May 262013

Busy day in analytic number theory

On May 13, 2013, Harald Andres Helfgott  uploaded to the arXiv his paper “Major arcs for Goldbach’s theorem” claimed that he has proved the ternary Goldbach conjecture, or odd Goldbach conjecture, asserts that every odd integer  \(n>5\) is the sum of three primes.

这论文仅仅证明了每个 \(>10^{30}\) 的奇数可以表示为三个质数之和. 至于 \(<10^{30}\) 的奇数, 已经通过计算机进行验证. 计算机实际上, 已经计算过, 对于 \(<8.875\cdot10^{30}\) 的奇数, Goldbach’s conjecture 都是对的. 这样, 奇数 Goldbach’s conjecture 彻底终结.

这文章采用是基于圆法 (Hardy–Littlewood circle method), 大筛法(the large sieve) and exponential sums 的一种途径.

Goldbach’s conjecture 已经有 \(271\) 年的历史了.

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