WonTae Hwang, Kyunghwan Song
For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the asymptotic behavior of prime numbers that are related to our construction.
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We disprove a conjecture of Schott that no Brazilian primes are Sophie Germain primes. We enumerate all counterexamples up to $10^{46}$.
Let $n\ge 5$ be an odd integer. It is shown that $\{1^{σ(1)},\ldots,n^{σ(n)}\}$ is a complete residue system modulo $n$ for some permutation $σ$ of $\{1,\ldots,n\}$ if and only if $\frac{1}{2}(n-1)$ is a Sophie Germain prime. Partial results are obtained also for the case $n$ even.
A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt function. The proof remains heuristic as interchange of certain limits has not been justified. Experimental evidence using computer calculations is provided for the plausibility of the result.
Sophie Germain Primes and the Totient of Fibonacci Numbers
研究了在模Pisano周期π(q)下，使得q整除φ(F_m)对所有m≡r mod π(q)的残差类集合S(q)，证明若q为赛普蒂默· Germain素数且z(2q+1)整除π(q)，则S(q)是非空的等差数列，且对于q>5，其基数为奇数且q≡8 mod 15。
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。
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…
A characterization of Sophie Germain primes - II
本文刻画了偶数 $n$ 使得 $\{1^{\sigma(1)},\ldots,n^{\sigma(n)}\}$ 为模 $n$ 完全剩余系的条件，等价于 $n/2$ 为素数且减一无平方因子，补充了奇数情形的已知结果。
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。
On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$
本文研究含索菲·热尔曼素数的一类丢番图方程，求解了p=2时的方程及奇素数下四类方程的所有非负整数解，验证了对应素数的存在性并提出相关无穷性猜想。
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。
The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${\bf Q}$, we give an easily testable condition which allows occasionally…
For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes.
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。
Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs
本文提出通过算术闭式生成和计数梅森素数、费马素数、孪生素数对及索菲·热尔曼素数的方法，结合威尔逊定理和特定测试技术。
Let $p$ be an odd prime number. Using modular arguments, we give an easy testable condition which allows often to prove Fermat's Last Theorem over the quadratic field ${\bf Q}(\sqrt{5})$ for the exponent $p$. It is related to the Wendt's resultant of the polynomials $X^n-1$ and $(X+1)^n-1$. We deduce Fermat's Last Theorem over this field in case one has $5\leq p<10^7$, and we obtain analogous results on Sophie Germain type criteria.
Toda primes
该研究提出Toda素数的定义，猜想每个正整数至少有两个Toda素数，给出至少存在一个的部分证明，并探讨其与Bernoulli数分母的关联及对Sophie Germain素数的推广。
Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.
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.
In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.
Indivisibility of ray class groups of real quadratic fields
研究实二次域中模ℓ的射线类群的负部分阶h_ℓ^-(Δ)被素数p整除的条件，利用半整数权模形式理论证明了下界，并应用于BSD猜想。
The Hurwitz problem for abelian differentials
研究固定亏格g≥2的阿贝尔微分平移群最大阶t(g)的取值，证明当不存在正则origamis时t(g)=2(g-1)，并给出存在时t(g)的精确形式，分类相关分层并构造无限族亏格。
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