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
前 n 个自然数的 m(m∈N)次幂的和的计算问题,是世界著名100个初等数学问题之一。本文得出一个新的计算公式,并且用差分方法。给出了严格证明。公式为:1~m+2~m+3~m+…+n^m=(n+1)sum from (k=1) to m sum from (r=0) to (m-1)((-1)~rC_k^...前 n 个自然数的 m(m∈N)次幂的和的计算问题,是世界著名100个初等数学问题之一。本文得出一个新的计算公式,并且用差分方法。给出了严格证明。公式为:1~m+2~m+3~m+…+n^m=(n+1)sum from (k=1) to m sum from (r=0) to (m-1)((-1)~rC_k^r(k-r)~m)/((k+1)1)n^(k)(k=1,2,…,m),m 为任意自然数(m,n∈N)。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.
文摘前 n 个自然数的 m(m∈N)次幂的和的计算问题,是世界著名100个初等数学问题之一。本文得出一个新的计算公式,并且用差分方法。给出了严格证明。公式为:1~m+2~m+3~m+…+n^m=(n+1)sum from (k=1) to m sum from (r=0) to (m-1)((-1)~rC_k^r(k-r)~m)/((k+1)1)n^(k)(k=1,2,…,m),m 为任意自然数(m,n∈N)。
