4,924 papers in this slice of arXiv.
Ahmed Gaber
This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer ℓ≥1, define Euler's ℓ
Stanley Rabinowitz, Ercole Suppa
Let ℓ be a line not passing through any vertex of a triangle ABC and not parallel to any side. Line ℓ meets the sidelines BC, CA, AB of △ABC at points D, E, F, respectively. We consider three of the triangles that are formed: △AEF, △BFD, and △CDE. Placing a fixed triangle center (such as the incenter, centroid, or orthocenter) in each of these three triangles determines a central triangle. We investigate when the reference triangle and its central triangle are perspective, i.e., when the lines AD, BE, and CF are concurrent. A computer search over the first 1000 centers in the Encyclopedia of Triangle Centers suggested numerous examples of concurrence. We give elementary geometric proofs for the circumcenter, orthocenter, and Clawson point, develop a general criterion for concurrence, and identify several operations, including isogonal and isotomic conjugation, that preserve this property. Our main result is a complete characterization of the center functions whose associated cevians are concurrent for every transversal ℓ. This yields an explicit normal form for such centers. We also show that concurrence depends only on the direction of the transversal, and we investigate the special case in which the transversal is parallel to the Euler line.
Kei Hisamoto
Let n be a natural number. It is proved that if the sum of divisors of n has sufficiently large relative to n, then n is some distinct sum of proper divisors of n.
Augustine L. Mahu, Benoît F. Sehba, Cecilia D. Williams
We introduce a reflection-substitution technique for sine inequalities that yields, via Hölder's inequality and its reverse, 2n−1 distinct upper bounds for products and lower bounds for ratios of weighted sines. The comparison between bounds reduces to simple sign conditions on linear combinations of the variables. We fully classify the two- and three-dimensional cases, providing explicit dominance criteria with detailed examples for each signature.
Pablo Fernández Refolio
We provide an explicit analytical evaluation of Ramanujan-type series for 1/π of signature 2. Focusing on the singular moduli kr for r∈{2,3,4,7}, we demonstrate that the underlying elliptic identities can be established through modular transformations of degree 2.
Thomas Fernique
This note explains how to count the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.
Sela Fried
This is the fourth work in a series devoted to proving conjectures recorded in the On-Line Encyclopedia of Integer Sequences (OEIS). The problems considered here concern elementary and multiplicative number theory, Fibonacci numbers, decimal concatenation, Diophantine and Pell equations, binary representations and bitwise operations, lattice paths, parity patterns, recurrences, and formal power series. Several of the results give complete characterizations of the relevant sequences; others establish exact identities, recurrences, generating functions, asymptotic estimates, integrality properties, or nonoccurrence results. The proofs use combinatorial bijections, congruences, Möbius inversion, valuations, Fibonacci identities, Pell-type arguments, Lucas' theorem, Riordan arrays, Lagrange inversion, and generating-function methods.
Taebong Kim, Youngsik Hong, Minsik Kim +3
This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law Ak(a)=log(1/a)+log(k−2)+B0+O(a), with B0=logq−2log2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a)∼(2a)3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ∗(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.
Fernanda Pereira
Sudoku is a widely popular puzzle whose complete grids have been enumerated computationally: there are approximately 6.67×1021 of them and, up to symmetry and renaming of digits, 5,472,730,538 essentially different ones. The classical enumeration reduces the count to 44 equivalence classes of the first band through a chain of ad hoc reductions, leaving the number 44 without any apparent structural explanation. We present an alternative derivation of these 44 classes, in which they arise as isomorphism classes of unordered triples (multisets) of column partitions under relabeling, a single invariant that replaces the original chain of reductions. This invariant makes it possible to apply Burnside's Lemma by hand: we recover 44 through a closed derivation requiring no computational enumeration.
Pinki Khatun
I investigate the Diophantine equation x13−x22x1+1=0
Pascal Ochem, Joshua Zelinsky
We show that if all the even exponents of an odd perfect number N are equal to 2 except for one, then 323,000,000,000 divides N .
Davit Kapanadze
This paper constructs a half-order differentiation operator on the graded monomial space spanned by non-negative integer and half-integer powers of x, with x > 0. The algebraic stage uses neither an integral kernel, a limiting process, nor the Gamma function. The operator is assumed to act in the form D1/2xβ=c(β)xβ−1/2. Requiring two successive applications of the operator to reproduce the ordinary first derivative yields the fundamental recurrence relation c(β)c(β−1/2)=β. Expanding the recurrence produces two linked coefficient families, one for integer powers and one for half-integer powers. Their formulas contain a free normalization constant c0=c(0), which cancels under composition. Thus, the compositional requirement fixes the relative coefficients but not the scale of each individual half-step. Ratios of even and odd double factorials lead naturally to a partial Wallis product; however, the Wallis product alone does not determine c0. Imposing compatibility with the monomial formula for the left Riemann-Liouville half-derivative with lower limit 0 selects the value c0=1/π. With this normalization, D1/2xβ=Γ(β+1/2)Γ(β+1)xβ−1/2 holds for every β∈{0,1/2,1,3/2,…}. The resulting monomial formula agrees with the corresponding Riemann-Liouville formula. For positive integer powers it also agrees with the Caputo formula, whereas for the constant function it agrees only with the Riemann-Liouville rule. The resulting linear compositional operator lowers the exponent by exactly one half and is fully specified on the stated graded monomial space. Extension to broader function spaces, construction of an integral or convolution kernel, and the analysis of non-locality remain open problems.
Bayaz Daraby, Hasan Haddadzadeh
In this paper, we propose a parameterized framework for vector spaces in which each vector is represented by a family of parameter-dependent realizations.
Francois Wolf, Marc Wolf
In this paper, we study the factorization of sums of two squares X2 + Y2. We show the existence of linear and quadratic progressions on X that generate both factorizations and alternative decompositions into sums of two squares. We are therefore naturally interested in the number of decompositions into sums of two squares of an integer n and relate it to its number of divisors d(n). Finally, we present an algorithm to obtain these decompositions using the sieve introduced in (1).
H. Y. Yuan
Russell's paradox was proposed in the early 20th century to address loopholes in set theory, which directly triggered the third mathematical crisis. This paper proposes and elaborates a perspective distinct from previous studies: paradoxes do not give rise to contradictions; instead, they constitute a Möbius strip-style self-consistent logical structure via self-reference and negation under the logical rules within a system. Gödel's incompleteness theorems indicate that such structures universally exist in formal logical systems. Turing proved the undecidability of the halting problem by first assuming the existence of a halting program and subsequently refuting this assumption through paradox construction, and this paper demonstrates flaws inherent to such proof strategy. Cases of paradoxes within three-valued logical systems are further discussed in this work, where the undecidability of paradoxes is rigorously proven. Finally, inspirations drawn from paradoxes for the real world are explored: two opposing factors can be integrated through the joint mechanism of self-reference and negation. A representative example is the wave-particle duality of light, whose essence may be interpreted as a paradox of waves and particles.
Mamadou Sadialiou Bah
Binary relations between finite sets N and X, families of subsets, and lattice homomorphisms from P(N) to P(X) are three faces of the same object. This self-contained monograph, merging and substantially expanding the author's ICTP preprint IC/97/180 (1997) and a 2026 companion note, enumerates every combinatorial type (arbitrary, injective, surjective, on either side), both raw and up to the actions of the symmetric groups S_N, S_X, and S_N x S_X, via the Cauchy-Frobenius-Burnside lemma. The generating functions of all counts are then determined, revealing e^{z+w+zw}, Bell numbers, Euler's partition product, and the quasi-polynomiality of the hardest, two-sided column of the table. All prerequisite tools -- elementary counting, group actions, Stirling and Bell numbers, formal power series, and Mobius inversion on the Boolean lattice -- are developed from first principles, making the book accessible after a first course in algebra and combinatorics. A documented correction to Theorem B of the 1997 preprint is included; all formulas have been verified by symbolic computation and brute-force enumeration.
Meysam Taheri-Dehkordi, Amir Hossein Nokhodkar, Gholam Hossein Fath-Tabar
The Diminished Sombor index (DSO) of a simple graph G is a recently introduced degree-based topological index, defined as DSO(G)=uv∈E(G)∑du+dvdu2+dv2 In this paper, we establish a collection of new sharp bounds for the DSO index in terms of several fundamental graph parameters and well-known topological indices. In all cases, the proposed bounds rigorously specify the equality conditions, thereby providing a complete characterization of the extremal graphs. These findings deepen the theoretical understanding of the DSO index and provide powerful analytical tools for interdisciplinary applications in mathematical chemistry.
Scott Duke Kominers
An Euler magic matrix is an integer matrix M with MMt=γI whose squared entries sum to γ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-4 proper example, and Müller settled orders 3 (none exist) and 8, leaving order 5 as the smallest open case. We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.
Jasem Hamoud
In this paper, a natural duality on the family of biperiodic Fibonacci words F(a,b) generated by the directive sequence (a,b,a,b,…). Both of F(a,b) and F(b,a) are related by the explicit morphism σa:0↦0a1, 1↦0, establishing a precise substitutional correspondence between them. We compute the exact letter frequencies, give a complete description of the return words for each letter, prove the existence of arbitrarily long palindromic prefixes, and determine the continued fraction expansion of the slope θ(a,b). These findings reveal that the apparent asymmetry in several invariants arises uniformly from the length-redistribution mechanism induced by the morphism σa.
Batoul M. Raddad, Ahmad Y. Al-Dweik, Marwan Aloqeili +1
Cartan's method classifies the class of linearizable system of two second-order ODEs into many branches. This paper investigates Branch I of the classification, characterized by a rank-one generalized Wilczynski invariant matrix and the vanishing of two relative invariants K1 and L1. It is demonstrated that any linearizable system belonging to this branch admits an eight-dimensional Lie point symmetry algebra. The canonical form for this class is provided and the invariant characterizations based on the obtained rank-zero invariant coframe and the corresponding constant structure equations are established. Also, a systematic procedure for constructing the linearizing point transformation is derived. The theoretical results are illustrated by several examples.