Tootfinder

NBCUniversal and Amazon reach distribution deals for Prime Video, Peacock, Fire TV, and Xfinity X1, including offering Peacock Premium Plus on Prime Video (Brian Steinberg/Variety)
https://variety.com/2025/tv/news/amazon-nbcu-peacock-available-prime-video-subscription-1236500831/
http://mediagazer.com/250828/p10#a250828p10

Exponential sums over primes are unbounded
Pierre-Alexandre Bazin
https://arxiv.org/abs/2508.18394 https://arxiv.org/pdf/2508.18394

Exponential sums over primes are unbounded
We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} Λ(m) \mathrm{e}(αm)\right|^2 \gg y\log y$$ whenever $y \ll x^{1/3-\varepsilon}.$ This answers a question of Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} Λ(m) \mathrm{e}(αm)\right| \gg x^{1/6 - \varepsilon}.$$

Hilbert-Kunz multiplicity and $F$-signature can disagree
Seungsu Lee, Suchitra Pande, Austyn Simpson
https://arxiv.org/abs/2508.19985 https://arxiv.org/pdf…

Hilbert-Kunz multiplicity and $F$-signature can disagree
We compute the $F$-signature function of the ample cone of any nontrivial ruled surface over $\mathbb{P}^1_k$ where $k$ is an algebraically closed field of prime characteristic. As an application, we construct a Noetherian $F$-finite strongly $F$-regular ring $R$ of prime characteristic admitting two maximal ideals $\mathfrak{n}_1,\mathfrak{n}_2\in \mathrm{Spec} R$ at which the Hilbert-Kunz multiplicity and $F$-signature measure different singularities; that is, $\operatorname{e}_{\operatorname…

Amazon says Prime Video in India caters to "subscription-ready" customers while Amazon MX Player is aimed at those transitioning from traditional media (Naman Ramachandran/Variety)
https://variety.com/2025/tv/news/amazon-india-st…

Amazon Preps Sets India Streaming Push With Prime Video, MX Player
Amazon targets India's diverse streaming market with Prime Video for premium subscribers and MX Player for mobile users.

Small Composite Numbers in Orbits of Linear Maps
Jose Reyes
https://arxiv.org/abs/2508.18305 https://arxiv.org/pdf/2508.18305

Small Composite Numbers in Orbits of Linear Maps
Generalized Cunningham chains are sets of the form $\{f^n(z)\}_{n\ge0}$ where all its elements are prime numbers and $f$ is a linear polynomial with integer coefficients. We generalize this definition further to include starting terms that are not prime, and we obtain the bound of $\ell(z)< z$ if $z$ is big enough, where $\ell(z)$ is the size of the generalized Cunningham chain. Unlike a direct generalization of previous results, which require $z$ to have a prime factor that does not divide the…

Cohomology of solvable saturable pro-$p$ groups and Lie algebras
Oihana Garaialde Oca\~na, Jon Gonz\'alez-S\'anchez, Lander Guerrero-S\'anchez
https://arxiv.org/abs/2507.18163

Cohomology of solvable saturable pro-$p$ groups and Lie algebras
Let $p$ be an odd prime, and let $n\in \N$ be an integer. We show that the $n^{\text{th}}$ mod-$p$ cohomology of a solvable saturable pro-$p$ group is isomorphic to the $n^{\text{th}}$ mod-$p$ cohomology of its associated $\Z_p$-Lie algebra $\g$ as a $\F_p$-vector space. Addittonally, we obtain that the $n^{\text{th}}$ mod-$p$ cohomology of $\g$ and of $\g/p\g$ are isomorphic as $\F_p$-vector spaces.

Finding congruences with the WZ method
Li-Quan Feng, Qing-Hu Hou
https://arxiv.org/abs/2506.19221 https://arxiv.org/pdf/2506.19221

Finding congruences with the WZ method
We utilize the Wilf-Zeilberger (WZ) method to establish congruences related to truncated Ramanujan-type series. By constructing hypergeometric terms $f(k, a, b, \ldots)$ with Gosper-summable differences and selecting appropriate parameters, we derive several congruences modulo $p$ and $p^2$ for primes $p > 2$. For instance, we prove that for any prime $p > 2$, \[ \sum_{n=0}^{p-1} \frac{10n+3}{2^{3n}}\binom{3n}{n}\binom{2n}{n}^2 \equiv 0 \pmod{p},\] and \[ \sum_{n=0}^{p-1} \frac{(-1)^n(20n^2+8n+…

Amazon plans to hold Prime Day 2025 from July 8 until July 11, lasting four days instead of the usual two and open for Prime members (Cameron Faulkner/The Verge)
https://www.theverge.com/news/685299/amazon-prime-day-2025-dates-when

Amazon Prime Day stretches to four days of deals this year
Amazon’s 2025 Prime Day will begin Tuesday, July 8th, and lasts through Friday, July 11th. There will be a ton of smart home, headphone, TV, laptop deals.

Power maps on General Linear groups over finite principal ideal local rings of length two
Saikat Panja, Ayon Roy, Anupam Singh
https://arxiv.org/abs/2508.18952 https://

Power maps on General Linear groups over finite principal ideal local rings of length two
Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of length two (e.g. $\mathbb{Z}/p^2\mathbb{Z}$ and $\mathbb F_q[t]/\langle t^2\rangle$). We study the power map $g\mapsto g^L$, where $L$ is a positive integer. We consider $L$ to be coprime to $p$ (an odd prime), the characteristic of the residue field $k$ of $\ma…

The mass of the exo-Venus Gliese 12 b, as revealed by HARPS-N, ESPRESSO, and CARMENES
Daisy A. Turner, Yoshi Nike Emilia Eschen, Felipe Murgas, Annelies Mortier, Thomas G Wilson, Jorge Fern\'andez Fern\'andez, Giuseppe Morello, Shreyas Vissapragada, Jos\'e A. Caballero, Stefan Dreizler, Jo Ann Egger, Alix Violet Freckelton, Nicole Gromek, Artie P. Hatzes, Ben Scott Lakeland, Evangelos Nagel, Luca Naponiello, Hugo M. Tabernero, Siegfried Vanaverbeke, Alexander Venner, Mar\…

The mass of the exo-Venus Gliese 12 b, as revealed by HARPS-N, ESPRESSO, and CARMENES
Small temperate planets are prime targets for exoplanet studies due to their possible similarities with the rocky planets in the Solar System. M dwarfs are promising hosts since the planetary signals are within our current detection capabilities. Gliese 12 b is a Venus-sized temperate planet orbiting a quiet M dwarf. We present here the first precise mass measurement of this small exoplanet. We performed a detailed analysis using HARPS-N, ESPRESSO, and CARMENES radial velocities, along with new…

Kenyon Theorem Revisited
Szymon G{\l}\k{a}b, Mateusz Kula
https://arxiv.org/abs/2508.19454 https://arxiv.org/pdf/2508.19454

Kenyon Theorem Revisited
We study sets $E(Σ,q)=\left\{\sum_{i=1}^\infty σ_iq^i\colon(σ_i)\inΣ^{\mathbb N}\right\}$ for a finite set $Σ\subset \mathbb R$ and $q\in(0,1)$. Under the assumption $q|Σ|=1$ we prove several new equivalent conditions for $E(Σ,q)$ to contain an interval. We give a full characterization, if additionally $|Σ|$ is prime.

Scarcity of partition congruences on semiprime progressions
Scott Ahlgren, Olivia Beckwith
https://arxiv.org/abs/2508.19512 https://arxiv.org/pdf/2508.1951…

Scarcity of partition congruences on semiprime progressions
In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \…

Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$
Pavel Turek, Jialin Wang
https://arxiv.org/abs/2507.15505 https://arxiv.org/pdf/2…

Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$
Let $p$ be a prime and $n\geq 2$ be a positive integer. We establish new formulae for the decompositions of the first $p-1$ symmetric powers of the Specht module $S^{(n-1,1)}$ and the irreducible module $D^{(n-1,1)}$ in characteristic $p$ as direct sums of Young permutation modules. As an application of the formulae, we show that these symmetric powers have Specht filtration and find the vertices of their indecomposable summands. Our main tool, constructed in this paper, is a lift of a splittin…

Barron regularity of many particle Schr\"odinger eigenfunctions
Pingbing Ming, Hao Yu
https://arxiv.org/abs/2508.17722 https://arxiv.org/pdf/2508.1772…

Barron regularity of many particle Schrödinger eigenfunctions
This work investigates the regularity of Schrödinger eigenfunctions and the solvability of Schrödinger equations in spectral Barron space $\mathcal{B}^{s}(\mathbb{R}^{nN})$, where neural networks exhibit dimension-free approximation capabilities. Under assumptions that the potential $V$ consists of one-particle and pairwise interaction parts $V_{i},V_{ij}$ in Fourier-Lebesgue space $\mathcal{F}L_{s}^{1}(\mathbb{R}^{n})+\mathcal{F}L_{s}^{α^{\prime}}(\mathbb{R}^{n})$ and an additional part $V_…

Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
Abbey Bourdon, \"Ozlem Ejder
https://arxiv.org/abs/2506.19560 https://

Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
Let $\ell$ and $n$ be positive integers with $\ell$ prime. The modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$ are algebraic curves over $\mathbb{Q}$ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main resu…

From dual canonical bases to positroidal subdivisions
Jian-Rong Li, Ayush Kumar Tewari
https://arxiv.org/abs/2506.19443 https://arxiv…

From dual canonical bases to positroidal subdivisions
The Grassmannian cluster algebra $\mathbb{C}[\text{Gr}(k, n)]$ admits a distinguished basis known as the dual canonical basis, whose elements correspond to rectangular semi-standard Young tableaux with $k$ rows and with entries in $[n]$. We establish that each such tableau induces a positroidal subdivision of the hypersimplex $Δ(k,n)$ via a map introduced by Speyer and Williams. For $\text{Gr}(2,n)$, we prove that non-frozen prime tableaux correspond precisely to the coarsest positroidal subdi…

Completions of Extremely Noncatenary Noetherian UFDs
Eli B. Dugan, S. Loepp
https://arxiv.org/abs/2508.18102 https://arxiv.org/pdf/2508.18102

Completions of Extremely Noncatenary Noetherian UFDs
Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$ of $A$ satisfying the following conditions: (1) $J_k = J_{\ell}$ if and only if $k = \ell$, (2) there exist positive integers $n \neq m$ such that for each $k \in \mathbb{N}$, there are two saturated chains of prime ideals of $A$ of the form $J_k \subsetneq J^…

A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
Sarah Meng Li (University of Amsterdam, University of Waterloo), Michele Mosca (University of Waterloo), Neil J. Ross (Dalhousie University), John van de Wetering (University of Amsterdam), Yuming Zhao (University of Copenhagen, University of Waterloo)
https://arxiv.org/abs…

A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer. This is the first completeness result for any fragment of quantum circuits in odd prime dimensions. We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting. Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form. We then simplify the rewrite rules in this procedure to a small natural set of rules, giving…

Computing the Level of a Fiber for Points on Modular Curves
Hailey Maxwell
https://arxiv.org/abs/2508.17463 https://arxiv.org/pdf/2508.17463

Computing the Level of a Fiber for Points on Modular Curves
The modular curves in the family $X_1(N)$ for natural numbers $N$ parametrize elliptic curves over the complex numbers with a distinguished point of order $N$. The purpose of this paper is to better understand how to calculate the degrees of points on $X_1(\ell^n)$ for a prime $\ell$ and arbitrary positive integer $n$. In analogy with the definition of the level of a Galois representation, we construct a new definition: the level of a fiber of a closed point on a modular curve. Using this defin…

An Arithmetic Characterization of 2-Generated Numbers
Bireswar Das, Kavita Samant, Dhara Thakkar
https://arxiv.org/abs/2508.16356 https://arxiv.org/pdf/250…

An Arithmetic Characterization of 2-Generated Numbers
A group $G$ is said to be $k$-generated if it has a generating set with $k$ elements. A positive integer $n$ is called a \emph{2-generated number} if every group of order $n$ is 2-generated. In this article, we establish an arithmetic characterization of 2-generated numbers expressed in terms of the prime factorization of $n$.

The Prime Focus Infrared Microlensing Experiment (PRIME): First Results
Takahiro Sumi, David A. H. Buckley, Alexander S. Kutyrev, Motohide Tamura, David P. Bennett, Ian A. Bond, Giuseppe Cataldo, Joseph M. Durbak, S. Bradley Cenko, Dale Fixsen, Orion Guiffreda, Ryusei Hamada, Yuki Hirao, Asahi Idei, Dan Kelly, Markus Loose, Gennadiy N. Lotkin, Eric I. Lyness, Stephen Maher, Shuma Makida, Noriyuki Matsunaga, Shota Miyazaki, Gregory Mosby, Samuel H. Moseley, Tutumi Nagai, Togo Nagano, Se…

The Prime Focus Infrared Microlensing Experiment (PRIME): First Results
We present the description of the instruments and the first results of the PRime-focus Infrared Microlensing Experiment (PRIME). PRIME is the first dedicated near-infrared (NIR) microlensing survey telescope located at the South African Astronomical Observatory (SAAO) in Sutherland, South Africa. Among its class, it offers one of the widest fields of view in the NIR regime. PRIME's main goals are (1) To study planetary formation by measuring the frequency and mass function of planets. In partic…

Emergence of charm-strange dibaryons with negative parity via baryon-baryon interactions
Yu-Yue Cui, Xiao-Mei Tang, Qi Huang, Rui Chen
https://arxiv.org/abs/2507.12958

Emergence of charm-strange dibaryons with negative parity via baryon-baryon interactions
Within the framework of the one-boson-exchange model, we systematically perform a coupled channel analysis of the $P-$wave interactions between a charm baryon and light baryon, the involved channels include $Ξ_cN$, $Λ_cΣ$, $Ξ_c^{\prime}N$, $Σ_cΛ$, $Ξ_c^*N$, $Σ_c^*Λ$, $Σ_cΣ$, and $Σ_c^*Σ$. Our results can predict several possible molecular candidates, such as a $Ξ_c^{\prime}N$ molecule with $I(J^P)=0(1^-)$, $Ξ_c^{*}N$ molecules with $0(0^-, 1^-, 2^-)$, $Σ_cΣ$ molecules with $0(1…

A Shiu Theorem for Larger and Smoother Functions
Thomas Wright
https://arxiv.org/abs/2508.17217 https://arxiv.org/pdf/2508.17217

A Shiu Theorem for Larger and Smoother Functions
In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $β<1$ such that $f(p^l)\ll (\log\log x)^{lβ}$ for every prime $p$ and every $l>1$, and if $f(n)\ll \max\{n^ε,(\log x)^ε\}$ for every $ε>0$, then $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{x}{ϕ(k)(\log x)^{1-ε_0}}\exp\left(\sum_{\substack{p\leq x \\…

Powers of 2 in High-Dimensional Lattice Walks
Nikolai Beluhov
https://arxiv.org/abs/2506.12789 https://arxiv.org/pdf/2506.12789

Powers of 2 in High-Dimensional Lattice Walks
Let $W_d(n)$ be the number of $2n$-step walks in $\mathbb{Z}^d$ which begin and end at the origin. We study the exponent of $2$ in the prime factorisation of this number; i.e., $w_d(n) = ν_2(W_d(n))$. We show that, for each $d$, there is a relationship between $w_d(n)$ and the number $s_2(n)$ of $1$s in the binary expansion of $n$. For example, $w_d(n) = s_2(n)$ if $d$ is odd and $w_d(n) = 2s_2(n)$ if $ν_2(d) = 1$; while $w_d(n) \ge 3s_2(n)$ if $ν_2(d) = 2$. The pattern changes further when …

Democrats must organize urgently for the 2026 midterm elections
and avoid a “it can’t happen here” mentality
to stop Donald Trump from staging a full-scale autocratic takeover,
a Hungarian opposition parliamentarian has said.
Katalin Cseh,
a critic of Hungary’s strongman prime minister, Viktor Orbšn,
told a forum on authoritarianism that
the central European country’s experience held vital lessons for Trump’s opponents
in their attempts to resi…

On a conjectural supercongruence involving the dual sequence $s_n(x)$
Chen Wang, Sheng-Jie Wang
https://arxiv.org/abs/2506.18287 https://

On a conjectural supercongruence involving the dual sequence $s_n(x)$
In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $$ s_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}=\sum_{k=0}^n\binom{n}{k}(-1)^k\binom{x}{k}\binom{-1-x}{k} $$ and investigated its congruence properties. In particular, Z.-W. Sun conjectured that for any prime $p>3$ and $p$-adic integer $x\neq-1/2$ one has \begin{equation*} \sum_{n=0}^{p-1}s_n(x)^2\equiv (-1)^{\langle…

On prime coprime graphs of certain finite groups
Ravi Ranjan, Shubh N. Singh
https://arxiv.org/abs/2507.15993 https://arxiv.org/pdf/2507.15993

On prime coprime graphs of certain finite groups
The prime coprime graph $Θ(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate Hamiltonicity, clique number, and vertex degree of $Θ(G)$ for cyclic, dihedral, and dicyclic groups $G$. We establish that $Θ(G)$ admits a $(k,1)$-partition for cyclic, dihedral, and dicyclic groups $G$ of specified orders.

Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small
Adam Chapman, Ilan Levin, Marco Zaninelli
https://arxiv.org/abs/2508.07451 https://arx…

Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small
A division ring $D$ is Amitsur-Small if for every $n$ and every maximal left ideal $I$ in $D[x_1,\dots,x_n]$, $I \cap D[x_1,\dots,x_{n-1}]$ is maximal in $D[x_1,\dots,x_{n-1}]$. The goal of this note is to prove that cyclic division algebras of odd prime degree over their center are never Amitsur-Small.

Generalized Schatunowsky theorem in a weak arithmetic
Hala King, Victor Pambuccian
https://arxiv.org/abs/2506.08256 https://arxiv.org…

Generalized Schatunowsky theorem in a weak arithmetic
Schatunowsky's 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai. In its generalized form, it states the finiteness of the set of all positive numbers $n$, which, for a fixed prime $p$, have the property that all of $n$'s totatives that are not divisible by any prime less than or equal to $p$ are prime numbers. It is this generalized form that we show holds in a weak arithmetic

Single layer clathrane: A potential superconducting two-dimensional (2D) hydrogenated metal borocarbide
Xiaoyu Wang, Warren E. Pickett, Matthew N. Julian, Rohit P. Prasankumar, Eva Zurek
https://arxiv.org/abs/2508.14371

Single layer clathrane: A potential superconducting two-dimensional (2D) hydrogenated metal borocarbide
We propose a new family of two-dimensional (2D) metal-borocarbide clathrane superconductors, derived from three-dimensional (3D) MM$^\prime$B$_6$C$_6$ clathrates. First-principles calculations reveal that hydrogen passivation and surface metal decoration stabilize the M$_2$M$^\prime$B$_8$C$_8$H$_8$ monolayers. These 2D systems exhibit tunable superconductivity governed by hole concentration, structural anisotropy, and electron-phonon coupling. We find that in-plane anisotropy competes with supe…

A Note on Deaconescu's Conjecture
Sagar Mandal
https://arxiv.org/abs/2507.02930 https://arxiv.org/pdf/2507.02930

A Note on Deaconescu's Conjecture
Hasanalizade [1] studied Deaconescu's conjecture for positive composite integer $n$. A positive composite integer $n\geq4$ is said to be a Deaconescu number if $S_2(n)\mid ϕ(n)-1$. In this paper, we improve Hasanalizade's result by proving that a Deaconescu number $n$ must have at least seventeen distinct prime divisors, i.e., $ω(n)\geq 17$ and must be strictly larger than $5.86\cdot10^{22}$. Further, we prove that if any Deaconescu number $n$ has all prime divisors greater than or equal to $…

On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
Anja Meyer
https://arxiv.org/abs/2506.04720 https://ar…

On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$
Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does no…

New upper bounds on the order of mixed cages of girth 6
Gabriela Araujo-Pardo, Lydia Mirabel Mendoza-Cadena
https://arxiv.org/abs/2506.20003 https://

New upper bounds on the order of mixed cages of girth 6
A $[z,r;g]$-mixed cage is a mixed graph of minimum order such that each vertex has $z$ in-arcs, $z$ out-arcs, $r$ edges, and it has girth $g$, and the minimum order for $[z,r;g]$-mixed graphs is denoted by $n[z,r;g]$. In this paper, we present an infinite family of mixed graphs with girth $6$, that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. \textit{On Mixed Cages of girth 6}, arXiv:2401.14768v2. In particular, if $q$ is an even prime power w…

The modified diagonal cycles of Hypergeometric curves
Payman Eskandari, Yusuke Nemoto
https://arxiv.org/abs/2508.06008 https://arxiv.org/pdf/2508.06008

The modified diagonal cycles of Hypergeometric curves
For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,λ}\}_{λ\in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric functions and the Fermat curve of degree $N$. In this paper, we study the Gross-Kudla-Schoen modified diagonal 1-cycles of these curves. We prove that if $p \ge 3$ is a prime, then for every $λ$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,λ}$ is…

Darboux's Theorem in $p$-adic symplectic geometry
Luis Crespo, \'Alvaro Pelayo
https://arxiv.org/abs/2508.15443 https://arxiv.org/pdf/2508.15443

Darboux's Theorem in $p$-adic symplectic geometry
Let $p$ be a prime number. We derive an analog of Moser's Path Method for $p$-adic analytic manifolds and use it to prove a $p$-adic analog of Darboux's Theorem. Hence, in the $p$-adic category there are also no local symplectic invariants other than the dimension, as it happens in the real case: locally near any point, a $2n$-dimensional $p$-adic analytic symplectic manifold is symplectomorphic to $((\mathbb{Q}_p)^{2n},\sum_{i=1}^n\mathrm{d} x_1\wedge\mathrm{d} y_i)$. We also prove a $p$-adic …

New Constructions of Full Flag Codes Based on Partial Spreads
Xiang Han, Xinran Li, Gang Wang
https://arxiv.org/abs/2506.15127 https://

New Constructions of Full Flag Codes Based on Partial Spreads
Flag codes are a class of multishot network codes comprising sequences of nested subspaces (flags) within the vector space $\mathbb{F}_q^n$, where $q$ is a prime power. In this paper, we propose a family of constructions for full flag codes based on partial spreads. The distances of this family include maximum distance (optimum distance flag codes), second-maximum distance (quasi-optimum distance flag codes), as well as other feasible values. The structure of these flag codes resembles that of …

Additive congruences with factorials modulo a prime
Moubariz Z. Garaev, Julio C. Pardo
https://arxiv.org/abs/2508.12127 https://arxiv.org/pdf/2508.12127

Additive congruences with factorials modulo a prime
Let $p$ be a large prime number. We prove that any integer $λ$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv λ\pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the exponent $1350/1351$ of Garaev, Luca and Shparlinski (2005). Furthermore, we prove that any integer $λ$ can be represented in the form $$ m_1!n_1! +m_2!n_2!+m_3!n_3! +m_4!n_4!+m_5!n_5! \equiv λ\pmod p $$ with $\max\{m_1,n_1,\ldots,m_5,n_5\}\le p^{97/113 +o(1)}.$ This i…

On nut graphs with two vertex and three edge orbits
Ksenija Rozman, Primo\v{z} \v{S}parl
https://arxiv.org/abs/2508.17842 https://arxiv.org/pdf/2508.17842

On nut graphs with two vertex and three edge orbits
Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \g…

Generalized ODE reduction algorithm with bounded degree transformation
Shaoxuan Huang
https://arxiv.org/abs/2508.09754 https://arxiv.org/pdf/2508.09754

Generalized ODE reduction algorithm with bounded degree transformation
As a generalization of our previous result\cite{huang2025algorithm}, this paper aims to answer the following question: Given a 2-dimensional polynomial vector field $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, how to find a rational transformation $y \to \frac{A(x,y)}{B(x,y)}$ with bounded degree numerator, the inverse of which transforms this vector field into a simpler form $y^{\prime}=\sum_{i=0}^nf_i(x)y^i$. Such a structure, often known as the generalized Abel equation and has been studied in variou…

A note on the Greek letter elements of the homotopy of the $L_n$-local spheres
Ryo Kato, Katsumi Shimomura, Mao-no-suke Shimomura
https://arxiv.org/abs/2507.02155

A note on the Greek letter elements of the homotopy of the $L_n$-local spheres
Let $BP$ denote the Brown-Peterson spectrum at a prime $p$, whose homotopy groups are isomorphic to the polynomial algebra generated by elements $v_i$'s for $i\ge 1$. We consider the homotopy groups of the $v_n^{-1}BP$-localized sphere spectrum $L_nS^0$ with $n^2\le 2p-1$, and show that the groups contain the $n$-th Greek letter family. For the proof of this, we further show the existence of the $v_n^{-1}BP$-localized Smith-Toda spectrum $W_n$ for the case $n^2\le 2p-1$. If the Smith-Toda spect…

The Exceptional Set in Goldbach's Problem with two Chen Primes
Lasse Grimmelt, Joni Ter\"av\"ainen
https://arxiv.org/abs/2508.16400 https://a…

The Exceptional Set in Goldbach's Problem with two Chen Primes
We show that all natural numbers $n\equiv 4\pmod 6$ are the sum of two Chen primes (primes $p$ such that $p+2$ has at most two prime factors), apart from a power-saving set of exceptions. This improves on various previous results and is optimal, barring substantial progress on the twin prime or binary Goldbach conjectures. The proof is based on constructing a non-negative model for the Chen primes in a suitable approximate sense. To do this, we develop an efficient sieving strategy that makes…

Prediction of synthesis parameters for N, Si, Ge and Sn diamond vacancy centers using machine learning
Zhi Jiang, Marco Peres, Carlo Bradac, Gil Gon\c{c}alves
https://arxiv.org/abs/2507.02808

Prediction of synthesis parameters for N, Si, Ge and Sn diamond vacancy centers using machine learning
Diamond and diamond color centers have become prime hardware candidates for solid state-based technologies in quantum information and computing, optics, photonics and (bio)sensing. The synthesis of diamond materials with specific characteristics and the precise control of the hosted color centers is thus essential to meet the demands of advanced applications. Yet, challenges remain in improving the concentration, uniform distribution and quality of these centers. Here we perform a review and me…

On the Inversion Modulo a Power of an Integer
Guangwu Xu, Yunxiao Tian, Bingxin Yang
https://arxiv.org/abs/2506.02491 https://arxiv.o…

On the Inversion Modulo a Power of an Integer
Recently, Koc proposed a neat and efficient algorithm for computing $x = a^{-1} \pmod {p^k}$ for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right shift per step. In this paper, we design an algorithm that computes $x = a^{-1} \pmod {n^k}$ for any integer $n>1$. The algorithm has a motivation from the schoolbook multiplication and achieves both efficiency and generality. The greater flexibility of our algorithm …

The Kodaira dimension of even-dimensional ball quotients
Shuji Horinaga, Yota Maeda, Takuya Yamauchi
https://arxiv.org/abs/2507.22203 https://arxiv.org/pdf…

The Kodaira dimension of even-dimensional ball quotients
We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for prime discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\operatorname{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we …

The hyperbolic lattice counting problem in large dimensions
Christos Katsivelos
https://arxiv.org/abs/2506.17753 https://arxiv.org/pd…

The hyperbolic lattice counting problem in large dimensions
For $n\geq 3$ and $Γ$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact sets of $Γ\backslash\mathbb{H}^n$. Our upper bound depends on the quantum variance and the spectral exponential sums appearing in the study of the Prime geodesic theorem. We also prove $Ω$-results for the mean value and the second moment of the error term.

On $A$-Groups with the Same Index Set as a Nilpotent Group
Wei Zhou, Ilya Gorshkov
https://arxiv.org/abs/2506.15250 https://arxiv.org…

On $A$-Groups with the Same Index Set as a Nilpotent Group
Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = Π_{p\in π(G)}|G||_p$. $G$ is said to be an $A$-group if all its Sylow subgroups are abelian. We prove that if $G$ is an $A$-group such that $N(G)$ contains $|G||_p$ for every $p\in π(G)$ as well as $|G||$, then $G$ must be abelian. T…

Hybrid Character Sums From Vectorial Dual-Bent Functions and Asymptotically Optimal Complex Codebooks With Small Alphabet Sizes
Ziling Heng, Peng Wang, Chengju Li
https://arxiv.org/abs/2506.23198

Hybrid Character Sums From Vectorial Dual-Bent Functions and Asymptotically Optimal Complex Codebooks With Small Alphabet Sizes
Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let $\gf_{p^m}$ be the finite field with $p^m$ elements for a prime $p$ and a positive integer $m$. Let $V_n^{(p)}$ be an $n$-dimensional vector space over $\gf_p$ for a prime $p$. In this paper, we study the hybrid character sums of the form \begin{eqnarray*} \sum_{x \in V_n^{(p)}}ψ\left(F(x)\right)χ_1\left(a x\right), \end{eqnarray*} where $F$ is a function fro…

Preference-restricted parking functions
Jasper Bown, Peter Kagey, Alan Kappler, Michael E. Orrison, Jayden Thadani
https://arxiv.org/abs/2507.11701 https:/…

Preference-restricted parking functions
A parking function is a function $π:[n]\to [n]$ whose $i$th-smallest output is at most $i,$ corresponding to a parking procedure for $n$ cars on a one-way street. We refine this concept by introducing preference-restricted parking functions, which are parking functions with codomain restricted to some $S\subseteq[n]$. Particular choices of $S$ yield new combinatorial interpretations of previous results about variant parking procedures, and new results too. In particular we consider prime parki…

The complexity of deciding characteristic formulae modulo nested simulation
Luca Aceto, Antonis Achilleos, Aggeliki Chalki, Anna Ingolfsdottir
https://arxiv.org/abs/2505.22277

The complexity of deciding characteristic formulae modulo nested simulation
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. …

On the Representation of Integers as Sums of Limited Prime Powers
Julius Stricker
https://arxiv.org/abs/2508.01686 https://arxiv.org/pdf/2508.01686

On the Representation of Integers as Sums of Limited Prime Powers
We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five prime powers p^k, where p is a prime number and k is an integer greater than or equal to 2. This conjecture is supported by comprehensive computational evidence covering all integers up to 10^7(exhaustively) and specific large numbers up to 10^12 (via sampling),…

On Fermat's Last Theorem over the $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and other fields
Luis Dieulefait, Franco Golfieri Madriaga
https://arxiv.org/abs/2507.16883

On Fermat's Last Theorem over the $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and other fields
The main result of the present article is a proof of Fermat's Last Theorem for sufficiently large prime exponents $p$ with $p \equiv 2 \pmod{3}$ over certain number fields. A particular case of these fields are the maximal real subfields of the cyclotomic extensions $\mathbb{Q}(ζ_{3^n})$ for every $n$. Our strategy consists in combining the modular method with a generalization of an arithmetic result of Pomey to these fields.

Genus Stability of $\mathbb Z_p^\times$-Towers
Shiruo Wang
https://arxiv.org/abs/2507.14247 https://arxiv.org/pdf/2507.14247

Genus Stability of $\mathbb Z_p^\times$-Towers
Let $p$ be a prime. Consider a tower of smooth projective geometrically irreducible curves over $\mathbb F_p$, $\mathscr C:\cdots\rightarrow C_n\rightarrow\cdots\rightarrow C_1\rightarrow C_0=\mathbb P^1$ whose Galois group is isomorphic to $\mathbb Z_p^\times$. In this paper, we study genus growth of the tower $\mathscr C$ and determine all the $\mathbb Z_p^\times$-towers with genus be a quadratic equation of $p^{n}$ when $n$ is sufficiently large.

Rough numbers between consecutive primes
Ayla Gafni, Terence Tao
https://arxiv.org/abs/2508.06463 https://arxiv.org/pdf/2508.06463

Rough numbers between consecutive primes
Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap, confirming a prediction of Erdős. In fact the number $N(X)$ of exceptional gaps with $p_n \in [X,2X]$ is shown to be at most $O(X/\log^2 X)$. Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic $N(X) \sim c X / …

A new characterization of the weakly Laskerian (FSF) modules
Ali Fathi
https://arxiv.org/abs/2507.04573 https://arxiv.org/pdf/2507.04…

A new characterization of the weakly Laskerian (FSF) modules
Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module such that the set of associated prime ideals of the quotient module $M/L$ is finite for all submodules $L$ of $M$. In this paper, it is shown that there is a finitely generated submodule $N$ of $M$ such that the set of associated primes of $M/N$ and the support of $M/N$ are equal.

Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors
Bhawesh Mishra, Paolo Santonastaso
https://arxiv.org/abs/2507.11828 ht…

Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors
Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$ prime factors if and only if $S$ corresponds to a $k$-blocking set of $\PG(\mathbb{F}_{q}^{n})$. Here, $n$ is the number of distinct primes that divides the $q$-free parts of elements of $S$. Consequently, this geometric connection enables us to utilize method…

On the irreducibility of slice algebraic Sets
Anna Gori, Giulia Sarfatti, Fabio Vlacci
https://arxiv.org/abs/2507.21674 https://arxiv.org/pdf/2507.21674

On the irreducibility of slice algebraic Sets
In the present paper we investigate the correspondance between irreducible algebraic sets in $\mathbb{H}^n$ and quasi prime right ideals of slice regular polynomials in $(\mathbb{H}[q_1,\ldots,q_n], +, *)$. We provide algebraic conditions on right ideals of slice regular polynomials for the irreducibility of the corresponding algebraic sets and show that radical ideals associated with irreducible algebraic sets are quasi prime. Furthermore we establish that this correspondence is an equivalence…

This https://arxiv.org/abs/2409.07077 has been replaced.
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Submonoid Membership in n-dimensional lamplighter groups and S-unit equations
We show that Submonoid Membership is decidable in n-dimensional lamplighter groups $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}^n$ for any prime $p$ and integer $n$. More generally, we show decidability of Submonoid Membership in semidirect products of the form $\mathcal{Y} \rtimes \mathbb{Z}^n$, where $\mathcal{Y}$ is any finitely presented module over the Laurent polynomial ring $\mathbb{F}_p[X_1^{\pm}, \ldots, X_n^{\pm}]$. Combined with a result of Shafrir (2024), this gives the first example of…

On the Monogenity of Polynomials with Non-Squarefree Discriminants
Rupam Barman, Anuj Narode, Vinay Wagh
https://arxiv.org/abs/2506.16496 https://

On the Monogenity of Polynomials with Non-Squarefree Discriminants
In 2012, for any integer $n \ge 2$, Kedlaya constructed an infinite class of monic irreducible polynomials of degree $n$ with integer coefficients having squarefree discriminants. Such polynomials are necessarily monogenic. Further, by extending Kedlaya's approach, for any odd prime $q$, Jones constructed a class of degree $q$ polynomials with non-squarefree discriminants. In this article, using a similar method provided by Jones, we present another infinite class of monogenic polynomials of de…

The $3$-sparsity of $X^n-1$ over finite fields, II
Kaimin Cheng
https://arxiv.org/abs/2507.10779 https://arxiv.org/pdf/2507.10779

The $3$-sparsity of $X^n-1$ over finite fields, II
Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the polynomial $X^n - 1 \in \mathbb{F}_q[X]$ is termed $3$-sparse over $\mathbb{F}_q$ if all its irreducible factors in $\mathbb{F}_q[X]$ are either binomials or trinomials. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $X^n - 1$ is 3-sparse over $\mathbb{F}_q$ when $q = 2$ and $q = 4$. Recently, the author provided a complete characterization for odd $q$. T…

Conjectures about Primes and Cyclic Numbers
Joel E. Cohen
https://arxiv.org/abs/2508.08335 https://arxiv.org/pdf/2508.08335

Conjectures about Primes and Cyclic Numbers
A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime to $n$. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose ana…

Complete characterization of $2$-near perfect numbers with exactly 2 prime factors
Richard Fearon, Henry Foushee, Benjamin Porosoff, Alexander Skula, Joshua Zelinsky, Kyle Zhang
https://arxiv.org/abs/2508.06651

Complete characterization of $2$-near perfect numbers with exactly 2 prime factors
Let $σ(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $σ(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd $2$-near perfect numbers with exactly two prime factors, and that all even $2$-near perfect numbers (i.e. those of the form $2^kp^m$, where $p$ is an odd prime) belong to a specific family, provided that $m$ is at least 3. In combination with prior work, these results prod…

Non-orientable regular maps with negative prime-power Euler characteristic
Marston Conder, Nick Gill, Jozef \v{S}ir\'a\v{n}
https://arxiv.org/abs/2507.03667

Non-orientable regular maps with negative prime-power Euler characteristic
In this paper we provide a classification of all regular maps on surfaces of Euler characteristic $-r^d$ for some odd prime $r$ and integer $d\ge 1$. Such maps are necessarily non-orientable, and the cases where $d = 1$ or $2$ have been dealt with previously. This classification splits naturally into three parts, based on the nature of the automorphism group $G$ of the map, and particularly the structure of its quotient $G/O(G)$ where $O(G)$ is the largest normal subgroup of $G$ of odd order.…

On the sum of a prime and two Fibonacci numbers
Ji-Zhen Xu, Yong-Gao Chen
https://arxiv.org/abs/2506.03631 https://arxiv.org/pdf/2506…

On the sum of a prime and two Fibonacci numbers
Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities.

On polynomial progressions via transference
Daniel Altman, Mehtaab Sawhney
https://arxiv.org/abs/2506.13010 https://arxiv.org/pdf/250…

On polynomial progressions via transference
We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-Ω_…

The $3$-sparsity of $X^n-1$ over finite fields
Kaimin Cheng
https://arxiv.org/abs/2507.06655 https://arxiv.org/pdf/2507.06655

The $3$-sparsity of $X^n-1$ over finite fields
Let $q$ be a prime power and $\mathbb{F}_q$ the finite field with $q$ elements. For a positive integer $n$, the binomial $f_n(X):= X^n-1\in\mathbb{F}_q[X]$ is called $3$-sparse over $\mathbb{F}_q$ if each irreducible factor of $f_n(X)$ over $\mathbb{F}_q$ is either a binomial or a trinomial. In 2021, Oliveira and Reis characterized all positive integers $n$ for which $f_n(X)$ is $3$-sparse over $\mathbb{F}_q$ when $q = 2$ and $q = 4$, and posed the open problem of whether, for any given $q$, th…

Locally Associated Quadratic Orders
Grant Moles, Talha Khan
https://arxiv.org/abs/2508.08447 https://arxiv.org/pdf/2508.08447

Locally Associated Quadratic Orders
In 2025, the concept of an order in a number field being associated, ideal-preserving, or locally associated was introduced in order to tackle problems in factorization. In this paper, we explore locally associated orders in real quadratic number fields of the form $\mathbb{Q}[\sqrt{p}]$, with $p\in\mathbb{N}$ prime. In particular, we develop strategies and produce results which make determining when a given order in such a number field is (or is not) locally associated much easier. We also hig…

An effective analytic recurrence for prime numbers: from asymptotics to explicit bounds
Benoit Cloitre
https://arxiv.org/abs/2508.02690 https://arxiv.org/p…

An effective analytic recurrence for prime numbers: from asymptotics to explicit bounds
We present an explicit and effective recurrence formula for prime numbers, bridging arithmetic and analytic approaches. Building upon foundational work by Gandhi (1971), Golomb (1976), and Keller (2007), we establish the effective bound $s_n \le 2p_n$ for all $n \ge 1$ within the Golomb-Keller analytic recurrence. This transforms their asymptotic formula into an explicit recurrence using twice the n-th prime as the exponent: $$ p_{n+1} = \left\lceil \left( -1 + ζ(2p_n) \prod_{j=1}^{n} \left(1 …

Notes and computations on forbidden differences
Christian Dean, Haley Havard, Elizabeth Hawkins, Patch Heard, Andrew Lott, Alex Rice
https://arxiv.org/abs/2508.03650 https://

Notes and computations on forbidden differences
We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that differ by an element of $X$? The set of forbidden differences, $X$, is called \textit{intersective} if $D(X,N)=o(N)$, with the most well-studied examples being $X=S=\{n^2: n\in \mathbb{N}\}$ and $X=\mathcal{P}-1=\{p-1: p\text{ prime}\}$. In addition to some ne…

Large values of character sums with multiplicative coefficients
Zikang Dong, Zhonghua Li, Yutong Song, Shengbo Zhao
https://arxiv.org/abs/2508.09750 https://

Large values of character sums with multiplicative coefficients
In this article, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)χ(n)$. We prove an Omega result in the region $\exp((\log q)^{\frac12+\varepsilon})\le N\le\sqrt q$, where $q$ is the prime modulus.

The essential $2$-dimension of the linear groups
Hannah Knight
https://arxiv.org/abs/2508.02859 https://arxiv.org/pdf/2508.02859

The essential $2$-dimension of the linear groups
In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear groups and quotients of it (such as the projective special linear groups) in the case case $q \equiv 1 \mod 4$, $s = v_2(q-1)$, and $Γ= \text{Gal}(k(ζ_{2^s})/k)$ is trivial.

Divisibility of Griesmer Codes
Haihua Deng, Hexiang Huang, Qing Xiang
https://arxiv.org/abs/2506.07846 https://arxiv.org/pdf/2506.078…

Divisibility of Griesmer Codes
In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if $p^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. In this paper, we show that if $q^e|d$, then $C$ has a basis consisting of $k$ codewords such that the first $\min\left\{e+1,k\right\}$ of them span a Griesmer subcode with constant weight $d$…

This https://arxiv.org/abs/2405.05727 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On Chen's theorem, Goldbach's conjecture and almost prime twins II
Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2}$, which is rather near to the asymptotic constant $2$ in Hardy--Littlewood conjecture for Goldbach's conjecture. We also get similar results on twin prime problem and additive representations of integers. The proof comb…

On the sum-of-squares function
Vitalii V. Iudelevich
https://arxiv.org/abs/2508.02701 https://arxiv.org/pdf/2508.02701

On the sum-of-squares function
In this paper, we derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\dfrac{r(n)}{r(n+1)} = {x}{(\ln x)^{-3/4}}(c+o(1)),\ \ x \to +\infty,$$ where $r(n)$ is the number of representations of $n$ as a sum of two squares, $c$ is a positive constant, and the prime indicates summation over those $n$ for which $r(n+1)\neq 0$.

On one of Erd\H{o}s' Problems -- An Efficient Search for Benelux Pairs
Christian Hercher
https://arxiv.org/abs/2506.01099 https://

On one of Erdős' Problems -- An Efficient Search for Benelux Pairs
Erdős asked for positive integers $m

The Hardy--Ramanujan inequality for sifted sets and its applications
Steve Fan
https://arxiv.org/abs/2508.06005 https://arxiv.org/pdf/2508.06005

The Hardy--Ramanujan inequality for sifted sets and its applications
The well-known Hardy--Ramanujan inequality states that if $ω(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$, \[\#\{n\le x\colonω(n)=k\}\ll\frac{x(\log\log x+C)^{k-1}}{(k-1)!\log x}.\] A myriad of generalizations and variations of this inequality have been discovered. In this paper, we establish a weighted version of this inequality for sifted sets, which generalizes an ea…

Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface
Sudhir Pujahari, Punya Plaban Satpathy
https://arxiv.org/abs/2506.07474

Erdős-Kac type theorem for the number of scattering geodesics on modular surface
In 1917, Hardy and Ramanujan showed that if $ω(n)$ is the number of distinct prime factors of a randomly chosen positive integer $n,$ then the normal order of $ω(n)$ is $\log \log \, n.$ This led Erdős and Kac to prove their celebrated result showing a Gaussian behaviour for $ω(n).$ In this article we prove an Erdős-Kac kind result for the number of scattering geodesics on the modular surface with a common sojourn time.

On Zel'manov's global nilpotence theorem for Engel Lie algebras
Michael Vaughan-Lee
https://arxiv.org/abs/2507.21677 https://arxiv.org/pdf/2507.216…

On Zel'manov's global nilpotence theorem for Engel Lie algebras
I give a proof of Zel'manov's theorem that if $L$ is an $n$-Engel Lie algebra over a field $F$ of characteristic zero then $L$ is (globally) nilpotent. This is a very important result which extends Kostrikin's theorem that $L$ is locally nilpotent if the characteristic of $F$ is zero or some prime $p>n$. Zel'manov's proof contains some striking original ideas, and I wrote this note in an effort to understand his arguments. I hope that my efforts will be of use to other mathematicians in underst…

On a conjecture about prime-detecting quasimodular forms
Ben Kane, Krishnarjun Krishnamoorthy, Yuk-Kam Lau
https://arxiv.org/abs/2507.00147 https://…

On a conjecture about prime-detecting quasimodular forms
Motivated by weighted partition of $n$ that vanish if and only if $n$ is a prime, Craig, van Ittersum, and Ono conjecture a classification of quasimodular forms which detect primes in the sense that the $n$-th Fourier coefficient vanishes if and only if $n$ is a prime. In this paper, we prove this conjecture by showing that Fourier coefficients of quasimodular cusp forms exhibit infinitely many sign changes.

Dyadically resolving trinomials for fast modular arithmetic
Robert Dougherty-Bliss, Mits Kobayashi, Natalya Ter-Saakov, Eugene Zima
https://arxiv.org/abs/2508.11043 https://

Dyadically resolving trinomials for fast modular arithmetic
Residue number systems based on pairwise relatively prime moduli are a powerful tool for accelerating integer computations via the Chinese Remainder Theorem. We study a structured family of moduli of the form $2^n - 2^k + 1$, originally proposed for their efficient arithmetic and bit-level properties. These trinomial moduli support fast modular operations and exhibit scalable modular inverses. We investigate the problem of constructing large sets of pairwise relatively prime trinomial moduli …

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums
Hai-Liang Wu, Li-Yuan Wang, Hao Pan
https://arxiv.org/abs/2506.14316 https://

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums
As a complement to our previous article, in this paper we determine the explicit values of $\det\left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1}$ and $\det \left[J_p(χ^{ki},χ^{kj})\right]_{0\le i,j\le n-1}$, where $p$ is a prime, $1\le k

Moments of the shifted prime divisor function
Mikhail R. Gabdullin
https://arxiv.org/abs/2505.24050 https://arxiv.org/pdf/2505.24050

Moments of the shifted prime divisor function
Let $ω^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}ω^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.

Choosing iteration maps for the parallel Pollard rho method
Finn Rudolph
https://arxiv.org/abs/2506.12844 https://arxiv.org/pdf/2506.…

Choosing iteration maps for the parallel Pollard rho method
Pollard's rho method finds a prime factor $p$ of an integer $N$ by searching for a collision in a map of the form $x \mapsto x^{2k} + c$ modulo $N$. This search can be parallelized to multiple machines, which may use distinct parameters $k$ and $c$. In this paper, we give an asymptotic estimate for the expected running time of the parallel rho method depending on the choice of $k$ for each machine. We also prove that $k = 1$ is the best choice for one machine, if nothing about $p$ is known in a…

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation
Sudarshan Kumaresan, Shipra Kumari, Neha Mishra
https://arxiv.org/abs/2508.02821 https:…

Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation
This paper introduces a novel class of prime-generating quadratic polynomials defined by $f_{Z,k,H}(n) = n^2 - (2Zk - 1)n + \frac{(2Zk - 1)^2 + H}{4}$, where $Zk \in \mathbb{Z}_{\geq 0}$ and $H$ belongs to the set of Heegner numbers. This form is closely related to the Euler-Rabinowitsch polynomials through specific substitutions. The structure enables algebraic tuning for prime-rich outputs and provides deeper insight into the impact of Heegner numbers on prime distribution. Using tools such a…

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

On the number of exceptional intervals to the prime number theorem in short intervals
Ayla Gafni, Terence Tao
https://arxiv.org/abs/2505.24017 https://

On the number of exceptional intervals to the prime number theorem in short intervals
For a fixed exponent $0 < θ\leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^θ} Λ(n) \sim x^θ$ as $x \to \infty$. From the recent zero density estimates of Guth and Maynard, this result is known for all $x$ for $θ> \frac{17}{30}$ and for almost all $x$ for $θ> \frac{2}{15}$. Prior to this work, Bazzanella and Perelli obtained some upper bounds on the size of the exceptional set where the prime number theorem in short intervals fails. We…

This https://arxiv.org/abs/2412.08600 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Chebotarev's theorem for roots of unity of square free order
Let $p$ be a prime number and $ζ_p$ a primitive $p$-th root of unity. Chebotarev's theorem states that every square submatrix of the $p \times p$ matrix $(ζ_p^{ij})_{i,j=0}^{p-1}$ is non-singular. In this paper we prove the same for principal submatrices of $(ζ_n^{ij})_{i,j=0}^{n-1}$, when $n=pr$ is the product of two distinct primes, and $p$ is a large enough prime that has order $r-1$ in $\mathbf{Z}_r^*$. As an application, an uncertainty principle for cyclic groups of order $n$ is establi…

This https://arxiv.org/abs/2212.00636 has been replaced.
link: https://scholar.google.com/scholar?q=a

Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors
Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \] For a positive integer $d$, let $B_{d}$ be the set of positive integers $M$ such that the number of prime divisors of $M$ is $d$. In this paper, we prove that for each positive integer $d$, the density of the set of positive integers $M$ for which Newman's Conj…

This https://arxiv.org/abs/2406.07575 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the largest prime factor of quadratic polynomials
Let $x$ denote a sufficiently large integer. We show that the recent result of Grimmelt and Merikoski actually yields the largest prime factor of $n^2 +1$ is greater than $x^{1.317}$ infinitely often. As an application, we give a new upper bound for the number of integers $n \leqslant x$ which $n^2 +1$ has a primitive divisor.

This https://arxiv.org/abs/2505.00493 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the greatest prime factor and uniform equidistribution of quadratic polynomials
We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq n^{1+o(1)}$ under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. We also prove a variant of the divisor problem for $ax^2+by^3$, which was used by the seco…

Wieferich primes in number fields and the conjectures of Ankeny--Artin--Chowla and Mordell
Nic Fellini, M. Ram Murty
https://arxiv.org/abs/2508.08472 https://

Wieferich primes in number fields and the conjectures of Ankeny--Artin--Chowla and Mordell
For a prime $p\equiv 1 \,(\bmod{4})$, let \[ \varepsilon = \frac{1}{2}\left( t + u\sqrt{p}\right) \] be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether $p$ can divide $u$. They suggested that this can never happen and this has since been called the Ankeny--Artin--Chowla (AAC) conjecture. We show that if $\mathfrak{p}$ is the prime above $p$ in $\mathbb{Q}(\sqrt{p})$, then the AAC conjecture is false if and only…

A new zero-density estimate for $\zeta(s)$ and the error term in the Prime Number Theorem
Chiara Bellotti
https://arxiv.org/abs/2508.02041 https://arxiv.or…

A new zero-density estimate for $ζ(s)$ and the error term in the Prime Number Theorem
We will provide a new type of zero-density estimate for $ζ(s)$ when $σ$ is sufficiently close to $1$. In particular, we will show that $N(σ,T)$ can be bounded by an absolute constant when $σ$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ ψ(x)-x \ll x\exp \left\{-(1-\varepsilon) ω(x)\right\},\qquad ω(x):=\min _{t \geq 1}\{ν(t) \log x+\log t\}, $$ where $ν(t…

A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications
Sourabhashis Das, Wentang Kuo, Yu-Ru Liu
https://arxiv.org/abs/2506.03215

A subset generalization of the Erdős-Kac theorem over number fields with applications
Let $ω(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erdős and Kac established that $ω(n)$ obeys the Gaussian distribution over natural numbers. In 2004, the third author generalized their theorem to all abelian monoids. In this work, we extend the work of the third author to any subset of the set of ideals of a number field satisfying some additional conditions. Finally, we apply this theorem to prove the Erdős-Kac theorem over $h$-free and over $h$-full…

Generalizations of Erd\H{o}s-Kac theorem with applications
Sourabhashis Das, Wentang Kuo, Yu-Ru Liu
https://arxiv.org/abs/2506.02432 https://

Generalizations of Erdős-Kac theorem with applications
Let $ω(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erdős and Kac established that $ω(n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this paper, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of $h$-free and $h$-full elements. We study generalizations of several arithmetic func…

This https://arxiv.org/abs/2504.00804 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The prime number theorem over integers of power-free polynomial values
Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is $k$-free infinitely often, if $f$ has no fixed $k$-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also hol…

On quantum ergodicity for higher dimensional cat maps modulo prime powers
Subham Bhakta, Igor E. Shparlinski
https://arxiv.org/abs/2507.00325 https://

On quantum ergodicity for higher dimensional cat maps modulo prime powers
A discrete model of quantum ergodicity of linear maps generated by symplectic matrices $A \in \mathrm{Sp}(2d,\mathbb{Z})$ modulo an integer $N\ge 1$, has been studied for $d=1$ and almost all $N$ by P. Kurlberg and Z. Rudnick (2001). Their result has been strengthened by J. Bourgain (2005) and subsequently by A. Ostafe, I. E. Shparlinski, and J. F. Voloch (2023). For arbitrary $d$ this has been studied by P. Kurlberg, A. Ostafe, Z. Rudnick and I. E. Shparlinski (2024). The corresponding equidis…

Counting linear congruence systems with a fixed number of solutions
Marcus Nilsson
https://arxiv.org/abs/2507.04688 https://arxiv.org…

Counting linear congruence systems with a fixed number of solutions
For a prime p and a positive integer s consider a homogeneous linear system over the ring Z_p^s (the ring of integers modulo p^s) described by an n x m-matrix. The possible number of solutions to such a system is p^j, where j=0,1,..., sm. We study the problem of how many n x m -matrices over Z_p^s there are given that we have exactly p^j homogeneous solutions. For the case s=1 (when Z_p^s is a field) George von Landsberg proved a general formula in 1893. However, there seems to be few published…

On Hypothesis H of Rudnick and Sarnak
Yujiao Jiang
https://arxiv.org/abs/2507.20653 https://arxiv.org/pdf/2507.20653

On Hypothesis H of Rudnick and Sarnak
We prove Hypothesis H in full generality for ${\rm GL}_n$ over any number field. This result is a consequence of our stronger effective bound on Euler products involving Rankin--Selberg coefficients at prime ideal powers. The proof rests on a new analytic method, which employs a power sieve over number fields and an iterative argument to bypass the functoriality barrier that had restricted prior results to $n\leq 4$. As applications, we unconditionally establish the GUE statistics for automorph…

This https://arxiv.org/abs/2504.21333 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the distribution of $αp^2$ modulo one over primes of the form $[n^c]$
Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote the distance from $x$ to the nearest integer. In this paper we show that whenever $α$ is irrational and $β$ is real then for any fixed $\frac{13}{14}<γ<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp^2+β\|< p^{\frac{13-14γ}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1/γ}]$.

This https://arxiv.org/abs/2412.01612 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Iwasawa theory for weighted graphs
Let $p$ be a prime number and let $d$ be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa's class number formula and that of Kida's formula for compatible systems of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas …