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An inverse theorem on sets with rich additive structure modulo primes
Ernie Croot, Junzhe Mao, Chi Hoi Yip
https://arxiv.org/abs/2510.08862 https://arxiv.o…

An inverse theorem on sets with rich additive structure modulo primes
In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$, where $(\log N)^C < y \leq N$. We also provide several constructions showing that our results cannot be improved. As an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has stro…

from my link log —
The Euclid-Mullin sequence of prime numbers.
https://blogs.scientificamerican.com/roots-of-unity/a-curious-sequence-of-prime-numbers/
saved 2019-05-26

A Curious Sequence of Prime Numbers
Euclid’s proof of the infinitude of primes opens the door to some interesting questions

Criteria for the presence of the maximal ideal in the set of associated primes
Mehrdad Nasernejad, Jonathan Toledo
https://arxiv.org/abs/2510.16566 https://

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

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Crosslisted article(s) found for math.CO. https://arxiv.org/list/math.CO/new
[1/1]:
- An inverse theorem on sets with rich additive structure modulo primes
Ernie Croot, Junzhe Mao, Chi Hoi Yip

A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes
Shao-Yuan Huang, Hsiu-Yu Wu
https://arxiv.org/abs/2510.10418 https://arxiv.org/p…

A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes
Let p1, p2,..., pn be distinct prime numbers, and let Nn be their product. We prove that, for any positive integer L that is divisible by the least common multiple of p1 minus one, p2 minus one, and so on, and for integers a1, a2,..., an satisfying that each ai is relatively prime to Nn and shares the same prime factor pi, a certain congruence relation holds among their Lth powers.

Bayes Meets Riemann Again: Large Prime Discovery and Re-emergence of the Bone of Contention
Durba Bhattacharya, Sucharita Roy, Sourabh Bhattacharya
https://arxiv.org/abs/2510.09651

Bayes Meets Riemann Again: Large Prime Discovery and Re-emergence of the Bone of Contention
Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable codes. Large prime-hunting, often brute-force in nature, is conceptually linked to the Riemann Hypothesis and the prime number theorem, which portrays prime distribution as essentially random. This motivates a statistical perspective, with Bayesian methodology pro…

Joint distributions of error terms for primes in arithmetic progressions modulo 11
K\"ubra Benl\.i, Greg Martin, Paul P\'eringuey
https://arxiv.org/abs/2510.05427 https…

Joint distributions of error terms for primes in arithmetic progressions modulo 11
We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $ψ(x;q,a)$ and $ψ(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for calculating the numerical values of such densities with rigorously bounded errors. We apply these formulas to the pairwise races in the case $q=11$, determining which pairs of residues $a$ and $b$ are more or less correlated in this way. The outcomes when $q=11$ p…

Deformation Theory of Galois Representations and the Taylor--Wiles Method
Ehsan Shahoseini
https://arxiv.org/abs/2510.12202 https://arxiv.org/pdf/2510.1220…

Deformation Theory of Galois Representations and the Taylor--Wiles Method
In this chapter, we want to have an overview of the Taylor--Wiles patching method. For this purpose, at the first, we recall Mazur's theory of deforming Galois representations and study both local and global deformation problems. Then, we go through the subject of Taylor-Wiles primes and examine the role that they play on the Galois side and the modular (automorphic) side. At the end, we arrive at the Taylor-Wiles patching method and use it to prove $R=\mathbb{T}$ in both minimal and non-minima…

On divisibility of Hecke eigenvalues of Ikeda lifts
Sanoli Gun, Sunil Naik
https://arxiv.org/abs/2510.07056 https://arxiv.org/pdf/2510.07056

On divisibility of Hecke eigenvalues of Ikeda lifts
In this article, we estimate the density of the set of primes $p$ such that the $p$-th Hecke eigenvalue of an Ikeda lift is divisible by a fixed positive integer. One of the main ingredients involves the study of abelian subfields of fixed fields of the kernel of Galois representations attached to elliptic Hecke eigenforms. Further, we study the distribution of Fourier coefficients of elliptic Hecke eigenforms in arithmetic progressions.

Verification of GIlbraith's conjecture up to 10^14
Simon Plouffe
https://arxiv.org/abs/2510.06688 https://arxiv.org/pdf/2510.06688

Verification of GIlbraith's conjecture up to 10^14
A calculation was performed to verify Proth-Gilbraith's conjecture for all prime numbers up to 10^14. The previous calculation was performed by Andrew Odlyzko in 1993 up to 10^13. This involves calculating the differences between consecutive primes in absolute value and starting over. The conjecture states that all lines except the first begin with 1. To prove it, it suffices to find a line beginning with 1 and followed only by 0 and 2.

Ratios of two exponents of van der Laan-Padovan numbers
Tomohiro Yamada
https://arxiv.org/abs/2510.06192 https://arxiv.org/pdf/2510.06192

Ratios of two exponents of van der Laan-Padovan numbers
The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0, 1, \ldots$. We determine all pairs $(P_m, P_n)$ satisfying $P_m^b/P_n^a=2^{g_1} 3^{g_2} 5^{g_3} 7^{g_4}$ for some integers $g_1, g_2, g_3, g_4$, $a$, and $b$. More generally, for a linear recurrence sequence $u_n$ satisfying the dominant root condition and a given set of primes $p_1, \ldots, p_k$, there exist only finitely many pairs $(u_m, u_n)$ satisfying $u_m^b/u_n…

Sums of the floor function related to class numbers of imaginary quadratic fields
Marc Chamberland, Karl Dilcher
https://arxiv.org/abs/2510.04387 https://a…

Sums of the floor function related to class numbers of imaginary quadratic fields
A curious identity of Bunyakovsky (1882), made more widely known by Pólya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, obtaining an identity in terms of the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. C…

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…

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Alexander J. Barrios, Chung Pang Mok
https://arxiv.org/abs/2510.00926 htt…

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N=N_+N_-$, where $N_+$ and $N_-$ are coprime and $N_-$ is squarefree. Let $D$ be a positive fundamental discriminant satisfying the modified Heegner hypothesis with respect to $(N_+,N_-)$: primes dividing $N_+$ (resp. $N_-$) split (resp. are inert) in $\mathbb{Q}(\sqrt{D})$; we denote by $E^D/\mathbb{Q}$ the quadratic twist of $E/\mathbb{Q}$ by $D$. In the first half of the paper we consider the situation where $N_-$ is a squarefree produc…

An Effective Version of the $p$-Curvature Conjecture for Order One Differential Equations
Florian F\"urnsinn, Lucas Pannier
https://arxiv.org/abs/2510.00892 https://…

An Effective Version of the $p$-Curvature Conjecture for Order One Differential Equations
We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Padé approximation. In conjunction with Honda's proof of the $p$-curvature conjecture for order one equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck $p$-curvature conjecture for order one equations. More precisely, we bound the number of primes for which the $p$-curvatu…