecture
The i = c Identity and the Structural Necessity of the Goldbach Conjecture Open
The Goldbach Conjecture asserts that every even integer N > 2 is the sum of two primes (N =p1+p2). This paper applies the Structural Unification Theory (SUT) to resolve this conjecture by structural necessity. We first establish the founda…
View article: The ABC Conjecture: Diophantine Bounds Beyond Fermat.
The ABC Conjecture: Diophantine Bounds Beyond Fermat. Open
The ABC conjecture, proposed independently by Joseph Oesterlé and David Masser in 1985, posits a profound relationship between the additive and multiplicative structures of integers. It states that for any three coprime positive integers $…
View article: Everything is a 9 Phase DoF Torus (Apple), nonlinearity is just winding.
Everything is a 9 Phase DoF Torus (Apple), nonlinearity is just winding. Open
1. Notation keySets and spaces ℝ – real numbers ℂ – complex numbers ℤ – integers T – 1D circle of phases (angles modulo 2π) T² – 2D torus = T × T (meta-phase space) T⁹ – 9D phase torus (Apple phase space) Hilbert spaces Hₐₚₚₗₑ ≅ ℂ⁹ – “Appl…
View article: PTO Window Calculus I — PTO–IAB1: Window–Level State–Sum, Integer Bridges, Holonomy, and Padma Harmonics (Pre–Stage–D Consolidation)
PTO Window Calculus I — PTO–IAB1: Window–Level State–Sum, Integer Bridges, Holonomy, and Padma Harmonics (Pre–Stage–D Consolidation) Open
This preprint is part of the PTO Window Calculus series and consolidates the window-level, scalar side of the framework for knot and link diagrams. It organises the single–diagram “present window” data into a layered state–sum hierarchy an…
View article: Integer Solutions of Pell Equation in a Closed Rotated Square Region
Integer Solutions of Pell Equation in a Closed Rotated Square Region Open
View article: Error Controllability Under Finite-Order Discipline: PSWF/DPSS Extremal Windows Uniqueness, Integer Leading Terms (Spectral Flow/Index of Projection Pairs), and $10^{-3
Error Controllability Under Finite-Order Discipline: PSWF/DPSS Extremal Windows Uniqueness, Integer Leading Terms (Spectral Flow/Index of Projection Pairs), and $10^{-3 Open
Under unified Fourier normalization \widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i t\xi}\,dt (frequency in cycles), we construct error discipline around time-limiting--band-limiting concatenated operators: windowing main leakage, multiplic…
View article: Diophantine Finiteness: Modular Methods, Radical Bounds, and the abc-Conjecture
Diophantine Finiteness: Modular Methods, Radical Bounds, and the abc-Conjecture Open
This paper explores the profound problem of Diophantine finiteness, focusing on the sophisticated interplay between modular methods, radical bounds, and the foundational abc-conjecture. Diophantine equations, equations where solutions are …
View article: Lattice-Based Cryptography: The Quantum-Resistant Successor to RSA
Lattice-Based Cryptography: The Quantum-Resistant Successor to RSA Open
The advent of quantum computing poses a significant existential threat to the security foundations of current public-key cryptography, most notably the widely adopted RSA algorithm. Shor's algorithm, specifically, can efficiently break the…
View article: Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay
Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay Open
By a theorem of Roberts, the integral closure of a regular local ring in a finite abelian extension of its fraction field is Cohen-Macaulay, provided that the degree of the extension is coprime to the characteristic of the residue field. W…
View article: Everything is a 9 Phase DoF Torus (Apple), nonlinearity is just winding.
Everything is a 9 Phase DoF Torus (Apple), nonlinearity is just winding. Open
1. Notation keySets and spaces ℝ – real numbers ℂ – complex numbers ℤ – integers T – 1D circle of phases (angles modulo 2π) T² – 2D torus = T × T (meta-phase space) T⁹ – 9D phase torus (Apple phase space) Hilbert spaces Hₐₚₚₗₑ ≅ ℂ⁹ – “Appl…
View article: An Elegant Proof for Fermat's Last Theorem for Odd n
An Elegant Proof for Fermat's Last Theorem for Odd n Open
This work presents an elementary algebraic argument showing that the equationx^n + y^n = z^n has no positive integer solutions for any odd exponent n > 2.The contradiction arises from the structure of the powered differences, whichforces i…