| New series of odd non-congruent numbers Dedicated to Professor Sheng GONG on the occasion of his 75th birthday |
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| 作者姓名： | FENG Keqin & XUE Yan |
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| 作者单位： | FENG Keqin & XUE Yan Department of Mathematical Science，Tsinghua University，Beijing 100084，China |
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| 摘 要： | We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and Sn 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}, Sn = {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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New series of odd non-congruent numbers Dedicated to Professor Sheng GONG on the occasion of his 75th birthday |
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| FENG Keqin & XUE Yan.New series of odd non-congruent numbers Dedicated to Professor Sheng GONG on the occasion of his 75th birthday[J].Science in China(Mathematics),2006(11). |
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| Authors: | FENG Keqin & XUE Yan |
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| Institution: | FENG Keqin & XUE Yan Department of Mathematical Science,Tsinghua University,Beijing 100084,China |
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| Abstract: | We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and Sn 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}, Sn = {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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| Keywords: | congruent number elliptic curves rank 2-descent odd graph |
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