New series of odd non-congruent numbers

摘要

We determine all square-free odd positive integers n such that the 2-Selmer groups S_n and S_n of the elliptic curve E_n: y~2 = x(x — n)(x — 2n) and its dual curve E_n: y~2 = x~3 +6nx~2 + n~2x have the smallest size: S_n = {1}, S_n = {1,2,n, 2n}. It is well known that for such integer n, the rank of group E_n(Q) of the rational points on E_n is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves E_n with rank zero and such series of integers n are non-congruent numbers.