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[3] viXra:2609.0093 [pdf] submitted on 2026-09-28 20:54:26
Authors: Deepak Ponvel Chermakani
Comments: 3 Pages. 3 theorems.
Consider the problem of finding the time T at which none of the n-1 moving runners are in the arc Z of length 2/n centered at the stationary runner 0 in an instance of the Shifted Lonely Runner Conjecture (SLRC). We show two important results for prime n and for large n. Firstly, there exist atleast 3 time intervals during which atmost 1 moving runner is in Z for the SLRC, and, there exist atleast 4(n-1) time intervals during which atmost 1 moving runner is in Z for the LRC. Secondly, for both the SLRC and the LRC considered separately, there are atleast 2 different moving runners which prevent the loneliness of runner 0 by being the atmost one runner in Z at some time.
Category: Algebra
[2] viXra:2609.0068 [pdf] replaced on 2026-09-29 01:49:42
Authors: Deng Ke
Comments: 35 Pages.
Current research on the distribution of prime numbers either narrows down to counting due to a focus on applied fields, meaning it only counts without knowing which numbers are prime, or it explores patterns but, because of a cause-from-effect order, leads to unclear results and only approximate rather than definitive distribution patterns. This paper attempts to discuss a common type of multiplication—multiplication whose factors do not contain 1, called skip-multiplication—introducing skip-numbers and skip-products, to explain the essence of prime numbers as full-skip numbers and their distribution patterns, and to achieve a fundamental grasp of prime numbers in terms of identity rather than merely quantity through the method of reverse selection from the skip-product table. Furthermore, because tables rely on enumeration, their role in application is at most auxiliary rather than a replacement for any tool; thus, this paper mainly provides a new number-theoretic understanding of prime and composite numbers in the pure mathematical theory domain. The appendix supplements some patterns of the skip-multiplication theory tables and discussions on the oddness of primes and the Riemann Hypothesis.
Category: Algebra
[1] viXra:2609.0047 [pdf] submitted on 2026-09-17 23:02:29
Authors: Theophilus Agama
Comments: 23 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
An addition chain of length $h$ that leads to a number $n$ is a sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$~($i>jgeq k$) for each $1leq ileq h$. We introduce a matrix-theoretic framework for studying the properties of arbitrary addition chains that lead to a target integer $ngeq 2$ by associating each chain and the sequence of dominant and lower weight summands that calculate each term in the chain an adjacency matrix that encodes the internal geometry. In this linear-algebraic and matrix theoretic framework, we investigate how rank, rank profiles, singular values, eigenvalues, matrix norms, predecessor depths, and track-interaction terms encode the combinatorial and arithmetic structure of addition chains.
Category: Algebra
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