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

作　　者：LI XiuMei ZENG JinXiang 

机构地区：[1]Department of Mathematical Sciences,Tsinghua University

出　　处：《Science China Mathematics》2014年第4期719-728,共10页中国科学：数学（英文版）

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.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：elliptic curve integer factoring Selmer group 

分 类 号：O187[理学—数学]