Search Results (692)

In recent years, artificial intelligence (AI) has been increasingly used in criminal justice systems across the world. To achieve objectives set out through Sustainable Development Goals (SDGs), adoption of technology is inevitable and undeniable. The press release dated 25 February 2025 from India’s Ministry of Law and Justice, quoting Prime Minister of India Narendra Modi to make a “justice system that will be fully future-ready”, confirmed that the Indian law enforcement agencies are integrating AI into policing and law enforcement to enhance crime detection, criminal investigation, etc. It is intended to enhance their capabilities in solving criminal cases and delivering justice speedily and more efficiently. However, the usage of AI tools in such contexts presents a double-edged sword, as evidenced by their application in a number of cases across the world like Christopher Gatlin, Nijeer Parks, the Harm Assessment Risk Tool (HART), and in India during the 2020 Delhi riots cases. As reported by the Washington Post, in Christopher Gatlin’s case it was found that the police arrested him on the basis of the facial recognition programme matching his face with the captured video footage. He spent 17 months in jail before his release by the court, observing that the police failed to conduct fair investigation. A similar incident was reported by NJ.com and CNN Business. In the investigations following the 2020 Delhi riots, Delhi Police effected over 1900 arrests in 758 riot-related cases, relying predominantly on AI-driven facial recognition matches. Subsequent court scrutiny in decided cases raised questions about reliability, leading to widespread acquittals and discharges of the accused in 82% of decided cases as of early 2025. In certain cases, AI-driven solutions have failed, leading to criminal prosecutions of innocent people based on AI-generated evidence. This study examines the reliability, validity, and ethics of AI technology in the criminal justice system in India’s unique socio-legal and political environment. The researchers analyse three interrelated axes. First, a comprehensive review of the international algorithmic policing literature to identify successes and failures. In addition, cases of AI-assisted investigations during the Delhi riots show how facial recognition systems and other AI techniques were used for inquiry. Finally, stakeholders’ perspectives, including a preliminary survey of 27 legal experts showing strong consensus on classifying AI-FRT outputs strictly as corroborative evidence and highlighting BSA insufficiencies for addressing opacity and explainability, help identify practical, procedural, and normative fault lines. Researchers noted that while AI has the potential to revolutionise resource-constrained investigative agencies, its unquestioning and uncritical adoption risks amplify pre-existing biases, undermine presumptions of innocence, and shift the burden of refuting algorithmic inference onto the accused. Independent algorithmic audits, transparent documentation of error rates and confidence thresholds, statutory guidelines on AI tool use and admissibility, and sustained capacity-building throughout the justice delivery chain are needed to integrate it into the Indian criminal justice system. Without such measures, the very tools designed and introduced to enhance accuracy threaten to undermine the fundamental norms of the criminal justice system such as fairness and due process. This fills a gap in doctrinal analysis of AI-specific evidentiary admissibility in non-Western contexts like India. This study aims to propose policy reforms, enhance judicial discourse, and promote a more circumspect trajectory for AI adoption in Indian law enforcement by mapping the potential and risks of algorithmic evidence in a non-Western legal order. Full article

This study introduces a novel matrix defined over the nonzero natural numbers, whose entries are governed by a rigorous closed-form expression. The matrix architecture replicates the topological properties of the Ulam spiral, mapping the integer sequence onto a structured lattice with a well-defined formulation. We investigate the interplay between the matrix’s linear algebraic properties and its number-theoretic implications. A primary focus is the established connection between the matrix’s lines, rows, diagonals, and antidiagonals, and the Hardy–Littlewood F-conjecture. By analyzing the matrix’s internal structure, this work provides a new analytical framework for further study of the conjecture. The matrix links its visual characteristics to quadratic polynomials, offering fresh insights into the distribution of prime numbers. Full article

This study presents a validation-aware EEG framework based on Chaotic Pattern of Prime Numbers (CPPN) features for depression treatment-response modelling across one SSRI cohort and two rTMS cohorts. CPPN features were evaluated through a seven-protocol validation hierarchy spanning random segment splitting, segment-level cross-validation, nested segment-level cross-validation, leave-N-subjects-out, fixed-feature leave-one-subject-out (LOSO), nested leave-N-subjects-out, and nested LOSO, with normalisation, NCA ranking, feature-count selection where applicable, and model fitting confined to the appropriate training partitions. In the representative K-nearest neighbour (KNN) comparison, segment-level 10-fold CV achieved accuracies of 98.79% for Mumtaz SSRI, 99.32% for small Atieh rTMS, and 99.42% for big Atieh rTMS, demonstrating strong discriminative structure in the CPPN feature space. In the available segment-level KNN comparison, CPPN features with fold-internal NCA-selected feature sets exceeded conventional statistical EEG features by 29.53, 16.33, and 11.45 percentage points across the three cohorts. Subject-wise validation produced lower and more cohort-dependent estimates, with the best fixed-feature LOSO accuracy of 80.00% and the best nested LOSO accuracy of 73.33% in the small Atieh rTMS cohort. These results show that CPPN provides a compact, inspectable and computationally accessible EEG feature representation, while the validation hierarchy gives a transparent account of how performance changes from segment-level separability to held-out-subject evaluation. The main contribution is methodological: this study combines an original CPPN feature representation with explicit validation-depth analysis, leakage-aware feature selection, and interpretable channel/bin inspection. It therefore provides a rigorous basis for future externally validated EEG treatment-response studies without claiming prospective clinical deployment from the present retrospective cohorts. Full article

In this paper we discover and prove a new identity that establishes an exact relationship between a certain weighted sum over prime numbers and the sum over nontrivial zeros of the Riemann zeta function of the reciprocal of $\rho (1-\rho )$. By analyzing a convergent series involving the natural logarithm of integers, and applying the fundamental theorem of arithmetic, we expand this series into an infinite expression that involves only primes, thereby eliminating any explicit appearance of composite numbers. The resulting identity is presented in two equivalent forms—a single sum and a double sum—each of which equates a purely prime-dependent infinite series to a combination of elementary constants together with the sum over the zeros. The derivation does not rely on the Riemann Hypothesis; it uses only classical analytic tools and the fundamental theorem of arithmetic. Numerical computations confirm the correctness of the identity. We also discuss its conceptual kinship with the Riemann–Weil explicit formula, illustrating how the identity reflects a dual relation between primes and zeros in a concrete setting, without being a direct special case of that formula. Full article

This study evaluated the effects of polyethylene glycol concentrations in enhancing the physiological performance of the rice varieties and their recovery ability after drought stress. The experiment comprised of IR64, NIL-SUB1DRO1, and NIL-DRO1. Seed priming was conducted by submerging 5 g of samples in petri dishes containing 100 mL of 5% and 10% PEG solutions. Drought stress significantly reduced all the growth traits, with the susceptible genotypes IR64 recorded highest reduction of shoot length 36%, tiller number 41.3%, shoot dry weight 77%, and root dry weight 72% compared to non-primed NIL-DRO1 and NIL-SUB1DRO1 with reduction in shoot length 34–35%, tiller number 34–45%, root dry weight 60–66%, and shoot dry weight (70–71%). Similar results were recorded for IR64, Pn, 63%, gs 78% E 66%, and RWC 66%, respectively, compared with NIL-DRO1 (55%, 60%, and 58%), while NIL-SUB1DRO1 showed reductions of 55%, 50%, and 54%. PEG 5% and 10% significantly enhanced primed IR64 Pn (29–57%), gs (70%), E (56–64%), and RWC 65%. During the recovery phase, primed seedlings showed a more rapid restoration of growth and photosynthetic efficiency than the non-primed seedlings. PEG 5% and 10% were effective in mitigating drought stress and enhanced recovery ability of rice. Full article

Let $\alpha$ be an irrational number, $\beta$ a real number, and $a_1,\dots ,a_s$ a set of distinct positive integers that do not form a reduced residue system modulo $p^r$ for any prime p. In this work, we establish that there are infinitely many prime numbers p that satisfy the condition $\Vert \alpha p^2+\beta \Vert <p^{-\theta }$ for a suitable $\theta$, and at the same time each of the numbers $p+a_1,\dots$, $p+a_s$ is r-free. Full article

This paper studies structural properties of semiprimes $N=pq$ in computational number theory, focusing on cases where the prime factors are close. We analyze the relationship between $\sqrt{N}$ and $\sqrt{\phi \left(N\right)}$ and show that, under a bounded prime gap condition, these quantities exhibit strong proximity. Specifically, assuming $|p-q|\le 2^{l/4}$ for an l-bit semiprime, we prove that the Euler totient function admits the exact representation $\phi \left(N\right)=N-1-2\lfloor \sqrt{N}\rfloor$. Based on this result, we develop an interval-based method for reconstructing $\phi \left(N\right)$ within a narrow neighborhood derived from square-root bounds, followed by a discriminant-based refinement step for recovering the prime factors. Experimental evaluation on large semiprimes, including RSA-type moduli of 4095 and 4096 bits, shows that the method operates efficiently under the stated structural condition using only elementary integer arithmetic. These results provide a theoretical characterization of semiprimes with small prime gaps and offer a framework for identifying structurally weak RSA moduli. This method, given its high efficiency when the prime factors are close to each other, can be regarded as an alternative to Fermat’s factorization method. In particular, for semiprime integers with a small prime gap (i.e., $|p-q|$ is small), the proposed approach exploits structural properties based on the proximity of square roots, thereby significantly accelerating the factorization process. Consequently, it not only aligns with the theoretical foundation of Fermat’s method but, under certain conditions, may also achieve comparable or even superior practical performance. Full article

Tanner quasi-cyclic low-density parity-check (QC-LDPC) codes form an important family of structured LDPC codes with favorable girth properties. This paper studies the girth of Tanner $(2, L)$-regular QC-LDPC codes (referred to as Tanner QC-LDPC cycle codes) for arbitrary integers $L>2$ and develops a novel algebraic number theoretic method to determine the girth for all sufficiently large primes p with $p\equiv 1\left(\mathrm{mod}2L\right)$. We first analyze the case $L=3$ and prove that the girth is 12 for every prime $p\equiv 1\left(\mathrm{mod}6\right)$ through exhaustive resultant computations. We then extend the method to arbitrary L and obtain a clear classification: when L is even, the girth is exactly 8 for all admissible primes; when L is odd, the girth attains the maximum value 12 for all sufficiently large admissible primes. The proof transforms cycle existence conditions into polynomial equations and applies resultant theory. This approach converts the infinite task of checking all primes into a finite set of algebraic checks. Numerical simulations show that the Tanner $(2, 5)$-regular non-binary code over $\mathrm{GF}\left(64\right)$ achieves a coding gain of approximately $0.2$ dB over the 5G LDPC code of equivalent binary length. Full article

Prime numbers are central to mathematics, yet popular discourse often treats particular primes as if they carried intrinsic messages, personalities, or moral charge. This study asks how that shift from legitimate curiosity to superstition-adjacent pattern making occurs and why it feels persuasive. Using qualitative content analysis of three widely circulated media examples, this paper maps how culturally specific number meanings are produced and transmitted, and how predictable cognitive biases support their plausibility. The analysis pairs anthropological mechanisms of symbolic association, prestige borrowing, community boundary marking, and meme-based diffusion with psychological mechanisms that include Type I error, apophenia, confirmation bias, availability, narrative fallacy, selection effects, survivorship, cultural priming, and authority or celebrity cueing. Across the cases, the results show a recurrent coupling: cultural schemas supply ready-made interpretive templates, while cognitive biases turn salience and coincidence into perceived significance, concentrating attention on narratively convenient primes and obscuring the many alternative patterns that could have been selected. This paper concludes that meanings such as 666 as evil are culture dependent rather than mathematical properties, and that improving public communication about primes requires making selection processes and interpretive frames explicit while preserving legitimate mathematical wonder. Full article

Let $\mathcal{P}$ be a system of generalized prime numbers, and ${\mathcal{N}}_{\mathcal{P}}$ the corresponding system of generalized integers. Assuming that ${\displaystyle \underset{m\in {\mathcal{N}}_{\mathcal{P}}}{{\displaystyle \sum _{m⩽x}}}}1-ax\ll x^{\beta }$ with $a>0$ and $0⩽\beta <1$, we consider the Beurling zeta-function ${\zeta }_{\mathcal{P}}\left(s\right)$, $s=\sigma +it$. Beurling zeta-functions constitute a wide class of non-standard zeta-functions which pose interesting mathematical problems. Numerous authors are searching for restrictions on the systems $\mathcal{P}$ and ${\mathcal{N}}_{\mathcal{P}}$ that the corresponding Beurling zeta-functions have some properties similar to those of classical zeta-functions. One of such properties is the functional independence which was initiated by O. Hölder and D. Hilbert, and, in the most general form, by S.M. Voronin. This is a motivation to obtain the functional independence in the Voronin sense for a certain class of Beurling zeta-functions. Under a certain additional condition involving the generalized von Mangoldt function, we obtain the functional independence of the function ${\zeta }_{\mathcal{P}}\left(s\right)$. We prove that the function ${\zeta }_{\mathcal{P}}\left(s\right)$ does not satisfy the equation ${\displaystyle \sum _{k=0}^r}{s}^k{F}_k\left({\zeta }_{\mathcal{P}}\left(s\right),{\zeta }_{\mathcal{P}}^{\prime }\left(s\right),\dots ,{\zeta }_{\mathcal{P}}^{(n-1)}\left(s\right)\right)=0$ with continuous functions $F_k$, $k=0,\dots ,r$. The proof is based on the universality property of ${\zeta }_{\mathcal{P}}\left(s\right)$ on approximation of analytic functions by shifts ${\zeta }_{\mathcal{P}}(s+i\tau )$, $\tau \in \mathbb{R}$. Full article

Integer factorization plays a foundational role in asymmetric cryptography systems, notably the Rivest, Shamir, and Adleman (RSA) cryptosystem. This paper presents an improvement of the integer factorization from both sequential and parallel computational perspectives. The algorithm is based on polynomial evaluation and the greatest common divisor. The objectives of these improvements are to decrease the execution time and memory consumption associated with the process of finding prime factors. Experimentally, we use different values of parameters (1) the number of bits n, (2) the difference between two factors, and (3) the number of processors in the parallel model. The experimental results indicate that both proposed methods, sequential and parallel, yield significant improvements regarding running time and memory usage when the difference between the two factors is $n^{1/3}$ and $n^{1/4}$. The average improvement observed is $99\%$ in running time, with memory consumption reduced to a constant. This characteristic is important for limited hardware devices. Furthermore, the proposed parallel method demonstrates scalability and achieves sublinear speedup. Full article

Gene amplification resulting from double strand breaks (DSBs) is a typical genetic alteration in tumorigenesis and drug-resistant progression. Amplified oncogenes and drug-resistant genes are present on extrachromosomal DNAs (ecDNAs), or chromosomal homogeneously staining regions (HSRs). Considering the role of mismatch repair (MMR) as a sensor of DSBs, we hypothesized that MMR may be involved in gene amplification. We used two MTX-resistant HT-29 colorectal cancer cell lines, which served as models with amplified genes mainly in HSRs or ecDNAs. Expression of MSH2, a key protein in MMR, was increased following the acquisition of MTX-resistant. MMR inhibition was achieved by depleting MSH2. Suppression of MMR led to decreased copy numbers of amplified genes as well as the quantity of ecDNAs and HSR. This was caused by the decreased efficiency of DSBs repair, which resulted from the reduced ability of MMR to recruit DSBs repair proteins. Additionally, it accelerated the formation of micronuclei (MN)/nuclear buds (NBUDs), which functioned to eliminate the amplified genes. Furthermore, the suppression of MMR was capable of inhibiting cell proliferation and enhancing MTX-sensitivity in ecDNA-containing cells. Conversely, suppression of MMR had no effect on gene amplification in HSR-containing cells. Our findings demonstrate that MMR plays a pivotal role in gene amplification through mediating DSBs repair pathways and facilitating the formation of MN/NBUDs in ecDNA-containing cells. MMR is likely to emerge as a prime therapeutic target worthy of in-depth exploration in future clinical investigations. Full article

The literature indicates that the qubit requirements for factoring RSA-2048 remain on the order of 1 million, under commonly assumed architectures and error-correction models, leaving a substantial gap between current resource estimates and near-term practical feasibility. Reducing this requirement to the low-thousand-qubit regime therefore remains an important open research objective. This work proposes a hybrid classical–quantum algorithm that uses a classical modular exponentiation subroutine with a Quantum Number Theoretic Transform (QNTT) circuit to increase the speed and reduce the required quantum resources relative to Shor’s algorithm for integer factorization, which underpins cryptographic systems like RSA and ECC. We evaluate multiple coprime numbers, the result of multiplication of two primes, in both simulation and real quantum hardware, using IBM’s reference Shor implementation as the baseline. Because Shor and proposed Jesse–Victor–Gharabaghi (JVG) use different register sizes for the same coprime N, the reported gate/depth reductions should be interpreted as end-to-end quantum-resource budgets for factoring the same N, rather than a per-qubit or transform-only efficiency claim. In simulation, the JVG algorithm achieved substantial practical reductions in computational resources, decreasing runtime from 174.1 s to 5.4 s, memory usage from 12.5 GB to 0.27 GB, and quantum gate counts by approximately 99%. On quantum hardware, JVG reduced the required runtime from 67.8 s to 2 s, and the quantum gate counts by over 98%. We showed that the proposed algorithm can address the relevant RSA-1024 case scenario, establishing that this method can provide validation for large-scale situations. Furthermore, extrapolation to RSA-2048 indicates that the JVG algorithm significantly outperforms Shor’s approach, requiring a projected quantum runtime of 29 h for ten thousand runs for factorization under identical scaling assumptions. Overall, these results support JVG as a more hardware-compatible and robust noise-tolerant substitute for Shor’s framework, offering a viable research direction toward practical quantum integer factorization on near-term Noisy Intermediate-Scale Quantum (NISQ) devices. Full article

With the growing importance of personal information security, numerous methods have been proposed for data encryption. To ensure system safety, ciphers must be unpredictable and robust. In modern Rivest–Shamir–Adleman (RSA) encryption systems, two prime numbers are required for key generation, and their randomness and unpredictability are essential for security. In this study, we propose a secure system for generating the prime numbers used in RSA encryption. The inherent properties of chaotic systems are employed as a Pseudo Random Number Generator (PRNG), while a Ring Oscillator is utilized as a True Random Number Generator (TRNG). The Miller–Rabin algorithm is further applied to verify the primality of the numbers produced by the PRNG. The entire design is implemented on a Field Programmable Gate Array (FPGA) to achieve a fully hardware system. Full article

The Zeta-Minimizer Theorem establishes that the Riemann zeta function $\zeta \left(s\right)$ and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space $\left(X,\mu ,G\right)$ equipped with helical operators. Three fundamental axioms—strict concave entropy maximization (Axiom 1), spectral Gibbs minima with non-vanishing ground states (Axiom 2), and irreducible bounded oscillations with flux conservation (Axiom 3)—allow for the selection of the non-proper Archimedean conical helix as the sole topology satisfying all constraints. Primes emerge as indivisible minimal cycles in the associated representation graph $\Gamma$ (via Hilbert irreducibility and Maschke’s theorem), while the Euler product is recovered through the spectral Dirichlet mapping of the helical eigenvalues. The partial zeta product, $Z\left(s\right)=\prod _j{\displaystyle \frac{1}{1-p_j^{-s}}},s\in \mathbb{R}\left\{0\right\},$ constitutes the exact grand partition function of any finite subsystem. Numerical inversion of this product directly recovers the mixture frequency $s$ from any experimental compressibility factor $Z_{\mathrm{m}\mathrm{i}\mathrm{x}}$. Mole fractions $x_i\left(s\right)$, interaction parameters $\Delta \left(x_i\right)$, and the Lyapunov spectrum $\lambda \left(x_i\right)$ then follow deductively via the helical transfer matrix and the closed-form linear ODE for $\Delta$. Occupation numbers $N\left(x_i\right)$ attain sharp maxima precisely at Fibonacci ratios $F_r/F_{r+1}$, leading to the molecular prime-ID rule. For twelve representative purely binary (irreducible) systems spanning atomic noble gases, simple diatomics, polar molecules, and an aromatic ring, the residuals satisfy $|Z\left(s\right)-Z_{\mathrm{m}\mathrm{i}\mathrm{x}}|<1.5\times {10}^{-8}$. The resulting $\lambda \left(x_i\right)$ curves accurately reproduce critical points, liquid ranges, and thermodynamic anomalies with zero adjustable parameters. The Riemann Hypothesis follows rigorously as a theorem: the unique fixed point of the duality functor $s\mapsto 1-s$ that preserves the orthogonality condition $\sum {\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^k=1$ is $\mathrm{R}\mathrm{e}\left(s\right)=1/2$, enforced by Axiom 1 concavity and Axiom 3 irreducibility. The framework is fully deductive and parameter-free and extends naturally to arbitrary mixtures and multiplicities through the helical representation graph. It provides a variational unification of analytic number theory, spectral geometry, thermodynamic phase behavior, and the Riemann Hypothesis from first principles. Full article