This paper presents the first complete and definitive disproof of Goldbach’s Conjecture, executed through a synthesis of deep mathematical insight, theoretical innovation, and structural rigour. Across a broad spectrum of modern mathematics, this work reveals foundational contradictions that undermine the conjecture’s claim to universality. The result is a clear and categorical conclusion that Goldbach’s Conjecture, while numerically resilient, collapses under formal scrutiny. This work not only resolves a centuries-old enigma but redefines the philosophical foundation of additive number theory. It proactively challenges the mathematical community to distinguish between empirical tradition and provable truth, and sets a new standard for resolving longstanding conjectures. In doing so, it transforms the landscape of number theory and establishes a model for multidisciplinary proof that enables future breakthroughs. This is not merely a mathematical achievement but a historic turning point as the era of Goldbach concludes, and the era of mathematical evolution begins.

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For over 160 years, the Riemann Hypothesis has stood as a cornerstone of modern mathematics: profound, elegant, and elusive. Revered for its connection to the distribution of prime numbers and the deeper structure of number theory, it has captivated generations of mathematicians, physicists, and philosophers alike. Yet despite its celebrated status, the RH has remained unproven, resting more on tradition, intuition, and partial evidence than on conclusive certainty. This work presents a definitive and comprehensive disproof of the Riemann Hypothesis. It is not another speculative exploration, but a conclusive argument grounded in mathematical rigour. Through eight independent and reinforcing lines of reasoning, including a holistic structural foundation, constructive counterexamples, contradictions from fundamental identities, breakdowns in spectral and statistical models, and failures of analytic criteria, this study demonstrates the untenability, implausibility, and inadmissibility of RH. Beyond its technical scope, the disproof invites a broader reflection on the nature of mathematical belief. The hypothesis has long been held up as a paragon of mathematical beauty, but this analysis reveals that its aesthetic appeal is no substitute for logical consistency. Every supporting structure of RH, be it core structural, analytic, probabilistic, or spectral, succumbs to scrutiny. The conclusion is both clear and consequential that the Riemann Hypothesis does not hold. What follows is not only a resolution to one of the most significant unsolved problems in mathematics, but a reaffirmation of the truth in mathematics, as in science, must ultimately rest on core foundational, verifiable and objective grounds. This marks the closing of a historic chapter and the beginning of a clearer understanding of the true landscape of real number theory.

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In this paper, we explain the logic for simultaneously solving linear congruences using the Chinese Remainder Theorem and apply it to develop an easy method for finding the solution. Mathematics Subject Classification [2020]: Primary 11A41.

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Let 𝑨 denote the multiplicative group {𝟏,−𝟏,𝒊, −𝒊}. In this paper, we define the notion of double uniform (𝑡1𝑙1,𝑡2𝑙2)-ply and prove that it is a group 𝑨-cordial with each path of length at least 5 by explicitly giving labellings for all the possible cases that arise. Mathematics Subject Classification [2020]: Primary 05C78

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In this paper, I investigate the method for constructing Maximum Distance Separable (MDS) codes using exponential sums. By utilizing the properties of the trace function over finite fields, I explicitly calculate linear codes associated with certain three-term exponential sums. The parameters, weight distributions, and distance properties of these codes are analyzed in detail. Furthermore, we analyse the syndrome-decoding process for these codes.

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This paper presents a straightforward and elementary proof of Fermat’s Last Theorem (FLT), asserting that there are no integer solutions to 𝒂 𝒏 + 𝒃 𝒏 = 𝒄 𝒏 for 𝒏 > 𝟐. Leveraging basic number theory and algebraic manipulations, we offer a concise demonstration that makes this fundamental result accessible to a broad mathematical audience.

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Many foundational rules in elementary and advanced mathematics are accepted as axiomatic, despite being introduced through heuristic or pedagogical arguments rather than strict logical or physical justification. This work undertakes a systematic analytical re-examination of several such widely accepted mathematical conventions to identify internal inconsistencies and clarify their conceptual basis. The study begins by reanalysing the four basic arithmetic operations—addition, subtraction, multiplication, and division—by reducing multiplication and division to repeated addition and subtraction. Within this framework, the conventional sign rules governing the multiplication of negative quantities are critically examined. It is shown that while the outcomes of these rules are operationally consistent, the standard logical justifications commonly provided are incomplete or internally inconsistent when interpreted in terms of direction, orientation, and repetition. Using geometric constructions and physically motivated examples, the work further examines the interpretation of signed quantities, demonstrating that positive and negative values naturally encode directionality rather than intrinsic magnitude. This perspective is extended to areas and volumes, which are typically assumed to be strictly non-negative. The analysis shows that signed areas and volumes can be meaningfully interpreted within a consistent mathematical–physical framework, thereby questioning one of the key assumptions underlying the introduction of imaginary quantities. The paper also revisits the treatment of fractional quantities, squaring operations, and dimensional comparisons, arguing that several commonly cited “paradoxes” arise from conflating quantities of different dimensions or from scaledependent representations rather than from inherent mathematical necessity. Additionally, division by zero and infinity is reinterpreted through the lens of repeated subtraction, yielding a physically intuitive understanding of divergence and null results. Finally, the exponential function and its Taylor series expansion are examined from combinatorial and geometric perspectives, offering an alternative interpretation of the exponential constant based on dimensional arrangements rather than abstract growth alone. Overall, this work does not seek to discard established mathematical tools, but to clarify their conceptual foundations by enforcing consistency across arithmetic, geometry, and physical interpretation. The results highlight the need for greater precision in distinguishing operational rules from their underlying logical and physical meanings.

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Jensen’s inequality is a fundamental result in probability and analysis, that offers simple boundsforthe convex function applied to a random variable. However, the classical form of this process does not include memory effects, which are very important in some physical and financial systems with long-range dependence. In this paper, we introduce a fractional-order generalisation of Jensen’s inequality involving memory effects that can be accounted for by means of fractional calculus. We concentrate on the exponential integral function Ei(x) because of its wide use. For a random variable Xwith mean µsupported on [1, ∞), and for every fractional-order α ∈ (0,1) we show the strict inequality Ei(µ) ≤ E[Ei(X)] − M(α), where the new quantity M(α) is a memory correction defined in terms of Riemann– Liouville fractional integrals of order 1 − αof the function e t /t. The correction term provides a whole family of bounds, controlled by the parameter α ∈ (0, 1], and depending on the specific behavior of X, as α→ 0+ , the bound reduces to Jensen’s gap, while for α → 1 —, the right-hand side approaches a new non-zero and pathdependent bound. The inequality is strict for non-degenerate X. (When M(α) ≥ 0, the bound is two-sided, Ei(µ) ≤ E[Ei(X)] − M(α) ≤ E[Ei(X)].) The proof is based on order-monotonicity properties of fractional integrals. An equivalent formulation of the result in terms of Caputo derivatives is also given. An illustrative interpretation is discussed in the context of economic utility, where the resulting bounds may be viewed as capturing nonlocal averaging effects in convex (risk-seeking) utility evaluations.

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In this paper, we show that a non-trivial zero of the Riemann zeta function occurs only when the complex number s = 𝒂/𝒃 + it, with 𝒂, 𝒃, 𝒕 ∈ 𝑹 and i² = -1 can be interpreted as a vector plus its inverse yielding zero displacement. We prove that for such a zero displacement to occur, the total distance covered by the vector and itsinverse must equal one unit, forcing the fundamental part of s to be 𝟏 𝟐 . We further show that no other fraction in the critical strip possesses this property. Consequently, no other fundamental part can host non-trivial zeros, thereby settling the Riemann Hypothesis.

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I will study the Sghiar’s function Ⓢ:(X,z)⟼∏_(p∈P)1/(1-X/p^z ), P the set of prime numbers. Which is an extension of the Riemann zeta function. The classical form of the Riemann zeta function and its Euler product are well known in analytic number theory, and I will show that : ζ(s)=0 and Re(s)>1/2⇒Ⓢ=0. We deduce the proof of the Riemann Hypothesis. MSC code: 11M26 ; 97F60 ; 32A10

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