ALT Let Qp,q(t)∈ℤ[t] be Sharipov's even monic degree-10 second cuboid polynomial, with coprime positive integers p≠ q. We prove that Qp,q(t) is irreducible over ℚ, and hence over ℤ. More generally, the normalized associated quintic Pₛ(x) and its composition Pₛ(t²) are irreducible over ℚ for every s∈ℚ∖{0,±1}. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-2 curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime 3. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are 3-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative r

ALT In the present paper, we investigate geometric properties of a Riemannian submersion for which the total manifold is endowed with hyperbolic and almost hyperbolic Ricci solitons. We establish conditions under which the fibers of the submersion admit hyperbolic and almost hyperbolic Ricci solitons. Further, we derive necessary conditions for the base manifold to be an Einstein and a hyperbolic Ricci soliton. Also, we determine the scalar curvature associated with the fibers and the base manifold. Moreover, when the total space admits an almost hyperbolic Ricci soliton with a torse-forming potential vector field, we obtain conditions ensuring that the extrinsic vertical and horizontal distributions are almost generalized quasi-Einstein. We also obtain necessary conditions for the fibers and base manifold to be an Einstein and almost generalized quasi-Einstein manifold. Finally, examples are included to illustrate the existence of hyperbolic and almost hyperbolic Ricci solitons in Riemann

ALT We study integers of the form N=(d+1)(3d+1)(4d+1). Using Korselt's criterion, we prove that, provided the three factors are prime, N is a Carmichael number if and only if d≡4±od12. We also show that the special case d=2^m produces only 1105 under the prime-factor hypothesis.

ALT The secondary zeta function is defined by Z(s)=∑γ>₀¹⁄γₛ, where tfrac12+iγ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of Z(s) at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann ξ-function. Define L(x)=log(ξ(x+tfrac12)/ξ(tfrac12)), ℰ=xᵈ⁄dₓ, Dₖ=ℰ∏ⱼ₌₁ᵏ(ℰ-2j). Then, for k≥1, Z(2k)=(-1)ᵏ/2ᵏ(k-1)!∫₀∞Dₖ L(x)/x²ᵏ⁺¹ dx and Z(2k+1)=(-1)ᵏ/π (2k-1)!!∫₀∞Dₖ L(x)/x²ᵏ⁺² dx. In the last part, we write these formulas out explicitly for Z(k), 2≤ k≤ 7, both in terms of L(x) and in terms of ξ(x+tfrac12) for reference. We also derive a limit formula for γ₁, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

ALT Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

ALT We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving 25 out of 30 two-variable equations of size up to 34 for which the problem of integer solubility remained open. We also resolve the remaining 5 equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to 35 and of length up to 12, with two exceptions. Before this work, there were 211 equations in this category that resisted the previously known methods, and we proved that 209 of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.

ALT A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients bₖ=eᵃ E[Yᵏ ]/[k!(d)ₖ ] for densities proportional to y⁽d-1) exp(-αyq+e̲tay), with known integer q>=2, d,α>0, and unrestricted real a,e̲ta. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs

ALT A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple {a,b,c,d} with a<b<c<d is called regular if d=a+b+c+2abc+2√((ab+1)(ac+1)(bc+1)). We prove that every irregular Diophantine quadruple satisfies c<10³³⁸ and d<10¹⁰²⁴.

ALT The 2026 measles resurgence in Bangladesh motivates a spatial modelling framework that accounts for heterogeneous local transmission, vaccination activity, and interdivisional connectivity. We formulate a multipatch susceptible-vaccinated-infectious-recovered model for the eight administrative divisions of Bangladesh. Division-specific parameters are estimated from reported measles cases and smoothed vaccination data using bounded nonlinear least squares, with uncertainty quantified through 1000 parametric bootstrap replicates. The fitted divisional systems are then connected through compartment-specific movement and cross-patch infection coefficients adjusted for connectivity and the source division's relative reproduction number. The fitted model captures the main division-specific epidemic patterns. Estimated local reproduction numbers range from 1.1386 in Khulna (95% CI:0.5703–1.3834) to 4.3491 in Dhaka (95% CI: 1.4449–5.6307), while Chattogram also exhibits high transmission poten

ALT Let K be an imaginary quadratic field with fundamental discriminant D=-5n, where n>0 is odd, and cyclic ideal class group Cl(K)=langle granglesimeq C₄M. Assume that the ramified prime class above 5 is the unique involution g²M. For the Kronecker eta invariant F(A)=|η(z_A)|²/√a_A, where z_A and a_A arise from a positive definite quadratic form representing the ideal class A, we show that the logarithms of the mirror ratios R_j=F(g²M-j)/F(g^j), 0≤ j<M, are related by an explicit discrete cosine transform to the odd class-character Hecke L(1)-values. A single level-5 generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove mathbbQ(R_j)=H^langlecrangle, R_j^{σ²M}=R_j^-1, mathbbQ(R_j+R_j^-1)=H^{langlec,σ²Mrangle}. Here H is the Hilbert class field, c denotes complex conjugation, and σ corresponds to g under the Artin map. Consequently [mathbbQ(R_j):mathbbQ]=4M, [mathbbQ(R_j+R_j^-

ALT For an odd prime p, we study the change in matrix order under lifting from Z/pᵏ Z to Z/p⁽k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl₂(Fₚ). We classify the full-order transfer for every q in GL₂(Fₚ): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U² = 0 and U!=0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX)!=0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruc

ALT We give a self-contained derivation of the closed form of the power sums Sₘ(n)=∑ᵢ₌₁ⁿ(a₀+(i-1)d)ᵐ in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope Sₘ'(0) of the polynomial at the origin and fixed by the value Sₘ(1)=a₀ᵐ. This yields a closed form for Sₘ(n) in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as dᵐBₘ(a₀/d); the progression starting at 0 reproduces that recurrence exactl

ALT The Diophantine equation Fₙˣ+Fₘʸ=2ᵃ, where Fₖ denotes the k-th Fibonacci number and n,m,a∈ ℤ≥₀, has attracted attention in recent years. The cases (x,y)=(1,1) and (x,y)=(2,2) were completely resolved by Bravo and Luca, and by Jeong and Park, respectively. In this note, we investigate the mixed exponent case (x,y)=(2,1) and we completely determine all nonnegative integer solutions.

ALT We study a family of special numbers and polynomials introduced by Simsek whose exponential generating function has as denominator a falling factorial in the exponential variable. Using the partial fraction decomposition of this denominator, each member of the family is expressed by a single finite sum involving Bernoulli polynomials, Apostol Bernoulli polynomials and two variable Fubini polynomials evaluated at reciprocals of integers. A second expansion, around the point where the denominator vanishes, expresses the same numbers through Stirling numbers of the second kind and generalized harmonic numbers, and yields closed formulas for the first few members. We construct a function, built from the Hurwitz zeta function and Lerch s transcendent, which interpolates the polynomials at the negative integers and gives a closed form of a multiple series recently introduced by Pokaz, Ribicic Penava and Simsek, we give an Euler type integral representation for it, and show that the numbers a

ALT We construct nonnegative bounded compactly supported functions on the real line that give the lower bound A≥ 0.4103779774321214877852987… for the minimum-autocorrelation constant. This improves the explicitly certified bound 2378625/5958277=0.39921356… reported by Russell. The construction combines a point mass with a monotone degree-ten polynomial density; a one-sided atom-removal argument then produces ordinary functions. Its numerical content reduces to exact rational polynomial identities and 20 Bernstein subinterval certificates. The finite certificate, actual overlap integrals, admissibility, mass formula, and ordinary-function witness bounds have been formalized in Lean 4.19.0; the pinned proof sources and recorded audit are supplied. The passage to the supremum constant is given separately as a mathematical argument. As applications, the same construction yields an upper bound below 1.5610183 for the large-multiplicity generalized-difference-basis coefficient and a feasible squ

ALT Ordinary angles and unsigned areas discard orientation, and much of the case analysis in plane geometry is the price of that loss. Signed quantities (directed angles, oriented areas and directed distances) keep the signs and so let one argument serve for every configuration. We illustrate this with results that hold uniformly for convex, concave and crossed quadrilaterals: the diagonal area formula, an identity expressing one diagonal in terms of the sides and the other diagonal, a test for a pair of parallel sides, a theorem on equidiagonal quadrilaterals (such a quadrilateral has a pair of equal opposite sides if and only if it has a pair of parallel opposite sides), and a single ratio of directed distances that decides whether the vertices lie on an ellipse, only on hyperbolas, or on a circle.

ALT We investigate the zeros of Leonardo polynomials and their distribution in the complex plane. We show that the structure of the zeros differs between even and odd degrees. For even degrees, all zeros are simple and purely imaginary, with imaginary parts lying in (-2,2). For odd degrees, there is exactly one positive real zero, whereas no negative real or purely imaginary zeros occur. We obtain two-sided bounds for the unique positive real zero, prove that it decreases strictly to zero as the degree increases through odd values, and determine its asymptotic behavior. Using the relation between Leonardo and Fibonacci polynomials together with a complex trigonometric parametrization, we transform the zero equation into a family of fixed-point problems. Suitable invariant disks are constructed, and the corresponding mappings are shown to be contractions. This yields a constructive description of the complete zero set, proves the simplicity of all zeros, and provides geometrically convergen

ALT We study the length of one round of Uno under the official rules, for N players who each start with M cards. The fastest possible round takes at least M turns with two players and at least 2M-1 turns with three or more, and both bounds are attained for small M. An exact search over random deals shows that real deals rarely allow them. Under the official rules a round has no maximum length. Under forced play the probability that a round lasts more than t turns decays exponentially in t at a rate that does not depend on M. Our main result concerns very large hands. If every card a player ever holds is eventually played, a round lasts (27/19)NM turns to first order, and the constant depends only on the composition of the deck. The official rule that a Wild Draw Four may only be played by a player who holds no card of the current colour breaks this law. Every large hand hoards Wild Draw Fours, M/23 of them per player, and the round ends with a dumping endgame. For an even number of players

ALT In this article, we mainly investigate when the binary operation T^[n]:[0, 1]²→ [0, 1] satisfies the associative law and thus becomes a triangular norm where T^[n](x,y)={ll fₙ(T(fₙ(x),fₙ(y))), if (x,y)∈[0,1)², min{x,y}, otherwise, fₙ:[0,1]→{0, ¹⁄ₙ, ²⁄ₙ, …, ⁿ⁻¹⁄ₙ, 1} given by fₙ(x)=¹⁄ₙ⌊nx⌋ with n a positive integer and T:[0, 1]²→ [0, 1] is a triangular norm. We present some sufficient conditions for T^[n] to be triangular norms, and for T^[n] to be never associative when T is strictly monotonic.

ALT Paulsen and West associate perfect cuboids with triples of rational points on a congruent-number elliptic curve and assert that the corresponding curve has rank at least two. The proof of their Theorem 4.2 contains a gap: translation of a generator by a point of order two interchanges only the odd-indexed terms of two coordinate sequences, rather than the entire sequences. We exhibit two explicit counterexamples to that intermediate assertion and give a complete proof of the rank obstruction. The division-polynomial argument reduces a hypothetical rank-one configuration to one remaining parity pattern. Ingram's primitive-divisor theorem then forces an additive relation among its indices, which is incompatible with a positivity lemma on the elliptic curve. More precisely, the three points of every positive cuboid configuration generate a subgroup of rank at least two, even when the ambient curve has larger rank. The proof is unconditional.