论文标题
最小的条件等于Fermat和Mersenne Primes的限制
Minimality conditions equivalent to the finitude of Fermat and Mersenne primes
论文作者
论文摘要
无论存在无限的Fermat Prime还是无限的许多复合费马特数字，它仍然是开放的。关于Mersenne数字的相同问题也未解决。从[9]中扩展了一些结果，我们从某些矩阵组的拓扑最小化方面表征了Fermat Primes和Mersenne Primes。这是通过显示出$ \ bbb {f} $是特征$ \ neq 2的本地领域的子场来完成的$ \ operatorName {sl}（n，\ bbb {f}）$ is。我们提供了$ \ operatorName {sl}（n，\ bbb {f}）$和$ \ permatatorName {st^+}（n，\ bbb {f}）的最小值（和总最小值）的标准。 令$ \ MATHCALF_π$和$ \ MATHCAL F_C $分别为Fermat Primes和一组复合Fermat编号。作为我们的主要结果，我们证明以下条件等于$ \ Mathcal {a} \ in \ {\ Mathcalf_π，\ Mathcal f_c \}：$ $ \ bullet \ \ Mathcal {a} $是有限的； $ \ bullet \ \ prod_ {f_n \ in \ Mathcal {a}} \ propatorAtorname {sl}（f_n-1，\ bbb {q}（i））$很少，其中$ \ bbb {q}（q}（q}（i）$是高斯$是高斯的理性字段; $ \ bullet \ \ prod_ {f_n \ in \ Mathcal {a}}} \ operatatorName {st^+}（f_n-1，\ bbb {q}（i））$是最小的。 同样，分别用$ \ MATHCALM_π$和$ \ MATHCAL M_C $分别用Mersenne Primes和一组复合Mersenne数字来表示，让$ \ Mathcal {B} \ in \ in \ {\ MathcalM_π，\ Mathcalm_π，\ Mathcal M_C \ \ $等于以下情况。 $ \ bullet \ \ Mathcal b $是有限的； $ \ bullet \ \ prod_ {m_p \ in \ mathcal {b}}} \ operatatorName {sl}（m_p+1，\ bbb {q}（i））$是最小的; $ \ bullet \ \ prod_ {m_p \ in \ mathcal {b}}} \ pereratatorName {st^+}（m_p+1，\ bbb {q}（i））$是最小的。
It is still open whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning the Mersenne numbers is also unsolved. Extending some results from [9], we characterizethe the Fermat primes and the Mersenne primes in terms of topological minimality of some matrix groups. This is done by showing, among other things, that if $\Bbb{F}$ is a subfield of a local field of characteristic $\neq 2,$ then the special upper triangular group $\operatorname{ST^+}(n,\Bbb{F})$ is minimal precisely when the special linear group $\operatorname{SL}(n,\Bbb{F})$ is. We provide criteria for the minimality (and total minimality) of $\operatorname{SL}(n,\Bbb{F})$ and $\operatorname{ST^+}(n,\Bbb{F}),$ where $\Bbb{F}$ is a subfield of $\Bbb{C}.$ Let $\mathcal F_π$ and $\mathcal F_c $ be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for $\mathcal{A}\in \{\mathcal F_π, \mathcal F_c\}:$ $\bullet \ \mathcal{A}$ is finite; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{SL}(F_n-1, \Bbb{Q}(i))$ is minimal, where $\Bbb{Q}(i)$ is the Gaussian rational field; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{ST^+}(F_n-1, \Bbb{Q}(i))$ is minimal. Similarly, denote by $\mathcal M_π$ and $\mathcal M_c $ the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let $\mathcal{B}\in\{ \mathcal M_π, \mathcal M_c\}.$ Then the following conditions are equivalent: $\bullet \ \mathcal B$ is finite; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{SL}(M_p+1, \Bbb{Q}(i))$ is minimal; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{ST^+}(M_p+1, \Bbb{Q}(i))$ is minimal.
