Abstract:

This article develops a structural framework that reduces Goldbach’s Strong Conjecture to a single short-interval analytic inequality. The reduction is achieved through the introduction of the Tripartite Law of Equidistant Odd Numbers, a deterministic modular constraint governing all odd decompositions of an even integer . Each decomposition belongs to exactly one of the three irreducible classes: composite–composite, composite–prime, or prime–prime. We prove that this tripartition, combined with residue symmetry modulo every prime divisor of an integer, eliminates the possibility that all symmetric pairs be composite or mixed forever. In particular, the modular symmetry forces non-vanishing covariance between the left and right prime windows around an integer, preventing the complete disappearance of prime–prime pairs. Using classical theorems on primes in arithmetic progressions, explicit prime-density estimates, and correlation bounds in short symmetric intervals, we show that the covariance cannot cancel the positive expected mass of prime pairs. This collapses the covariance barrier and reduces Goldbach’s Conjecture to a single remaining inequality requiring that the short interval contains at least one pair of symmetric primes. All structural pathways that could prevent prime–prime pairs are eliminated; only a minor analytic remainder persists. Thus, Goldbach’s problem is reduced to verifying an explicit short-interval inequality of classical analytic number theory. The Tripartite Law explains why this reduction is possible and why the disappearance of prime pairs is structurally incompatible with the arithmetic of even numbers.

Abstract: This paper presents a unified analytic–conditional framework for resolving Goldbach’s Conjecture. The first part develops an analytic demonstration based on the mirror-law symmetry of prime densities around E/2 and the overlap of symmetric windows derived from the Prime Number Theorem. This analytic structure shows that the density of primes on both sides ofE/2 necessitates the existence of at least one symmetric prime pair for every sufficiently large even integer E. The second part formulates Goldbach’s Conjecture as an equivalent conditional theorem requiring the validity of only two explicit lemmas: a local symmetry lemma (Lemma C) and a global overlap lemma (Lemma S). Appendix 8 provides a mathematical derivation of Lemma C, while Appendix 8B establishes a partial reduction strategy for Lemma S. Appendix 9 identifies an explicit threshold E₀ beyond which the analytic overlap is guaranteed. The resulting hybrid manuscript gives both (1) an analytic justification for the structure underlying Goldbach’s identity, and (2) a precise blueprint for reducing the conjecture to two explicit, finitely verifiable lemmas. This combined approach significantly narrows the gap between heuristic,analytic, and conditional pathways to a full resolution.

Abstract:

We study the additive and fractal structure of digit-restricted subsets of the unit interval, \( A_D=\Bigl\{ \sum_{n=1}^\infty a_n b^{-n} : a_n\in D\subseteq\{0,\dots,b-1\},\ |D|\ge2\Bigr\} \) defined by allowing only digits from D in base-b expansions. These sparse-digit sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous carry-propagation framework for base-b digit arithmetic, give sharp combinatorial criteria for intervals in \( A_D+A_D \) and \( A_D-A_D \), expand the proof of the similarity-dimension formula, and strengthen the dimension-jump theorem for iterated sumsets \( A_D^{(k)} \) by providing full justification and relevant references. A new Carry Stabilization Lemma, an expanded interval-criterion proof, and a related-work section situate these results within the literature on sumsets of Cantor sets, fractal addition theorems, and digit-based self-similar sets.

Abstract: This work investigates the relationship that exists between the parastrophes of some notion of inverses in quasigroups. Our findings revealed that, of the 5 parastrophes of LIP quasigroup, (23)- parastrophe is a LIP quasigroup, (12)- and (132)- parastrophes are RIP quasigroups, while (13)- and (132)- parastrophes are an anti-commutative quasigroup. Similarly, the (12)- and (132)-parastrophes of a RIP quasigroup are LIP quasigroups; the (13)-parastrophe of a RIP quasigroup is an RIP quasigroup, while the (23)- and (123)-parastrophes are anti-commutative quasigroups. As for the CIP quasigroup, only (12)- parastrophe is a CIP quasigroup; other parastrophes are symmetric quasigroups of order 2. Finally, (12)-parastrophe of WIP quasigroup is an IP quasigroup, (13)-,(23)- and (132)-parastrophes of WIP quasigroup are CIP quasigroups, while (123)-parastrophe of WIP quasigroup is a WIP quasigroup.

Abstract: We develop a complete operator-theoretic and spectral framework for the Collatz map using its backward transfer operator acting on weighted Banach spaces of arithmetic functions. The associated Dirichlet transforms form a holomorphic family that isolates a zeta-type pole at s = 1 and a holomorphic remainder, while on a finer multiscale space adapted to the Collatz preimage tree we prove a Lasota–Yorke inequality with an explicit contraction constant λ < 1, yielding quasi-compactness, a spectral gap, and a Perron–Frobenius theorem in which ρ(P) = 1, the eigenvalue 1 is algebraically and geometrically simple, no other spectrum lies on the unit circle, and the unique invariant density is strictly positive with a c/n decay profile. The fixed-point relation Ph = h is converted into an exact multiscale recursion for the block averages c_j, showing that mass propagation along the preimage tree is governed by a rigid two-scale coupling with exponentially small error terms. The spectral classification then forces every weak-star limit of the Cesàro averages Λ_N(f) of any hypothetical infinite forward orbit to be either 0 or a scalar multiple of the Perron–Frobenius functional, and convergence to 0 occurs precisely under the Block–Escape Property, an extreme transience condition that drives the block index to infinity. The forward map always satisfies an unconditional exponential upper bound, whereas Block–Escape combined with linear block growth along a subsequence would create a contradictory exponential lower bound. All analytic and spectral components of the proof are therefore complete, and the Collatz conjecture is reduced to a single forward-dynamical problem: to exclude infinite forward orbits that satisfy the Block–Escape Property without forcing the kind of linear block growth incompatible with the spectral bounds.

Abstract: This work establishes a complete arithmetic resolution of the Collatz Conjecture by decomposing the odd–to–odd dynamics into two complementary structures: a local residue–phase automaton and a global affine counting system. The reverse map R(n;k)=(2^{k}n-1)/3 is shown to act on the live residues {1,5} mod 6 through a finite residue–phase state space, while every admissible exponent k=c+2e induces an affine expansion factor $2^{k}$ whose inverse coincides exactly with the dyadic slice weight 2^{-k}.From this, every odd integer is seen to belong to a unique dyadic slice {S}{c,e}, forming a disjoint partition of {N}odd. Independently, the introduction of the zero–state index Z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic ladder m -> 4m+1 whose union also partitions the odd integers without overlap. We prove that these two partitions coincide exactly, yielding a unified global structure in which all odd integers arise from admissible lifts above anchors {1,5}.The locked forward–reverse equivalence T(n)=(3n+1)/2^{\nu_2(3n+1)} and R(T(n);k)=n then implies that forward trajectories cannot branch or diverge: each forward iterate lies on a single admissible ladder descending toward its zero–state origin at 1. Because the residue–phase automaton is finite and every ladder has a uniquely determined forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. All constructions, residue frameworks, and affine decompositions used in this paper are original to this work. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every forward trajectory converges to 1.

Abstract:

A theta--regularized inner product identity in rank one is established, linking a mixed theta--weighted Eisenstein pairing on \( \Gamma \)\H to the \( \sigma \)--derivative of \( \log|\xi(s)| \), up to explicit Euler factor correction terms arising from the \( G\times G \) doubling formalism. More precisely, for \( s=\tfrac12+\sigma+t \) it is shown that \( \frac{\partial}{\partial\sigma}\log\left|\big\langle\Theta(\cdot)E(\cdot,s),\ \Theta(\cdot)E(\cdot,1-\overline{s})\big\rangle_{\mathrm{reg}}\right|=2\,\mathrm{Re}\,\frac{\xi'(s)}{\xi(s)}\ -\ 2\,\mathrm{Re}\,\frac{\zeta'(2s)}{\zeta(2s)}\ +\ 2\,\mathrm{Re}\,\frac{\zeta'(2-2\overline{s})}{\zeta(2-2\overline{s})} \),as an identity of tempered distributions in t. On the critical line \( \sigma=0 \) the Euler corrections cancel and a particularly simple formula is obtained:\( \frac{\partial}{\partial\sigma}\log\big\langle\Theta(\cdot)E(\cdot,\tfrac12+\sigma+t),\ \Theta(\cdot)E(\cdot,\tfrac12-\sigma+ t)\big\rangle_{\mathrm{reg}}\Big|_{\sigma=0}=2\,\frac{\partial}{\partial\sigma}\log\left|\xi\bigl(\tfrac12+\sigma+t\bigr)\right|\Big|_{\sigma=0} \). Fejér--windowed versions of these identities are then obtained, and a Fejér--windowed "strip bridge'' is proved: a harmonic operator identity expressing the short--band component of \( \partial_\sigma\log|\xi(1/2+\sigma+t)| \) at an interior latitude via a linear combination of Fejér--smeared edge data, with a power--saving \( O(H^{-\eta}) \) remainder after short--band freezing, uniformly for \( |\sigma^\star|\ge \sigma_0>0 \). A sharp truncation stability result is also established. After subtracting the finitely many Zagier--Arthur cusp counterterms, the Fejér--smeared \( \sigma \)--derivative of the logarithm of the truncated mixed theta--Eisenstein pairing agrees with its regularized version up to \( O(H^{-A}) \) for any prescribed \( A>0 \), provided the truncation height \( Y=H^{B(A)} \) is chosen sufficiently large. A brief discussion is included of numerical checks in a sample region, and a short Fourier--analytic proof note is given for the renormalization estimate that underlies the strip bridge.

Abstract: I introduce an n-dimensional canonical number system (CNS) on modules: a finite base frame b, an E-linear shift T, and a finite digit set D yield unique finite expansions along places \( \{T^{k}(b_i)\} \). For a finitely generated ring \( R=\mathbb Z[a_1,\dots,a_n] \), I use the presentation \( R\cong \mathbb Z[x_1,\dots,x_n]/\ker(\phi) \) and compute \( \ker(\phi) \) via a graph-ideal elimination scheme. When the quotient has finitely many standard monomials, choosing a non–zero-divisor prime p equips R with a base-p CNS (pre-folding) that uses those monomials as places. A digit-folding lemma then compresses coordinates whenever some place has only finitely many powers, preserving uniqueness. This provides a constructive pipeline from presentations of finitely generated, countable rings to explicit multi-dimensional CNS representations, supporting the conjecture that every such ring is isomorphic to an n-dimensional CNS.

Abstract: We present several constructions of classical constants such as e, π, and the golden ratio φ using only prime numbers or prime–based sequences (for instance, prime indices in the Fibonacci and Lucas sequences). Our goal is not numerical efficiency but to highlight the pedagogical connection between prime arithmetic and the analytic nature of fundamental constants. We show how the Prime Number Theorem and Euler’s product for the Riemann zeta function allow us to reinterpret e and π as “prime-generated” limits or products, and we suggest classroom activities for calculus, real analysis, and introductory number theory courses.

Abstract: The binary Goldbach conjecture states that every even integer greater than 2 is the sum of two primes. We analyze a variant of this conjecture, positing that every even integer 2N ≥ 8 is the sum of two distinct primes P and Q. We establish a novel equivalence between this statement and a geometric construction: the conjecture holds if and only if for every N ≥ 4, there exists an integer M ∈ [1, N − 3] such that the L-shaped region N2 − M2 (between nested squares) has a semiprime area P · Q, where P = N − M and Q = N + M. We define the set DN = n Q−P 2 2 < P < N < Q < 2N, P, Q prime o of half-differences arising from prime pairs straddling N with Q < 2N. The conjecture is equivalent to the non-emptiness of DN ∩ {N − p | 3 ≤ p < N, p prime}. We conduct a computational analysis for N ≤ 214 and define a gap function G(N) = log2(2N) − ((N − 3) − |DN|). Our experimental results show that the minimum of G(N) is positive and increasing across intervals [2m, 2m+1]. This result, G(N) > 0, establishes that |DN| > (N − 3) − log2(2N). Under this bound, the pigeonhole principle applied to the cardinality of the candidate set {N − p | 3 ≤ p < N, p prime} (of size π(N − 1) − 1) and the bad positions (of size (N − 3) − |DN| < log2(2N)) implies a non-empty intersection for all N ≥ 4, yielding a proof of the conjecture. Our work establishes a novel geometric framework and demonstrates its viability through extensive computation.

Abstract: The Riemann Hypothesis, one of the most celebrated open problems in mathematics, addresses the location of the non-trivial zeros of the Riemann zeta function and their profound connection to the distribution of prime numbers. Since Riemann’s original formulation in 1859, countless approaches have attempted to establish its truth, often by examining the asymptotic behavior of arithmetic functions such as Chebyshev’s function θ(x). In this work, we introduce a new criterion that links the hypothesis to the comparative growth of θ(x) and primorial numbers. By analyzing this relationship, we demonstrate that the Riemann Hypothesis follows from intrinsic properties of θ(x) when measured against the structure of primorials. This perspective highlights a striking equivalence between the distribution of primes and the analytic behavior of ζ(s), reinforcing the deep interplay between multiplicative number theory and analytic inequalities. Beyond its implications for the hypothesis itself, the result offers a fresh framework for understanding how prime distribution governs the analytic landscape of the zeta function, thereby providing new insight into one of mathematics’ most enduring mysteries.

Abstract: This paper explains why the critical line sits at real part equal to one-half by treating it as an intrinsic boundary of a reparametrized complex plane (“z-space”), not a mere artifact of functional symmetry. In z-space the real part is defined by a geometric-series map that induces a rulebook for admissible analytic operations. Within this setting we rederive the classical toolkit—eta–zeta relation, Gamma reflection and duplication, theta–Mellin identity, functional equation, and the completed zeta—without importing analytic continuation from the usual s-variable. We show that access to the left half-plane occurs entirely through formulas written on the right, with boundary matching only along the line with real part one-half. A global Hadamard product confirms the consistency and fixed location of this boundary, and a holomorphic change of variables transports these conclusions into the classical setting.

Abstract: The Riemann Zeta function ζ(s) lies at the heart of analytic number theory, encoding the distribution of primes through its non-trivial zeros. This paper introduces a direct computational framework for evaluating Z(t) = eiθ(t)ζ( 1/2 + it) at large imaginary parts t, employing a real-to-complex number conversion and a novel valley scanner algorithm. The method efficiently identifies zeros by tracking minima of |Z(t)|, achieving stability and precision up to t ≈ 10^20 with moderate computational cost, using AWS EC2 computation. Results are compared against known Andrew Odlyzko zero datasets, validating the method’s accuracy while simplifying the high-t evaluation pipeline.

Abstract: This paper advances the Collatz conjecture by analyzing binary representations of natural numbers through fractional parts. We introduce a direct non-recursive relation for intermediate mantissas $\sigma_j$ in binary decompositions and prove their equidistribution using Weyl's theorem. The self-correcting dynamics of $\sigma_j$ ensure a balance between 1s and 0s, leading to an asymptotic density of 1/2 for 1s in binary expansions of $3^n$. This yields a probabilistic estimate: in approximately half of all cases, the binary expansions have many leading zeros, ensuring rapid descent. Theorems estimate zero density in powers of three and demonstrate sequence decrease for large $n$. Numerical verifications and updated figures support the findings, providing strong evidence for convergence in large cases.

Abstract: Goldbach’s conjecture—every even number greater than two is the sum of two primes—has stood for nearly three centuries as one of mathematics’ most captivating challenges. This review presents an accessible narrative of the Bahbouhi λ-analytic program, an approach that translates Goldbach’s statement into a problem of symmetry and density. The key idea is to represent the local distribution of primes by a continuous “λ-field,” expressing prime density through the analytic form λ(x) ≈ 1 / ln x. Two mirrored density functions, λ₁(E / 2 − t) and λ₂(E / 2 + t), evolve on opposite sides of the midpoint E / 2 of an even integer E. Their intersection corresponds to the existence of symmetric prime pairs. Through this lens, Goldbach’s conjecture becomes a theorem about the inevitable meeting of two analytic curves. The paper explains how this formulation unifies heuristic intuition with rigorous analysis: continuity ensures intersection, monotonicity fixes its uniqueness, and the reduction theorem converts analytic equality into the existence of genuine primes under mild covariance hypotheses. The review also connects this λ-framework to classical theorems—the Prime Number Theorem, Hardy–Littlewood conjectures, and explicit bounds of Dusart—and outlines future extensions linking λ-symmetry with the Riemann ζ-function. The narrative is written for mathematicians, teachers, and students interested in understanding the path from intuition to analytic proof.

Abstract: The $n$-cube graphs (or the $n$-hypercube graphs) $\mathcal{Q}_n$ could be seen as the Cayley graph $\caay(G,\mathcal{S})$ for $G=\bigoplus\limits_r\mathbb Z_{2}^{r^2}$ and $\mathcal{S}:=\{E_{ij}=(e_{op})_{1\le o,p\le n} \ | \ e_{ij}=1 \ \& \ e_{op}=0 \ \text{for} \ o\neq i \ \text{and} \ p\neq j\}$. It is a well-known result that the quantum automorphism group $G_{\qaut}(\mathcal{Q}_n):=H_{n}^{+}$ is isomorphic with the anticommutative orthogonal quantum group $\mathcal{O}(O_{r}^{-1})$, for some $r$ dependent on $n$ (here in this paper we will try to specify $r$) but our knowledge helps, as we talk, the main generators of $G_{\qaut}(\mathcal{Q}_n)$ have not been observed yet. So, in this paper we tried to study them along with the Cayley graphs such as $\caay(\mathbb K_2,\mathcal{S})$ for the free group on two generators $\mathbb K_2$, and the folded $n$-cube graphs $\mathsf{F}\mathcal{Q}_{n}$ and their generalized versions $\prescript{3}{\Delta}{\mathsf{F}\mathcal{Q}_{n}}$. In the last part of this paper, we tried to study the quantum symmetric groups of infinite dimensional spaces, by looking at them as arbitrary matrix spaces with points considered arbitrary matrices!We believe that this way of observation might help providing a partial answer to the open problem on the unit distance graphs and their quantum symmetries!

Abstract: The advances related to Modular Elliptic Curves and the Fermat – Wiles’ theorem in recent years has opened the door to entirely new approaches to numerous problems and techniques. This paper studies the linking between a family of exponential function and the Fermat – Wiles’ theorem. In this family, the convergence of the coordinates of the abscissas of the maxima and the equation determined by the fulfilment of the maximum condition leads to a singular connection with the Fermat – Wiles’ theorem.

Abstract: For nearly three centuries, Goldbach’s Strong Conjecture has resisted both analytic proof andcomplete computational verification. In this work a new perspective is introduced—the*Goldbach Circle*—which unites analytic number theory, geometric symmetry, and empiricalregularity into a single predictive law. Every even number E is represented as a circle ofdiameter E, with its midpoint E / 2 serving as the axis of mirror symmetry. The analyticprime-density function λ(x) = 1 / (x ln x) defines two mirrored density fields,λ₁(E / 2 − t) and λ₂(E / 2 + t). Their continuous equality determines a bounded overlapwindow Δ(E) ≈ K (ln E)², with K ≈ 0.1, in which at least one symmetric pair of primes(p, q) exists such that p + q = E and |p − E / 2| = |q − E / 2| = t.Geometrically, this corresponds to two arcs on the circle whose angular separationφ = 2 t / E = 2 K (ln E)² / E shrinks inversely with E, ensuring persistent intersection.The resulting *Predictive Goldbach Law*, p = E / 2 − K (ln E)², q = E / 2 + K (ln E)²,links analytic continuity, geometric curvature, and empirical stability acrosstested domains up to 10¹⁸.The model transforms Goldbach’s problem from an additive conjecture into a symmetry theorem:prime distribution behaves as a continuous field of mirror equilibrium. The analytic andgeometric forms converge to the same invariant, revealing that Goldbach’s Conjecture is thenatural expression of the constant curvature of the prime field.

Abstract: In some previous works we gave algorithms for determining generators of power integral basis in sextic fields with a quadratic subfield, under certain restrictions. The purpose of the present paper is to extend those methods to the general case, when the relative integral basis of the sextic field over the quadratic subfield is of general form. This raises several technical difficulties, that we consider here.

Abstract: We propose a novel framework utilizing a full binary tree structure to systematically represent the set of natural numbers, which we classify into three subsets: pure odd numbers, pure even numbers, and mixed numbers. Within this framework, we employ a binary string representation for natural numbers and develop a comprehensive composite methodology that integrate both odd- and even-number functions. Our investigation centers on the itera- tive dynamics of the Collatz function and its reduced variant, which effectively serves as a pruning mechanism for the full binary tree, enabling rigorous exam- ination of the Collatz conjecture’s validity. To establish a robust foundation for this conjecture, we ingeniously incorporate binary strings into an algebraic formulation that fundamentally captures the intrinsic properties of the Col- latz sequence. Through this analytical framework, we demonstrate that the sequence generated by infinite iterations of the Collatz function constitutes an eventually periodic sequence, thereby providing a discuss of this long-standing mathematical conjecture that has remained unresolved for 87 years.