Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an int...Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.展开更多
为了研究对任意素数模p的一类广义Kloosterman和的四次均值,利用初等与解析方法、Gauss和以及三角和的转换性质引入了当素数p≡1 mod 4时该均值的计算问题,并将该类均值转化为特征和的简易形式。从计算结果上对均值的估计具有充分性,从...为了研究对任意素数模p的一类广义Kloosterman和的四次均值,利用初等与解析方法、Gauss和以及三角和的转换性质引入了当素数p≡1 mod 4时该均值的计算问题,并将该类均值转化为特征和的简易形式。从计算结果上对均值的估计具有充分性,从计算方法上对广义Kloosterman和各种形式的四次均值研究具有重要的参考价值。此外,这也为指数和均值计算问题提供了一种新的转化思路与方法,必将对有关问题的进一步探索起到推动作用。展开更多
文摘Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.
文摘为了研究对任意素数模p的一类广义Kloosterman和的四次均值,利用初等与解析方法、Gauss和以及三角和的转换性质引入了当素数p≡1 mod 4时该均值的计算问题,并将该类均值转化为特征和的简易形式。从计算结果上对均值的估计具有充分性,从计算方法上对广义Kloosterman和各种形式的四次均值研究具有重要的参考价值。此外,这也为指数和均值计算问题提供了一种新的转化思路与方法,必将对有关问题的进一步探索起到推动作用。
