In this research a neccessary and sufficient condition for the proof of the Binary Goldbach conjecture is established. It is established that the square of all natural numbers greater or equal to 2 have an additive partition equal to the sum of the square of a natural number greater or equal to zero and a Goldbach partition semiprime. All Goldbach partition semiprimes are odd except 4. This finding is in itself proof all composite even numbers have at least Goldbach partition. The result of the proof of the Binary Goldbach conjecture is used to prove the Andrica and Legendre conjectures. The Riemann hypothesis is examined and sources of non trivial zeroes outside the critical strip are discussed. An example example of a non-trivial zero outside the critical strip is given

The Goldbach Conjecture, which is frequently termed as “1 + 1”, has been a fascinating goal for many mathematicians over centuries. A Chinese mathematician, Dr. Jingrun Chen, proved 1 + 2, which is a great success and the best result so far achieved. Although there were several attempts proving the conjecture, these attempts are either tediously long, complicated, or logically imperfect, thus not widely accepted. Taking advantage of the periodicity of primes, here the author provides a straight forward rigorous proof for the Conjecture.

A proof is given showing the Collatz Conjecture is true for all positive integers. The proof was possible once the perspective of the proof was changed from looking at the pattern of the positive integers to looking at the conjecture rules. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and pathways to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers ultimately reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer.

The paper presents two novel mathematical approaches to prove Goldbach's conjecture. The conjecture states that every even number greater than or equal to 6 can be expressed as the sum of two odd prime numbers. We propose that the conjecture has only four genetic codes, and as long as one of them remains valid, the conjecture continues to hold. Through these new mathematical ideas, we aim to prove the validity of Goldbach's conjecture comprehensively.

This paper addresses the Collatz Conjecture, an open question in mathematics that postulates all positive integers will eventually reach one when a pair of specific operations are repeatedly applied. Despite its apparent simplicity, the conjecture lacks a formal proof. To tackle this enigma, we introduce Algebraic Inverse Trees (AITs), data structures that trace inverse operations of the Collatz sequence. This new approach not only elaborates our unique methodology but also sheds light on the underlying complexities of the Collatz Conjecture.

This article presents a novel resolution of the Collatz Conjecture, centered on the concept of Generative Completeness of the inverse Collatz function. We introduce and rigorously prove that for all N ∈ ℕ⁺, there exists a minimal generator m_N = 1 such that all positive integers up to N can be generated through successive applications of the inverse Collatz function G. This key property, which we term "Generative Completeness", forms the cornerstone of our proof. Building upon this foundation, we establish several crucial results: The boundedness of all Collatz sequences The existence and uniqueness of cycles in Collatz sequences The nature of the unique cycle as {1, 4, 2} We then present three distinct approaches to resolving the Collatz Conjecture, all fundamentally rooted in the Generative Completeness property. These diverse methods not only prove the conjecture but also provide deep insights into the structure of Collatz sequences.

In this article, we present solutions for one of the oldest mathematical problems, the Collatz Conjecture, and provide a proof for the Riemann Hypothesis, utilizing a new number theory based on newly discovered number properties, which will also be presented in this paper. This is made possible through the Sophy-Peter mathematical framework, built upon this new number theory. The Collatz Conjecture will be disproven, but as an alternative, the Oscillating Theorem will be introduced, with its correctness proven within this article. Furthermore, we present the general version of the Riemann zeta function, developed based on the new number theory. The correctness of this function is verified by comparing it with existing results of the zeta function. Using this approach and the Sophy-Peter framework, we have successfully proven the Riemann Hypothesis, long considered a millennium problem. Moreover, since the general zeta function is proven, this implies the correctness of the new number theory and the Sophy-Peter framework.

This article integrates existing theory from distributed computing and cryptology with anecdotal material from the cryptocurrency industry, to provide a comprehensive description of the minimum requirements of the hypothetical academic blockchain. The paper argues that such a community could significantly reduce the biases and misconduct that now exist in the academic peer review process. Theory suggests such a system could operate effectively as a distributed encrypted telecommunications network where nodes are anonymous, do not trust each other, and there is minimal central authority. To incentivize the academic community to join such a proposed community, the paper proposes a pseudo-cryptocurrency called litcoin (literature coin). This litcoin-based system would create economic scarcity based on proof of knowledge (POK), which is a synthesis of the proof of work (POW) mechanism used in bitcoin, and the proof of stake (POS) mechanism used in various altcoin communities. The paper argues that the proposed POK system would enable the academic community to more effectively develop the research it finds valuable.

This paper explores the geometric impossibility of angle trisection within Euclidean geometry, with a particular focus on the 90° angle. It presents a rigorous proof that highlights the fundamental limitations of various approaches to angle trisection, highlighting their failure to achieve universality. The paper critiques modern proofs that allow for the trisection of certain angles while dismissing others, arguing that such methods are inherently flawed due to their lack of geometric consistency. The paper introduces a new perspective through a new property named “inconsistent property”, a concept derived from geometric operations involving proportional magnitudes. This property reveals contradictions within the framework of Euclidean geometry, paralleling the logical challenges faced in proving the impossibility of trisecting an arbitrary angle. By drawing comparisons between traditional and contemporary proofs, the paper demonstrates that inconsistencies arise not only from specific cases but also from broader geometric principles. The findings reaffirm the robustness of Euclidean geometry in addressing the trisection problem and challenge the validity of alternative proofs that do not adhere to universal geometric principles. This paper contributes to a clearer understanding of the limitations of angle trisection methods and encourages further investigation into the technical implications and broader impact of the angle impossibility proof.

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by conjugate complex roots. Finally, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$ with solution $\alpha_i= \frac{1}{2}, i \in \mathbb{N}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Therefore, a proof of the Riemann Hypothesis is achieved.

This short technical note proves some observed novel corollaries of the Collatz Conjecture. For this, a unique ‘Collatz Representation’ of Hailstone numbers has been employed. After that, an extension of this corollary is explained using the fact that only certain differences in the last digit of the ‘Collatz Representation’ gives a whole number difference between numbers. A more general result is obtained. The final proof is for a corollary that some numbers the same number of steps to reach 1 if the Collatz Conjecture is true and all variables are whole numbers.

This paper presents a new proof of the infinite nature of prime numbers. Grounded in fundamental Set theory and Analysis, the proof uses the method of contradiction to demonstrate the absurdity of assuming the set of all prime numbers to be finite.

In the realm of autonomous driving tort, a significant disparity exists in the parties' access to autonomous driving data and essential technical information, resulting in challenges in unilateral proof. The traditional burden of proof framework in driving litigation is inadequate for direct application in the autonomous driving sphere. As we approach the era of widespread autonomous driving operations, there is an urgent need to clarify and redefine the allocation of the burden of proof in specific litigations. Utilizing comparative legal analysis and case studies, this paper delves into the disparities in legislative provisions concerning the burden of proof for autonomous driving in Germany and China. China can learn from Germany's legislative precedence in shifting the burden of proof for 'product defect' and 'fault' onto the manufacturer, thereby requiring the infringed party to merely furnish preliminary evidence indicating a 'causal relationship between the defect and the damage.' This approach mitigates the evidentiary burden on the aggrieved party, clarifies litigation procedures, incentivizes manufacturers to enhance technology, reinforces risk management, and ultimately, facilitates the progression of autonomous driving technology.

This study builds on our previous Systematic Literature Review (SLR) that assessed the applications and performance of zk-SNARK, zk-STARK, and Bulletproof Non-Interactive Zero-Knowledge Proof (NIZKP) protocols. To address identified research gaps, we designed and implemented a benchmark comparing these three protocols using a dynamic MiMC hash application. We evaluated performance across four general-purpose programming libraries and two programming languages. Our results show that zk-SNARK produced the smallest proofs, while zk-STARK generated the largest. In terms of proof generation and verification times, zk-STARK was the fastest, and Bulletproof the slowest. Interestingly, zk-SNARK proofs verified marginally faster than zk-STARK, contrary to other findings. These insights enhance our understanding of the functionality, security, and performance of NIZKP protocols, providing valuable guidance for selecting the most suitable protocol for specific applications.

Adjunctive treatment of periodontitis lacks solutions allowing enough time for wound healing in the periodontal pockets by avoiding a fast re-colonization. Such a solution might be an antibiotic containing formulation with a controlled release over weeks. Here, a recently described minocycline containing approach is qualified for further clinical development by focusing on proof-of-concept, systemic burden, resistance development and degradation studies. Animal studies were done in two different (mouse-chamber, rat Porphyromonas gingivalis challenging) models, including effects on inflammation markers, bone loss and bone structure. Also, serum concentrations of minocycline after local application were determined by HPLC-MS/MS. The resistance status of bacterial clinical isolates against minocycline was investigated and the degradation of the formulation was characterized by laser scanning- and scanning electron microscopy. Animal studies clearly demonstrated the applicability of the new formulation in the investigated models. Inflammation markers decreased dosage dependent and reduced bone loss compared to non-treated group was observed. Thereby, the systemic burden of the antibiotic was neglectable. Minocycline is still effective against oral pathogens; resistance development was not seen. The biodegradable thread was first swollen and subsequently degraded over weeks. The results support the continued clinical development of this new formulation. A phase I clinical trial is planned to further evaluate its safety and efficacy.

In the era of cloud computing, guaranteeing the safety and effectiveness of data management is of utmost importance. This investigation presents a novel approach that amalgamates sharding concept, encryption, zero-knowledge proofs (zkp), and blockchain technology for secure data retrieval and data access control to improve data security, efficiency in cloud storage and migration. Further we utilize user-specific digital wallets for secure encryption keys in order to encrypt the file before storing into the cloud. As Large files ( greater than 50 MB) or Big data file ( grater than 1 TB) computational complexity is more we leverage on sharding concept to enhance both space and time complexity in cloud storage, hence the large files are divided into shards and stored in different database servers. We also employed blockchain smart contract to enhance secure retrieval of the file and also a secure access method which ensures the privacy of the user. The zk-snark protocol is utilized to ensure the safe transfer of data between different cloud services. By utilizing this approach, data privacy is preserved, as only the proof of the data’s authenticity is shared with the verifier at the destination cloud, rather than the actual data itself. The suggested method tackles important concerns related to data protection, privacy, and efficient resource utilization in cloud computing settings by ensuring it meets all the cloud policies require to store data. As the environment maintains privacy of the user data and by not storing the raw data of the user anywhere, the entire environment is setup as a Zero trust model.

Zero-Knowledge Proofs ZKP provide a reliable option to verify that a claim is true without giving detailed information other than the answer. A classical example is provided by the ZKP based in the Graph Isomorphism problem (GI), where a prover must convince the verifier that he knows an isomorphism between two isomorphic graphs without publishing the bijection. We design a novel ZKP exploiting the NP-hard problem of finding the algebraic ideal of a multivariate polynomial set, and consequently resistant to quantum computer attacks. Since this polynomial set is obtained considering instances of GI, we guarantee that the protocol is at least as secure as the GI based protocol.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

According to the Global Report on Internal Displacement (GRID, 2022), the number of displaced people worldwide at the end of 2021 was 59.1 million, with 53.2 million of them being the result of conflicts and violence, and 5.9 million caused by climate-related disasters. Furthermore, the In-ternational Organization for Migration (IOM) predicts that the climate crisis will be solely re-sponsible for one billion migrants by 2050. In this context, the research examines the issue of coastal multi-risk arising from a combination of heavy rainfall, river's overflow, and sea-level rise phenomena, analysing it through the disciplinary perspective of urban planning. Specifically, the study focuses on the need to upgrade the Local Urban Plan from a climate-proof perspective by defining a method aimed at updating the territory's cognitive framework through the preparation of a multi-risk map (which considers different time horizons), as well as the prescriptive apparatus of the Plan through the definition of a toolkit of site-specific design actions oriented towards urban adaptation. The proposed methodology is verified on the case study of Isola Sacra, Fiumicino (Italy), and it is intended to be applicable to other territorial contexts. The research aims to outline guidelines for updating urban planning tools from a climate-proof perspective to promote long-term resilient urban development in risk-prone areas.

We consider a random dynamical system arising in a model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend relatively long time visiting attractor's ruins.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

This paper first discusses a fractional-order Liu system of order as low as 2.7 and shows its chaotic characteristics by carrying out numerical simulations such as Lyapunov exponents, bifurcation diagrams and phase portraits. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, the fractional-order system is implemented on a Field Programmable Gate Array (FPGA) and the results obtained show that the fractional-order Liu system is indeed chaotic.

Cryptocurrencies like Bitcoin rely on a proof-of-work system to validate transactions and prevent attacks or double-spending. A new proof-of-work is introduced which seems to be the first number theoretic proof-of-work unrelated to primes: it is based on a new metric associated to the Collatz algorithm whose natural generalization is algorithmically undecidable: the inflation propensity is defined as the cardinality of new maxima in a developing Collatz orbit. It is numerically verified that the distribution of inflation propensity slowly converges to a geometric distribution of parameter $0.714 \approx \frac{(\pi - 1)}{3}$ as the sample size increases. This pseudo-randomness opens the door to a new class of proofs-of-work based on congruential graphs.

Blockchain technology has been recognized as a promising solution to enhance the security and privacy of Internet of Things (IoT) and Edge Computing scenarios. Taking advantage of the Proof-of-Work (PoW) consensus protocol, which solves a computation intensive hashing puzzle, Blockchain assures the security of the system by establishing a digital ledger. However, the computation intensive PoW favors members possessing more computing power. In the IoT paradigm, fairness in the highly heterogeneous network edge environments must consider devices with various constraints on computation power. Inspired by the advanced features of Digital Twins (DT), an emerging concept that mirrors the lifespan and operational characteristics of physical objects, we propose a novel Miner-Twins (MinT) architecture to enable a fair PoW consensus mechanism for blockchains in IoT environments. MinT adopts an edge-fog-cloud hierarchy. All physical miners of the blockchain are deployed as microservices on distributed edge devices, while fog/cloud servers maintain digital twins that periodically update miners’ running status. By timely monitoring miner’s footage that is mirrored by twins, a lightweight Singular Spectrum Analysis (SSA) based detection achieves to identify individual misbehaved miners that violate fair mining. Moreover, we also design a novel Proof-of-Behavior (PoB) consensus algorithm to detect byzantine miners that collude to compromise a fair mining network. A preliminary study is conducted on a proof-of-concept prototype implementation, and experimental evaluation shows the feasibility and effectiveness of proposed MinT scheme under a distributed byzantine network environment.

The rapid advancement in artificial intelligence (AI) and wide deployment of Internet of Video Things (IoVT) enable situation awareness (SAW). Robustness and security of the IoVT systems are essential to a sustainable urban environment. While blockchain technology has shown great potentials to enable trust-free and decentralized security mechanisms, directly embedding crypto-currency oriented blockchain schemes into resource-constrained Internet of Video Things (IoVT) networks at the edge is not feasible. Leveraging Electrical Network Frequency (ENF) signals extracted from multimedia recordings as region-of-recording proofs, this paper proposes EconLedger, an ENF-based consensus mechanism that enables secure and lightweight distributed ledgers for small scale IoVT edge networks. The proposed consensus mechanism relies on a novel Proof-of-ENF (PoENF) algorithm where a validator is qualified to generate a new block if and only if a proper ENF-containing multimedia signal proof is produced within the current round. Decentralized database (DDB) is adopted to guarantee efficiency and resilience of raw ENF proofs on the off-chain storage. A proof-of-concept prototype is developed and tested in a physical IoVT network environment. The experimental results validated the feasibility of the proposed EconLedger to provide a trust-free and partially decentralized security infrastructure for IoVT edge networks.

The integration of Unmanned Aircraft Systems (UAS) into the current airspace poses significant challenges in terms of safety, security and operability. As an example, the European Union has defined in 2019 a set of rules to support the digitalization of UAS Traffic Management (UTM) systems and services, namely the U-Space regulations. Current propositions opted for a centralized and private model, concentrated around governmental authorities (e.g., AlphaTango provides the Registration service and depends on the French government). In this paper, we advocate in favor of a more decentralized and transparent model in order to improve safety, security and operability among UTM stakeholders. As such, we propose DFly, a Publicly Auditable and Privacy-preserving UAS Traffic Management system on Blockchain, and developed two initial services: Registration and Flight Authorization. We demonstrate that the use of a blockchain guarantees the public auditability of the two services and corresponding service providers’ actions. In addition, it facilitates the comprehensive and distributed monitoring of airspace occupation and the integration of additional functionalities (e.g., the creation of a live UAS tracker). The combination with zero knowledge proofs enable the deployment of an automated, distributed, transparent and privacy-preserving flight authorization service, performed on-chain thanks to the blockchain logic. In addition to its construction, this paper details the instantiation of the proposed UTM system with the Ethereum Sepolia’s testnet and the Groth16 ZK-SNARK protocol. On-chain (gas cost) and off-chain (execution time) performance analyses confirm that the proprosed solution is a viable and efficient alternative in the spirit of digitalization and offers additional security guarantees.

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

With the emergence of blockchain and smart contracts, traditional digital applications 1 such as identity management, supply chain management, banking and finance etc. are expected 2 to be transformed into Decentralized Applications (DApps). Blockchain interoperability is a new 3 and exciting aspect of blockchain technology that is quickly gaining popularity in many industries. 4 However, the widespread use of blockchains has not yet been achieved because DApps running on the 5 blockchain using smart contracts require access to authentic off-chain data. Users are more conscious 6 of their personal data privacy and consequently are more reluctant to share their data, posing a 7 challenge in the adoption of DApps. To solve this challenge, we present a novel quantum-resistant, 8 data authentication and zero-knowledge proof scheme named zk-DASTARK. Based on our proposed 9 scheme, we present a novel framework which can be used to feed authenticated off-chain data to 10 DApps without compromising user data privacy. Our proposed framework is quite efficient against 11 well-known off-chain data authentication frameworks. The proposed framework is implemented on 12 state-of-the-art quantum-resistant IOTA Blockchain and is highly efficient as it takes approximately 13 60 ms to generate a proof and approximately 10 ms to verify a proof.

Blockchain technology has been gaining great interest from a variety of sectors including healthcare, supply chain, and cryptocurrencies. However, Blockchain suffers from its limited ability to scale (i.e. low throughput and high latency). Several solutions have been proposed to tackle this. In particular, sharding proved that it is one of the most promising solutions to Blockchain’s scalability issue. Sharding can be divided into two major categories: (1) Sharding-based Proof-of-Work (PoW) Blockchain protocols, and (2) Sharding-based Proof-of-Stake (PoS) Blockchain protocols. The two categories achieve good performances (i.e. good throughput with a reasonable latency), but raise security issues. This article focuses on the second category. In this paper, we start by introducing the key components of sharding-based PoS Blockchain protocols. Then, we briefly introduce two consensus mechanisms, namely PoS and practical Byzantine Fault Tolerance (pBFT), and discuss their use and limitations in the context of sharding-based Blockchain protocols. Next, we provide a probabilistic model to analyze the security of these protocols. More specifically, we compute the probability of committing a faulty block and measure the security by computing the number of years to fail. Finally, we evaluate the effectiveness of the proposed model via numerical analysis.

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

Methods are currently lacking to prove artificial general intelligence (AGI) safety. An AGI ‘hard takeoff’ is possible, in which first generation AGI1 rapidly triggers a succession of more powerful AGIn that differ dramatically in their computational capabilities (AGIn≪AGIn+1). No proof exists that AGI will benefit humans or of a sound value-alignment method. Numerous paths toward human extinction or subjugation have been identified. We suggest that probabilistic proof methods are the fundamental paradigm for proving safety and value-alignment between disparately powerful autonomous agents. Interactive proof systems (IPS) describe mathematical communication protocols wherein a Verifier queries a computationally more powerful Prover and reduces the probability of the Prover deceiving the Verifier to any specified low probability (e.g., 2-100). IPS procedures can test AGI behavior control systems that incorporate hard-coded ethics or value-learning methods. Mapping the axioms and transformation rules of a behavior control system to a finite set of prime numbers allows validation of ‘safe’ behavior via IPS number-theoretic methods. Many other representations are needed for proving various AGI properties. Multi-prover IPS, program-checking IPS, and probabilistically checkable proofs further extend the paradigm. In toto, IPS provides a way to reduce AGIn↔AGIn+1 interaction hazards to an acceptably low level.