Abstract
We determine all square-free odd positive integers n such that the 2-Selmer groups S n and Ŝ n of the elliptic curve E n : y 2 = x(x − n)(x − 2n) and its dual curve Ê n : y 2 = x 3 + 6nx 2 + n 2 x have the smallest size: S n = {1}, Ŝ 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.
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Dedicated to Professor Sheng GONG on the occasion of his 75th birthday
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Feng, K., Xue, Y. New series of odd non-congruent numbers. SCI CHINA SER A 49, 1642–1654 (2006). https://doi.org/10.1007/s11425-006-2048-7
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DOI: https://doi.org/10.1007/s11425-006-2048-7