This study aimed to evaluate the performance of Saudi students in the International Mathematical Olympiad (IMO) from 2019 to 2023, through benchmarking their results against peers from G20 countries. The objective was to identify the extent of competitiveness of Saudi participants at the international level and to highlight performance gaps that require intervention. The descriptive-analytical method was employed as the research approach, supported by statistical techniques including arithmetic mean, standard deviation, Z-score, and T-score. The study sample consisted of 30 Saudi students (28 males and 2 females) who officially participated in the IMO under the supervision of the King Abdulaziz and His Companions Foundation for Giftedness and Creativity (Mawhiba). Six students participated annually, carefully selected after several qualifying stages and examinations. Raw scores were standardized to enable valid comparisons and classification of performance into high, medium, or low levels. The findings indicated that Saudi students generally performed at an “average” level, with the exception of 2020, which recorded a “low” level. Although some improvement appeared in 2022, negative Z-scores confirmed that overall performance remained below the global average, reflecting a persistent gap with G20 countries. Performance varied significantly by problem type: students achieved stronger results on Problem 1 (P1), while struggling with more complex problems such as P3 and P4. Differences also emerged across mathematical domains, with “average” outcomes in number theory, polygons, probability, and circles, but “low” outcomes in sequences, truth tables, and quadrilaterals. In light of these results, the study recommends more specialized training programs, instructional strategies aligned with IMO complexity, and enriched curricula with challenging problems across diverse mathematical domains to bridge performance gaps and strengthen competitiveness.

This paper introduces the framework of generalized hyperbolic Pandita numbers constructed over the bidimensional Clifford algebra of hyperbolic numbers, contributing a novel class of structured sequences to the expanding domain of number theory. Anchored in the principles of hyperbolic systems, these constructs pave the way for exploring algebraic symmetries and recursive behaviors beyond classical formulations. Special attention is devoted to notable cases, including the hyperbolic Pandita and hyperbolic Pandita-Lucas numbers, whose properties are meticulously examined. To deepen understanding and facilitate computation, we derive explicit closed-form representations using Binet-type formulas, construct generating functions through formal power series, and establish summation expressions with broad applicability. Additionally, matrix-based representations are developed to offer an algebraic lens through which structural dynamics can be modeled and analyzed. These formulations not only enrich the theoretical foundations of discrete mathematics and symbolic computation but also highlight promising applications in engineering disciplines— particularly in the modeling of iterative systems, signal transformations, the analysis of complex networks, and cryptographic systems. The insights presented herein lay the groundwork for future exploration into hybrid sequence systems and their role in interdisciplinary problem solving.

Abstract Mathematical proof plays a key role in studying mathematics at the university level. In this exploratory study, we investigate mathematics students’ perception of the role of proofs during their degree at a UK university. Nineteen semi-structured interviews were conducted with students in Years 2–4 of their degree. We found that students recognized proofs as an important part of their degree. However, they perceived them as difficult, especially when encountering them initially. Further, students’ enjoyment in working with proofs was linked to their understanding. In the transition to working with proofs, the second year of study was identified as an important period by more than half of participants while the first year’s importance was also recognized. Seeing examples in class, continuous practise and self-study were frequently mentioned aspects of study that aided the transition. Key modules mentioned in this context were Real Analysis (Years 1–2) and a second-year module on Algebra and Number Theory. We observed evidence that suggests proofs are initially difficult when first encountered and propose further investigation on whether proof constitutes a threshold concept in mathematics.

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if f f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L\cdot V) converge in distribution, as L → ∞ L\to \infty , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (f(L\cdot V+t))_{t\in \mathbb {R}} in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.

This research article presents ten pioneering principles introduced by Professor Dr. FazalRehman for rapidly calculating the sums of even consecutive numbers. These principles provide elegant shortcuts that eliminate traditional step-by-step addition and replace it with direct computational formulas based on positional terms. Each principle is supported with ten fully solved examples, demonstrating its universality and effectiveness across small and large even integers. The resulting framework contributes a significant new direction in elementary number theory and computational mathematics by simplifying sequence summations into compact algebraic forms.

We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many recent advances in number theory and combinatorics. We define a family of "quantum measures" on linear transformations on a Hilbert space, that reduce in the case of diagonal matrices to the uniformity norms introduced by Timothy Gowers. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum computation. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the [Formula: see text] level of the Clifford hierarchy.

We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of zeroes of the Riemann zeta-function. These results were obtained by creating the Analytic Number Theory Exponent Database to collect results and relationships between these exponents, and then systematically optimising these relationships to obtain the new bounds. We welcome further contributions to the database, which aims to allow easy conversion of new bounds on these exponents into optimised bounds on other related exponents of interest.

We build a parallel theory of simple Hurwitz numbers for the reflection groups G(m,1,n) . We study analogs of the cut-and-join operators. An algebraic description as well as a description of Hurwitz numbers in terms of ramified coverings is provided. An explicit formula for them in terms of Schur polynomials is given. In addition, the generating function of G(m,1,n) -Hurwitz numbers is shown to give rise to an m independent-variables \tau -function of the KP hierarchy. Finally, we provide an ELSV-type formula for these new Hurwitz numbers. These results extend the results of Fesler (2023).

The Horadam sequence {Hn(a,b;p,q)}n⩾0 has been widely studied in combinatorics and number theory. In this paper, we find that the context-free grammar G={x→px+y,y→qx} can be used to generate Horadam sequences. Using this grammar, we deduce several identities, including Cassini-like identities. Moreover, we investigate the relationship between two distinct Horadam sequences Hn(a,b;p,q) and Hn(c,d;p,q) with (a,b)≠(c,d) and provide an approach to derive identities, which can be illustrated by the Fibonacci and Lucas sequences as well as the two kinds of Chebyshev polynomials.

ABSTRACT Discrete Mathematics is an important and challenging course for computer science and engineering students. It includes topics, such as logic, sets, proofs, number theory, graphs, trees, computation, relations, functions, and basic algorithmic concepts. These topics require strong analytical reasoning and consistent effort. As a result, many students find this course challenging to perform well. The aim of this study is to predict student performance in a Discrete Mathematics course at a reputed private university located in Bangladesh. Data were collected from both course instructors and students during the spring and summer semester of 2025. Instructors provided academic records, such as attendance, quizzes, assignments, and midterm scores. Students provided additional information, which included daily study time, subject interests, and use of learning platforms. The final data set included records for 240 students. K ‐means clustering with the Davies–Bouldin method was used to group similar students. Then, four machine learning (ML) models were trained and tested: Support Vector Machine (SVM), Decision Tree, K ‐Nearest Neighbors, and Naïve Bayes. The models were implemented using Python's scikit‐learn library, with stratified sampling and fivefold cross‐validation. Among the models, SVM achieved the highest accuracy of 96% after parameter tuning. Naïve Bayes had the lowest accuracy due to the assumption of feature independence. Key predictors of performance included mean score, attendance, and daily study hours. Findings show that ML can help instructors identify at‐risk students early, provide focused academic support, and improve learning outcomes. While the results are promising, the study is limited by sample size and does not include psychological or emotional factors. Future work will explore larger data sets and apply interpretable Artificial Intelligence techniques for better model transparency.

The role of pure mathematics in resolving complex biological problems: Applications to F1-ATPase, achievements, and future directions.

Pierre de Fermat first stated around 1637 that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he said the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a4 + b4 = c4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a3 + b3 = c3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integerssatisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x3 + y3 = z3, and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.

In this paper, we introduce and develop the concept of generalized dual hyperbolic Pierre numbers, a novel class of number sequences that extends the structural framework of classical Pierre-type sequences through duality and hyperbolic transformations. This generalization offers a unified approach that encompasses both established and newly constructed numerical models. As distinguished special cases, we examine the dual hyperbolic Pierre numbers and their Lucas-type counterparts, emphasizing their algebraic relationships and unique structural features. Our study presents a comprehensive set of mathematical results, including closed-form identities, matrix representations, and recurrence relations that define the behavior of these sequences. We further derive Binet-type formulas for explicit term computation and construct generating functions that capture their combinatorial and analytical properties. Additionally, we explore Simson’s formulas and establish various summation identities that reveal deeper interconnections among sequence elements. This investigation contributes to the broader theory of Pierre-type sequences and offers new tools for research in discrete mathematics, algebraic structures, and computational number theory.

Group theory is a very important concept in mathematics with many interesting theories that have been widely applied in other areas of mathematics. As one of the fundamental tools in abstract algebra, it provides a unifying language for studying symmetries, structures, and transformations, making it central to both theoretical and applied mathematics. This paper proves the orbit stability theorem based on the theory of group actions. Then, this paper introduces the application of the orbit stabilizer in other parts of mathematics and its full proof. Among these theorems, compared with other proof methods, the orbit stabilizer theorem is more concise and easier to understand. These examples show the wide application of the orbit stability theorem in mathematics, proving its practicality. Furthermore, the theorem serves as a foundation for exploring topics such as combinatorics, number theory, and geometry, where orbit-stabilizer arguments simplify otherwise complex counting and classification problems. In this way, the study highlights how group theory not only develops its own framework but also contributes essential insights to broader mathematical investigations.

Abstract This study introduces Fibonacci Cartan and Lucas Cartan numbers, extending the classical Fibonacci and Lucas sequences into the framework of Cartan numbers. By leveraging algebraic and geometric properties, we establish recurrence relations, generating functions, Binet-like formulas, and fundamental identities such as Catalan’s, Cassini’s, and d’Ocagne’s identities for these novel number sequences. Furthermore, summation formulas and additional properties are explored to provide a comprehensive mathematical characterization. The findings contribute to the ongoing development of number theory and its applications in algebra and geometry.

This paper investigates D-finite discrete generating series and their sections. The concept of D-finiteness is extended to multidimensional discrete generating series and its equivalence to p-recursive sequences is rigorously established. It is further shown that sections of the D-finite series preserve D-finiteness, and that their generating functions satisfy systems of linear difference equations with polynomial coefficients. In the two-dimensional case, explicit difference relations are derived that connect section values with boundary data, while in higher dimensions general constructive methods are developed for obtaining such relations, including cases with variable coefficients. Several worked examples illustrate how the theory applies to solving difference equations and analyzing multidimensional recurrent sequences. The results provide a unified framework linking generating functions and recurrence relations, with applications in combinatorics, number theory, symbolic summation, and the theory of discrete recursive filters in signal processing.

In this paper we examine two arithmetic identities, which come from distinct chapters of elementary number theory. They are: “the Honsberger Fibonacci number identity” and the “Busche-Ramanujan μ -identity”. The interplay between the two is just the beginning of bridging the gap between the theory of Fibonacci numbers and the theory of multiplicative arithmetic functions.

Most concrete analyses of lattice reduction focus on the BKZ algorithm or its variants relying on Shortest Vector Problem (SVP) oracles. However, a variant by Li and Nguyen (Cambridge U. Press 2014) exploits more powerful oracles, namely for the Densest rank- k Sublattice Problem (DSP k ) for k ≥ 2 . We first observe that, for random lattices, DSP 2 –and possibly even DSP 3 – seems heuristically not much more expensive than solving SVP with the current best algorithm. We indeed argue that a densest sublattice can be found among pairs or triples of vectors produced by lattice sieving, at a negligible additional cost. This gives hope that this approach could be competitive. We therefore proceed to a heuristic and average-case analysis of the slope of DSP k -BKZ output, inspired by a theorem of Kim (Journal of Number Theory 2022) which suggest a prediction for the volume of the densest rank- k sublattice of a random lattice. Under this heuristic, the slope for k = 2 or 3 appears tenuously better than that of BKZ, making this approach less effective than regular BKZ using the “Dimensions for Free” of Ducas (EUROCRYPT 2018). Furthermore, our experiments show that this heuristic is overly optimistic. Despite the hope raised by our first observation, we therefore conclude that this approach appears to be a No-Go for cryptanalysis of generic lattice problems.

A variety of sets of equivalent resistances can be constructed by combining equal resistors in different configurations using series, parallel or bridge connections. The orders of such sets are traditionally obtained manually for small numbers and computationally, when the numbers are not small. Due to available computer memory, it has not been possible to analyze these sets beyond the case of 30 equal resistors. In this article, we analytically derive a strict lower bound and a strict upper bound for the orders of these sets using inequalities, integer sequences and certain number theoretic techniques. The strict lower bound is obtained using combinatorial arguments and the resulting inequalities. The strict upper bound is obtained by using techniques from number theory, which make use of the Haros-Farey sequence with Fibonacci numbers as their argument. The lower and upper bounds thus obtained hold true for any number of equal resistors as long as they are combined in series and parallel combinations. The various sequences arising in this study are presented in detail with references to the “The On-Line Encyclopedia of Integer Sequences”.

Abstract Let $$\mathbb {C}$$ C be the set of complex numbers, and let $$\mathcal P$$ P be a collection of complex polynomial maps in several variables. Assuming at least one $$P\in \mathcal P$$ P ∈ P depends on at least two variables, we classify all possibilities for the structure $$(\mathbb {C};\mathcal P)$$ ( C ; P ) up to definable equivalence. In particular, outside a short list of exceptions, we show that $$(\mathbb {C};\mathcal P)$$ ( C ; P ) always defines $$+$$ + and $$\times $$ × . Our tools include Zilber’s Restricted Trichotomy, as well as the classification of symmetric non-expanding pairs of polynomials over $$\mathbb C$$ C from arithmetic combinatorics. Along the way, we also give a new condition for a reduct "Equation missing" of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of M.