一类有关虚二次域理想类群的指数丢番图方程  

AN EXPONENTIAL DIOPHANTINE EQUATION CONCERNING IDEAL CLASS GROUPS OF IMAGINARY QUADRATIC FIELDS

作　　者：乐茂华[1] 

机构地区：[1]湛江师范学院

出　　处：《数学杂志》1997年第1期69-71,共3页Journal of Mathematics

基　　金：国家自然科学基金;广东省自然科学基金

摘　　要：设Ｄ１、Ｄ２是适合Ｄ１≥１，Ｄ２＞１，ｇｃｄ（Ｄ１，Ｄ２）＝１的无平方因子正整数，ＩＫ是虚二次域Ｋ＝Ｑ（－Ｄ１Ｄ２）的理想类群。本文证明了：如果存在正整数ａ、ｂ、ｋ以及奇素数ｐ，可使Ｄ１ａ２＋Ｄ２ｂ２＝ｋｐ，ｇｃｄ（ａ，ｂ）＝１，ｂ∈Ｎｐ，２ｋ，其中Ｎｐ是所有不含２ｔｐ±１之形素因数的正整数的集合，则当ｐ＞５且ｍａｘ（Ｄ１，Ｄ２）≥１０１０１９７时。Let D 1,D 2 be positive integers such that D 1≥1,D 2>1, gcd (D 1,D 2)=1 and D 1D 2 is square free, and let I k denote the ideal class group of the imaginary quadratic field K=Q(-D 1D 2). Let p be an odd prime ,and let N p be the set of positive integers n such that n has no prime factor q with q≡±1 (mod p). In this paper we prove that if there exist positive integers a,b,k such that D 1a 2+D 2b 2=k p, gcd (a,b)=1, b∈N p,p>5,2k and  max (D 1,D 2)≥10 10 197 , then I k has a cyclic subgroup of order p.

关 键 词：虚二次域 理想类群 循环子群 丢番图方程 

分 类 号：O156.7[理学—数学]