Ricardo Adonis Caraccioli Abrego
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View article: Layers of Prime Gaps and Spectral Inheritance of Noise: An Analytic–Computational Study
Layers of Prime Gaps and Spectral Inheritance of Noise: An Analytic–Computational Study Open
This paper presents an analytic and computational framework to study noise in the distribution of prime numbers through layers based on multi-step prime gaps. Given the n-th prime pn, we consider the differences g(k,n) := p(n+k) − p(n), gr…
View article: Layers of Prime Gaps and Spectral Inheritance of Noise: An Analytic–Computational Study
Layers of Prime Gaps and Spectral Inheritance of Noise: An Analytic–Computational Study Open
This paper presents an analytic and computational framework to study noise in the distribution of prime numbers through layers based on multi-step prime gaps. Given the n-th prime pn, we consider the differences g(k,n) := p(n+k) − p(n), gr…
View article: Normalized Coefficient Linear Combinations (NCLC): A Unifying Framework for Discrete-Time Filters, Control, and Signal Analysis
Normalized Coefficient Linear Combinations (NCLC): A Unifying Framework for Discrete-Time Filters, Control, and Signal Analysis Open
This paper presents the Normalized Coefficient Linear Combination (NCLC) as a unifying structural framework for analyzing systems in digital signal processing (DSP), discrete-time control, and numerical sequence analysis. While traditional…
View article: Normalized Coefficient Linear Combinations (NCLC): A Unifying Framework for Discrete-Time Filters, Control, and Signal Analysis
Normalized Coefficient Linear Combinations (NCLC): A Unifying Framework for Discrete-Time Filters, Control, and Signal Analysis Open
This paper presents the Normalized Coefficient Linear Combination (NCLC) as a unifying structural framework for analyzing systems in digital signal processing (DSP), discrete-time control, and numerical sequence analysis. While traditional…
View article: Explicit Stability for Mills-Type Prime-Generating Constants
Explicit Stability for Mills-Type Prime-Generating Constants Open
Mills proved in 1947 that there exists a constant A>1 such that \( \lfloor A^{3^n}\rfloor \) is prime for all \( n\ge 0 \), using deep results about primes in short intervals. Later work (for example by Caldwell--Cheng) made this constr…
View article: Exact Identities for the Binary Hamming Weight Under Arithmetic and Bitwise Operations
Exact Identities for the Binary Hamming Weight Under Arithmetic and Bitwise Operations Open
We collect and prove exact identities for the binary digital sum S2(n)—the Hamming weight wt (n)—under elementary arithmetic and bitwise operations. For x, y ≥ 0 we derive explicit carry/borrow decompositions of wt(x + y) and wt (x − y) in…
View article: On Representations of Even Integers as a Sum of Two Semiprimes
On Representations of Even Integers as a Sum of Two Semiprimes Open
Representations of a large even integer N as a sum of two semiprimes (products of two primes, squares allowed) are studied. Using a smooth bilinear weight W localized on uv ≍ N and the Hardy–Littlewood circle method, an asymptotic formula …
View article: Continuous Universal Analog Logic (LCUA): Smooth Functional Equivalents of Digital Gates Without Comparators
Continuous Universal Analog Logic (LCUA): Smooth Functional Equivalents of Digital Gates Without Comparators Open
We present a framework to implement Boolean logic using only smooth functions (sums, products, and low-cost nonlinearities) without hard thresholds or explicit comparators, while exactly preserving truth tables on the Boolean vertices {0,1…
View article: Asymptotic Classification of Diophantine Equilibrium in the Base {2, 3, 5}: Lattice Geometry, Harmonic Proof, and Explicit Residues
Asymptotic Classification of Diophantine Equilibrium in the Base {2, 3, 5}: Lattice Geometry, Harmonic Proof, and Explicit Residues Open
We study nonnegative integer solutions (a,b,c) to 2a+3b+5c = n that minimize the dispersion of coefficients, equivalently the quadratic form Q(a,b,c) = 3(a^2+b^2+c^2) − (a+b+c)^2. We prove an asymptotic classification theorem: for sufficie…
View article: Exact Projections of the Real Part of ζ(s) Arithmetic Grids, Identities, Bounds, and a Mean-Square Formula in ℜ(s) > 1
Exact Projections of the Real Part of ζ(s) Arithmetic Grids, Identities, Bounds, and a Mean-Square Formula in ℜ(s) > 1 Open
We present a deterministic framework for decomposing the real part of the Riemann zeta function Re(zeta(s)) in the region Re(s) > 1 by means of _arithmetic grids_ (structured subsets of N). We deduce exact identities for multiplicative,…
View article: Algebraic Fractal Structure, the Principle of Least Effort, and the Strong Goldbach Conjecture
Algebraic Fractal Structure, the Principle of Least Effort, and the Strong Goldbach Conjecture Open
We present a conditional framework that combines two complementary ideas: (i) a "gearbox" identity that chains ternary representations from the Weak Goldbach Conjecture to produce Goldbach pairs, and (ii) a Principle of Least Additive Effo…
View article: Pathway Towards a Proof of the Riemann Hypothesis via Spectral Transfer
Pathway Towards a Proof of the Riemann Hypothesis via Spectral Transfer Open
The Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Rie- mann Zeta function lie on the critical line R(s) = 1/2, remains one of the most pro- found open problems in mathematics. This paper presents a potential pathw…
View article: A Heuristic Approach to the Strong Goldbach Conjecture Based on a Minimum Additive Prime Product Principle
A Heuristic Approach to the Strong Goldbach Conjecture Based on a Minimum Additive Prime Product Principle Open
This paper presents a heuristic exploration of the Strong Goldbach Conjecture from an optimization principle perspective. We postulate that the representation of an even number n as a sum of primes pi is governed by an "additive effort," d…
View article: Six Exact Formulations of the Sieve of Eratosthenes and Their Algebraic Equivalence
Six Exact Formulations of the Sieve of Eratosthenes and Their Algebraic Equivalence Open
We present and compare six mathematically exact formulations of the Sieve of Eratosthenes, ranging from arithmetic expressions with the floor function to elegant algebraic products using roots of unity, primorials, and factorials. We demon…
View article: The Inverse–Li Residue Sieve: A New Local-Analytic, Memory-Light Method for Computing the N-Th Prime
The Inverse–Li Residue Sieve: A New Local-Analytic, Memory-Light Method for Computing the N-Th Prime Open
We propose the Inverse–Li Residue Sieve (ILIRS), a novel local-analytic algorithm for com- puting the n-th prime P (n) that: • stores only O(1) floats (no bitmaps, no tables); • needs at most O(log n) deterministic Miller–Rabin tests per i…
View article: High-Precision Empirical Formula for Approximating the Next Prime Number
High-Precision Empirical Formula for Approximating the Next Prime Number Open