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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://

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.

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

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…

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…