Ramanujan - Fourier Series and the Density of Sophie Germain Primes

索菲·热尔曼，非凡的数学家：比小说更离奇的故事

Sophie Germain, mathématicienne extraordinaire: A story stranger than fiction

本文回顾索菲·热尔曼在数论领域的贡献，揭示她对费马大定理的深入研究及与拉格朗日、勒让德和高斯的互动，重新评估其在数论中的重要性。

Hopf-Galois Structures on Non-Normal Extensions of Degree Related to Sophie Germain Primes

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree $n$. We show that $G$ has derived length at most $4$, but that many permutation groups of squarefree degree and of derived length $2$ cannot occur. We then investigate in detail the case where $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, a…

关于索菲热尔曼循环数的科恩次可加性猜想的一个反例

A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers

研究针对科恩关于索菲热尔曼循环数的次可加性猜想，通过给出 \(m = 31\)，\(n = 3928\) 时 \(C_\sigma(3959) > C_\sigma(31) + C_\sigma(3928)\) 的反例进行反驳，论证简短且经Lean 4内核验证。

$Kp^n+1$数的质数测试及安全质数和索菲·热尔曼质数的推广

A primality test for $Kp^n+1$ numbers and a generalization of Safe Primes and Sophie Germain Primes

本文对形式为$Kp^n+1$的整数推广了普罗特定理，提出了一种仅需一次模幂运算的质数测试方法，无需计算任何GCD。同时提供了两种测试以提高验证$Kp^n+1$数质数的可能性。作为这些测试的推论，我们给出了三个整数家族$N$，其质数性只能通过证明$a^{N-1} \equiv 1 \pmod N$（费马测试）来确认。还推广了安全质数，定义为$a$-安全质数，因其$N-1$具有与安全质数相同的大型质因数。我们讨论了这些数的分布问题，并提出了关于相关数$a$-索菲·热尔曼质数分布的猜想。

通过 Rogue Primes 研究 Cunningham 界

A Study of Cunningham Bounds through Rogue Primes

本文研究 Cunningham 链长度的界限，通过提出 rogueness 框架分析素数链行为，显著改进了现有界限，推进了 Sophie Germain 素数研究。

The algorithm for the $2d$ different primes and Hardy-Littlewood conjecture

We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers between the $2d$ different primes. We may conclude that there exist infinitely many $2d$ different primes including the twin primes in case of $d=1$ because we can give the lower bounds of the existence density for the $2d$ different primes in this algorithm. We also discuss the Hardy-Littlewood conjecture and the Sophie Germain primes.