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On some conjectural supercongruences involving the sequence $t_n(x)$
Hui-Li Han, Chen Wang
https://arxiv.org/abs/2510.11338 https://arxiv.org/pdf/2510.1133…

On some conjectural supercongruences involving the sequence $t_n(x)$
In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{n=0}^{p-1}t_n(x)^2$ and $\sum_{n=0}^{p-1}(n+1)t_n(x)^2$ modulo $p^2$; for example, we establish that \begin{align*} \sum_{n=0}^{p-1}t_n(x)^2\equiv\begin{cases} \left(\dfrac{-1}{p}\right)\pmod{p^2},&\text{if }2x\equiv-1\pmod{p},\\[8pt] (-1)^{\langle x\rangle…

Amazon Prime Video pulls its AI recaps, which it began testing in November, for Fallout, Bosch, and other shows after users noticed errors in the Fallout recaps (Emma Roth/The Verge)
https://www.theverge.com/news/842978/amazon-prime-video-ai-fallout-recap<…

Amazon Prime Video pulls AI-powered recaps after Fallout flub
Amazon Prime Video has pulled its AI-powered Video Recaps after viewers noticed errors in Fallout’s season one summary.

Amazon Prime Video pulls its AI recaps, which it began testing in November, for Fallout, Bosch, and other shows after users noticed errors in the Fallout recaps (Emma Roth/The Verge)
https://www.theverge.com/news/842978/amazon-prime-video-ai-fallout-recap<…

Amazon Prime Video pulls AI-powered recaps after Fallout flub
Amazon Prime Video has pulled its AI-powered Video Recaps after viewers noticed errors in Fallout’s season one summary.

On the distinction between Kloosterman sums and multiplicative functions
Yang Zhang
https://arxiv.org/abs/2510.10721 https://arxiv.org/pdf/2510.10721

On the distinction between Kloosterman sums and multiplicative functions
Let $S(a,b;n)$ be the Kloosterman sum. For any integers $a\neq0,b\neq0,k\geqslant2$, any given complex number $η\not=0$ and multiplicative function $f: \mathbb{N} \rightarrow \mathbb{C}$, we show that \[ |\{n\leqslant X:S(a,b;n)=ηf(n), n \text{ square-free }k\text{-almost prime}\}|\leqslant π(\dfrac{X}{L_k})+O(X^{1-\frac{1}{2k}}) \] and \[ |\{n\leqslant X:S(a,b;n)=ηf(n), n \text{ square-free}\}|\leqslant π(X)+O(Xe^{-\sqrt{\log X}}) \] as $X\rightarrow \infty$. Here, $L_…

Good Morning #Canada
There is a lot of news coverage about a USA president that lusts after a Nobel Prize like he attacks a family sized bucket of KFC. So let's celebrate that OTD in 1957, the first Canadian won the peace prize - Lester B. Pearson. Before he became our 14th Prime Minister, he spent the majority of his career as a diplomat. He was one of the founders of the United Nations and almost ended up as its first Secretary General. He was instrumental in ending the Suez Canal crisis by suggesting and helping to implement the first U.N. Peacekeeping Force.
#CanadaIsAwesome #CanadianHeroes
https://www.nobelprize.org/prizes/peace/1957/pearson/facts/

On 2-absorbing submodules and their Amalgamations
Abuzer Gunduz
https://arxiv.org/abs/2510.09863 https://arxiv.org/pdf/2510.09863

On 2-absorbing submodules and their Amalgamations
Let $R_1$ and $R_2$ be commutative rings with $1\neq 0,\;M$ and $N$ be unitary $R_1-$module and $R_2-$module, respectively. $f:R_1\rightarrow R_2$ be a ring homomorphism and $φ: M\rightarrow N$ be an $R-$module homomorphism. This article studied $2-$absorbing submodule that is a generalization of the concept of prime submodule. Firstly, some characterizations of $2-$absorbing submodule are presented. Then, we examine the notion of $2-$absorbing submodule in amalgamation module $M \bowtie^φ JN…

Europe’s populist worries will intensify today
when right-wing billionaire Andrej Babiš becomes Czech prime minister.
Combined with Hungary’s Viktor Orbšn and Slovakia’s Robert Fico,
Babiš has the potential to jam up the legislative machinery in Brussels as it works on key files.
Czech President Petr Pavel is set to appoint Babiš to the position
after resolvinglongstanding conflict-of-interest issues related to the PM-elect’s conglomerate, Agrofert.
Babiš…

New Czech PM Babiš is poised to aggravate Brussels’ populist headache
Andrej Babiš could find common cause with Hungary’s Viktor Orbán and Slovakia’s Robert Fico on several issues.

Sum of short exponential sums with prime numbers
Firuz Rakhmonov
https://arxiv.org/abs/2510.09102 https://arxiv.org/pdf/2510.09102

Sum of short exponential sums with prime numbers
For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = αu^n + α_{n-1}u^{n-1} + \ldots + α_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $α$ is a real number such that $\left|α- \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ and $\mathscr{L} = \ln x$, an estimate of the form $$ \sum_{k=1}^K \left| \sum_{x - y < p \le x} e(kf(p)) \right| \ll K y \left( \frac{1}{q} + \frac…

Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
Xiaorui Li, Hui Liu, Wei Wang
https://arxiv.org/abs/2510.06597 h…

Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
There are two long-standing conjectures in Hamiltonian dynamics concerning Reeb flows on the boundaries of star-shaped domains in $\mathbb{R}^{2n}$ ($n \geq 2$). One conjecture states that such a Reeb flow possesses either $n$ or infinitely many prime closed orbits; the other states that all the closed Reeb orbits are irrationally elliptic when the domain is convex and the flow possesses finitely many prime closed orbits. In this paper, we prove that for dynamically convex Reeb flow on the boun…

On the ternary Estermann problem with almost proportional summands
Firuz Rakhmonov
https://arxiv.org/abs/2510.05602 https://arxiv.org/pdf/2510.05602…

On the ternary Estermann problem with almost proportional summands
For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ in the form $p_1+p_2+m^n=N$, where $p_1$, $p_2$ $-$ prime numbers, $m$ $-$ natural number satisfying the conditions $$ \left|p_k-μ_kN\right|\le H, \quad k=1,2,\qquad \left|m^n-μ_3N\right|\le H,\qquad H \ge N^{1-\frac1{n(n-1)}} {\mathscr{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $μ_1+μ_2+μ_3=1, \ \ μ_i >0, \mathscr{L} = \ln{N}. $ Keywords: Estermann…

Some formulae relating modular representations of elementary abelian $p$-groups
Jonathan Elmer, Kazal Kadr
https://arxiv.org/abs/2510.07939 https://arxiv.o…

Some formulae relating modular representations of elementary abelian $p$-groups
Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that $V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$ provided $i$ is not divisible by $p$, and that $V_2 \otimes V_p$ is indecomposable if $n>1$. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order $p$. We show how o…

On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function
Tomohiro Yamada
https://arxiv.org/abs/2510.11252 https://arxi…

On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function
We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$.

Every sufficiently large odd integer is a sum of two positive perfect squares and a prime
Ricardo Adonis Caraccioli Abrego
https://arxiv.org/abs/2509.14246 https://

Every sufficiently large odd integer is a sum of two positive perfect squares and a prime
We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain the asymptotic formula R(n) = S(n) C n / log n, where S(n) > 0 is the singular series and C > 0 is the singular integral. The proof uses the Hardy-Littlewood circle method: on the major arcs we extract the main term, and on the minor arcs we combine an L2 bound…

Further results on bent partitions
Jiaxin Wang, Yadi Wei, Fang-Wei Fu
https://arxiv.org/abs/2509.16911 https://arxiv.org/pdf/2509.16911

Further results on bent partitions
Bent partitions of $V_{n}^{(p)}$ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the finite field $\mathbb{F}_{p}$, $n$ is an even positive integer, and $p$ is a prime. For bent partitions, there remains a challenging open problem: Whether the depth of any bent partition of $V_{n}^{(p)}$ is always a power of $p$. Notably, the depths of all current known bent part…

JCMT detection of HCN emission from 3I/ATLAS at 2.1 AU
Iain M. Coulson, Yi-Jehng Kuan, Steven B. Charnley, Martin A. Cordiner, Yo-Ling Chuang, Yueh-Ning Lee, Min-Kai Lin, Stefanie N. Milam, Bannawit Pimpanuwat, Nathan X. Roth, Micha{\l} \.Z\'o{\l}towski
https://arxiv.org/abs/2510.02817

JCMT detection of HCN emission from 3I/ATLAS at 2.1 AU
We report the detection of HCN ($J=3-2$) rotational emission from comet 3I/ATLAS at a heliocentric distance of 2.13 AU with the James Clerk Maxwell Telescope (JCMT). Observations were conducted from 07 August 2025 (UT) using the $^{\prime}\overline U^{\prime}\overline u$ heterodyne receiver and ACSIS spectroscopic backend. The HCN line was detected at $>5σ$ on 14 Sep 2025 (UT) and a production rate of $Q({\rm HCN})=(4.0\pm1.7)\times10^{25}\ {\rm s}^{-1}$ was derived by non-LTE radiative transf…

On 1-absorbing prime and weakly 1-absorbing prime subsemimodules
Mohammad adarbeh, Mohammad Saleh
https://arxiv.org/abs/2509.15398 https://arxiv.org/pdf/25…

On 1-absorbing prime and weakly 1-absorbing prime subsemimodules
In this paper, we introduce the concepts of 1-absorbing prime and weakly 1-absorbing prime subsemimodules over commutative semirings. Let S be a commutative semiring with 1 \neq 0 and M an S-semimodule. A proper subsemimodule N of M is called 1-absorbing prime (weakly 1-absorbing prime) if, for all nonunits a, b \in S and m \in M, abm \in N (0 \neq abm \in N) implies ab \in (N :_{S} M) or m \in N. We study many properties of these concepts. For example, we show that a proper subsemimodule N of …

Non-Simple knots in Contact 3-Manifolds
Ipsita Datta, Tanushree Shah
https://arxiv.org/abs/2510.02932 https://arxiv.org/pdf/2510.02932

Non-Simple knots in Contact 3-Manifolds
We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are topologically isotopic to the $n$-twist knot in a Lens space $L(α, β)$ than in $S^3$. We also include connect sum formulas for rational variants of classical invariants, $\mathrm{tb}_\mathbb{Q}$, $\mathrm{rot}_\mathbb{Q}$, and $\mathrm{sl}_\mathbb{Q}$, whi…

The Complexity of Deciding Characteristic Formulae Modulo Nested Simulation (extended abstract)
Luca Aceto, Antonis Achilleos, Aggeliki Chalki, Anna Ing\'olfsd\'ottir
https://arxiv.org/abs/2509.14089

The Complexity of Deciding Characteristic Formulae Modulo Nested Simulation (extended abstract)
This paper studies the complexity of determining whether a formula in the modal logics characterizing the nested-simulation semantics is characteristic for some process, which is equivalent to determining whether the formula is satisfiable and prime. The main results are that the problem of determining whether a formula is prime in the modal logic characterizing the 2-nested-simulation preorder is coNP-complete and is PSPACE-complete in the case of the n-nested-simulation preorder, when n>=3. T…

Lepton models from non-holomorphic $A^{\prime}_{5}$ modular flavor symmetry
Cai-Chang Li, Gui-Jun Ding
https://arxiv.org/abs/2509.15183 https://arxiv.org/p…

Lepton models from non-holomorphic $A^{\prime}_{5}$ modular flavor symmetry
In the framework of non-holomorphic modular invariance approach, we have systematically constructed all minimal lepton models based on the non-holomorphic $A^{\prime}_{5}$ modular symmetry from a bottom-up approach. In these models, the Yukawa couplings are described by polyharmonic Maaß forms of integer weights at level $N=5$. Under the assumption of Majorana neutrinos, both the Weinberg operator and the type-I seesaw mechanism are considered for neutrino mass generation. All minimal models a…

Homological torsion growth in non-normal chains of graphs of free groups
Dario Ascari, Jonathan Fruchter
https://arxiv.org/abs/2509.15075 https://arxiv.org…

Homological torsion growth in non-normal chains of graphs of free groups
Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1 \ge G_2 \ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, $\liminf_{n\rightarrow \infty} \frac{\log \vert \mathrm{Tor}_p(G_n^{\mathrm{ab}})\vert}{[G:G_n]}…

Ukraine sees its EU membership bid advancing before the end of this year
thanks to “creative solutions” to overcome Hungary’s opposition,
Ukrainian Deputy Prime Minister Taras Kachka told POLITICO.
EU leaders could sign off on opening as many as six negotiating “clusters”
— legal steps on the path to membership in the bloc
— in December as “political momentum” builds
and Hungarian Prime Minister Viktor Orbšn faces growing pressure to remove his veto, he sa…

Ukraine’s EU membership bid set for breakthrough in December, says deputy PM
Taras Kachka believes “creative solutions” will overcome Hungary’s opposition to Kyiv’s joining the EU.

Small Prime Primitive Roots in Arithmetic Progressions
N. A. Carella
https://arxiv.org/abs/2509.19309 https://arxiv.org/pdf/2509.19309

Small Prime Primitive Roots in Arithmetic Progressions
Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a

Lifting of cycles in functional graphs
Tadahisa Nara
https://arxiv.org/abs/2509.16234 https://arxiv.org/pdf/2509.16234

Lifting of cycles in functional graphs
For a given function from a set to itself, we can define a directed graph called the functional graph, where the vertices are the elements of the set, and the edges are all the pairs of inputs and outputs for the function. In this article we consider functional graphs on Z/mZ with respect to polynomial functions. The main result describes the behavior of cycles in functional graphs on Z/p^nZ while $n$ is increasing, where $p$ is a prime number.

Sometimes \(2^n 1\) is not prime. but let me just check that on #Google.
#Odometer #AISlop #math

Discrepancy And Fair Division For Non-Additive Valuations
Max Dupre la Tour, Kaito Fujii
https://arxiv.org/abs/2509.16802 https://arxiv.org/pdf/2509.16802

Discrepancy And Fair Division For Non-Additive Valuations
We extend the notion of combinatorial discrepancy to \emph{non-additive} functions. Our main result is an upper bound of $O(\sqrt{n \log(nk)})$ on the non-additive $k$-color discrepancy when $k$ is a prime power. We demonstrate two applications of this result to problems in fair division. First, we establish a bound for a consensus halving problem, where fairness is measured by the minimum number of items that must be transferred between the two parts to eliminate envy. Second, we improve the u…

On distributional topological complexity of groups and manifolds
Alexander Dranishnikov
https://arxiv.org/abs/2509.16759 https://arxiv.org/pdf/2509.16759…

On distributional topological complexity of groups and manifolds
We prove the equality $\dTC(Γ)=\TC(Γ)$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. For the distributional topological complexity of lens spaces we prove the inequality $\dTC(L^n_p)\le 2p-1$ and for the distributional LS-category the inequality $d\cat(L^n_p)\le p-1$ which turns into equality for prime $p$ and $n>p$. We use these inequalities to bring counter-examples to the product formula for $d\cat$ and $\dTC$.

Weighted uniform distribution of subpolynomial functions along primes and applications
Vitaly Bergelson, Grigori Kolesnik, Younghwan Son
https://arxiv.org/abs/2509.19722 https:/…

Weighted uniform distribution of subpolynomial functions along primes and applications
Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $p_n$ denotes the $n$-th prime. This extends the main result of [4] to the weighted setting and leads to new applications in uniform distribution theory, ergodic theory, and additive combinatorics.

Products of strictly hyperbolic conjugacy classes in symplectic groups
Klaus Nielsen
https://arxiv.org/abs/2509.26237 https://arxiv.org/pdf/2509.26237

Products of strictly hyperbolic conjugacy classes in symplectic groups
We call a conjugacy class of the symplectic group Sp$(2n, K)$ over a field $K$ strictly hyperbolic if its minimal polynomial is of the form $q(x) q^*(x)$, where the polynomial $q(x)$ is prime to its reciprocal $q^*(x) := x^n q(x^{-1})$. It is shown that the product of 2 cyclic, strictly hyperbolic conjugacy classes of Sp$(2n, K)$ contains all nonscalar elements of Sp$(2n, K)$. It follows that the projective symplectic group has a conjugacy class of covering number 2, i.e. PSp$(2n,K) = Ω^2$ for…

The Number of Parts in the (Distinct) Partitions With Parts From a Set
A. David Christopher
https://arxiv.org/abs/2509.22618 https://arxiv.org/pdf/2509.226…

The Number of Parts in the (Distinct) Partitions With Parts From a Set
The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the number of prime divisors of $n$, $p$-adic valuation of $n$, the number of Carlitz-binary compositions of $n$ and the Hamming weight function. Finally, we obtain an asymptotic estimate for the number of parts in the partitions of $n$ with parts from a finite se…

Primes of the form $ax by$ in certain intervals with small solutions
Yuchen Ding, Takao Komatsu, Honghu Liu
https://arxiv.org/abs/2510.01781 https://arxiv.…

Primes of the form $ax+by$ in certain intervals with small solutions
Let $1

BOWIE-ALIGN: Weak spectral features in KELT-7b's JWST NIRSpec/G395H transmission spectrum imply a high cloud deck or a low-metallicity atmosphere
Eva-Maria Ahrer, Charlotte Fairman, James Kirk, Hannah R. Wakeford, Joanna K. Barstow, Anna B. T. Penzlin, Lili Alderson, Richard A. Booth, Duncan A. Christie, Alastair B. Claringbold, Emma Esparza-Borges, Carlos Gasc\'on, Mercedes L\'opez-Morales, N. J. Mayne, Mason McCormack, Annabella Meech, Paul Molli\`ere, James E. Owen, Vats…

BOWIE-ALIGN: Weak spectral features in KELT-7b's JWST NIRSpec/G395H transmission spectrum imply a high cloud deck or a low-metallicity atmosphere
Hot Jupiters and their atmospheres are prime targets for transmission spectroscopy due to their extended atmospheres and the corresponding large signal-to-noise, providing the best possible constraints for the atmospheric carbon-to-oxygen (C/O) ratio and metallicity of exoplanets. Within BOWIE-ALIGN, we aim to compare JWST spectra of a sample of orbitally aligned and misaligned hot Jupiters orbiting F-type stars to probe the link between hot Jupiter atmospheres and planet formation history. Her…

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
G. Kavya Keerthana, S. Ananya, Ranganatha D
https://arxiv.org/abs/2509.17705 https://arxi…

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
Let $\overline{p}_{k}(n)$ denote the number of overpartition $k$-tuples of $n$. In 2023, Saikia \cite{saikia} conjectured the following congruences: \begin{align*} \overline{p}_{q}(8n+2)& \equiv 0 \pmod{4},\quad \overline{p}_{q}(8n+3)\equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+4) \equiv 0 \pmod{2},\\ \overline{p}_{q}(8n+5)& \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+6) \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+7)\equiv 0 \pmod{32}, \end{align*} where $n\geq0$ and $q$ is prime. Recently, Sell…

On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
Takamichi Sano
https://arxiv.org/abs/2510.01601 https://arxi…

On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
Let $f \in S_{2r}(Γ_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $χ$ be an anticyclotomic Hecke character of $K$ which is crystalline at the primes above $p$ and such that $L(f,χ,r)\neq 0$. We prove that the Tamagawa number conjecture for the critical value $L(f,χ,r)$ follows from the Iwasawa main conjecture for the Bertolini-Darmon-Prasanna $p$-adic $L$-functi…

$p$-adic hypoerbolicity for Shimura varieties and period images
Benjamin Bakker, Abhishek Oswal, Ananth N. Shankar, Zijian Yao
https://arxiv.org/abs/2509.25461 https://

$p$-adic hypoerbolicity for Shimura varieties and period images
We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let $\mathsf{D}^{\times}\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Th…

Dold-Gauss Congruences, Norm Descent, and Rational Rigidity
Hartosh Singh Bal
https://arxiv.org/abs/2509.25038 https://arxiv.org/pdf/2509.25038

Dold-Gauss Congruences, Norm Descent, and Rational Rigidity
We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove \emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms $N_{K/\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Tw…