Abstract
We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and Ŝn of the elliptic curve En: y2 = x(x − n)(x − 2n) and its dual curve En: y2 = x3 + 6nx2 + n2x have the smallest size: Sn = {1}, Ŝn = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(ℚ) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.
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