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[7] ai.viXra.org:2601.0120 [pdf] submitted on 2026-01-30 16:30:00
Authors: Anish Sola
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We study representations of integers as differences of prime powers, n = p^a − q^b,with distinct prime bases p ̸= q and distinct exponents a ̸= b. We focus on the positive-exponent setting (a, b ≥ 1) and on the proper-prime-power variant (a, b ≥ 2), for which the problem is closer in spirit to Goldbach- and Pillai-type questions. We prove elementary structural constraints (notably parity restrictions), propose first-moment heuristics, and outline a computational program.
Category: Number Theory
[6] ai.viXra.org:2601.0106 [pdf] submitted on 2026-01-26 21:01:05
Authors: Silvio Gabbianelli
Comments: 12 Pages.
This paper explores a deterministic mapping ϕ : Nodd → Z + that defines an informational lattice for the study of prime distribution. By analyzing the topological exclusion of composite generating functions y(x, k), we identify a structural symmetry within the manifold. Computational verification through a mapping Probe confirms density alignment with the Gram series up to 10^50. The results suggest thatcertain symmetries, such as the critical line equilibrium and rotational invariance,are emergent properties of the lattice’s geometric rigidity.
Category: Number Theory
[5] ai.viXra.org:2601.0087 [pdf] submitted on 2026-01-22 11:46:52
Authors: Bahbouhi Bouchaib
Comments: 27 Pages. Original study showing a predictive search method of Goldbach's prime pairs.
We introduce a new geometric and algorithmic framework for the search of Goldbach pairs based on the organization of admissible deviations around the midpoint of an even integer. By embedding admissible deviations into a sunflower phase space and a normalized helical geometry, we observe a robust concentration of Goldbach-successful deviations along narrow geometric lanes. This structure persists across multiple scales, from 10^9 up to at least 10^26, and leads naturally to a phase-guided algorithm that reduces the number of primality tests required to find a Goldbach pair. Extensive numerical experiments demonstrate a stable efficiency gain relative to random search strategies. The results suggest the presence of a universal geometric organization underlying the Goldbach conjecture and provide a new perspective on additive problems involving prime numbers.
Category: Number Theory
[4] ai.viXra.org:2601.0072 [pdf] submitted on 2026-01-18 22:20:03
Authors: Budeeu Zaman
Comments: 22 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references)
The identity is explored using matrix and tensor formulations, integral representations,graph-theoretic methods,and a deep dive into the logarithmic analysis of prime sums these methodsshed light on fresh structural clues about how primes are spread out and their deep number-theoretic correlations.
Category: Number Theory
[3] ai.viXra.org:2601.0058 [pdf] submitted on 2026-01-16 02:12:46
Authors: Joshua Christian Elfers
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We introduce and analyze a carry-coupled skew-product extension of the Collatz dynamical system, in which whole-integer dynamics are coupled to modular arithmetic through an explicit arithmetic carry term. The central result establishes that coupling parameter K = 4 produces a unique invariant structure characterized by identity evolution of the return map. This coupling realizes, in purely arithmetic terms, the mathematical mechanism underlying conservation laws and coherent evolution in physical systems. We provide rigorous proofs of the main theorems, numerical verification, and discuss the group-theoretic foundations of this invariance.
Category: Number Theory
[2] ai.viXra.org:2601.0053 [pdf] submitted on 2026-01-14 15:22:34
Authors: Shanzhong Zou
Comments: 16 Pages.
The Twin Prime Conjecture, the Goldbach Conjecture, and the Polignac Conjecture are all conjectures concerning the distribution of prime numbers. This paper introduces a novel sieve-based framework termed the " Comb Sieve Method ". This approach not only unifies the three conjectures by treating them as different manifestations of an underlying fundamental problem but also circumvents the need to estimate error terms, a common challenge in traditional sieve methods. We believe this framework will open new avenues for research in this area of number theory.
Category: Number Theory
[1] ai.viXra.org:2601.0005 [pdf] replaced on 2026-01-04 13:55:51
Authors: Birke Heeren
Comments: 24 Pages.
This paper introduces the "Cosmos Automaton" (CA), a deterministic fractal automaton that generates prime numbers through symbolic operations rather than direct primality testing. By treating the sequence of natural numbers as a dynamic process, we show that primality emerges from the constructive interference of "pulse trains" (periodic symbolic words). We demonstrate that the CA’s structures are isomorphic to the set of natural numbers and that its evolutionary steps correspond to arithmetic progressions. This approach provides a visual and algorithmic bridge between automata and the distribution of primes, leading to a definition of primality based on geometric expansion. This provides a new algorithmic perspective on the structure of primes.
Category: Number Theory