On the elliptic curve y^2=x^3-2r Dx and factoring integers

查看全文 作　　者：LI [1]XiuMei;ZENG [1]JinXiang 机构地区：[1]Department of Mathematical Sciences,Tsinghua University 出　　处：《Science China Mathematics》索引2014年第57卷第4期,共10页 基　　金：supported by National Natural Science Foundation of China (Grant No. 11271212) 摘　　要：Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q. 关 键 词：因子分解 椭圆曲线 整数 黎曼假设 奇素数 无穷多 发电机 极小值