科学与数学plus.maths.org
拉马努金再次带来惊喜
Ramanujan surprises again
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导读
这篇 Plus Maths 文章围绕拉马努金留下的手稿与笔记本，讲述他去世多年后仍不断给数学界带来惊喜：他对接近费马大定理反例的构造，意外为现代椭圆曲线研究提供了方法。文章也串起他的笔记本、1729 出租车数等著名故事，说明他的直觉为何至今仍在启发数学。
为什么经久不衰：它把拉马努金的传奇直觉与当代数学成果连在一起，展示了纯数学研究难以预料的长远价值，所以多年后读来仍令人惊叹。
HN 评论精选
拉马努金的旧工作让现代数学家几乎不费力地追平椭圆曲线问题的世界纪录。
这正是拉马努金提出的东西。他关于 K3 曲面的工作，为 Ono 和 Trebat-Leder 提供了一种方法，能产生不止一条、而是无穷多条需要两三个解来生成所有其他解的椭圆曲线。这不是首个被发现的方法，但几乎不费力。Ono 说：“我们在这个问题上追平了世界纪录，却没做任何重活；除了认出拉马努金所做的工作，我们几乎什么都没做。”
chris_wot查看英文原文
...this is exactly what Ramanujan came up with. His work on the K3 surface he discovered provided Ono and Trebat-Leder with a method to produce, not just one, but infinitely many elliptic curves requiring two or three solutions to generate all other solutions. It's not the first method that has been found, but it required no effort. "We tied the world record on the problem [of finding such elliptic curves], but we didn't have to do any heavy lifting," says Ono. "We did next to nothing, expect recognise what Ramanujan did." There's a joke in Dr Who where the 11th Doctor (Matt Smith) gives a proof of Fermat's Last Theorem to geniuses on earth in order to convince them he can prevent annihilation by the Atraxi. He basically gives them "Fermat’s Theorem. The proof. And I mean the real one. Never been seen before. Poor old Fermat got killed in a duel before he could write it down. My fault. I slept in." That always makes me wonder - was there and is there indeed a better, shorter and more elegant proof than the one we have? Who knows, maybe Ramanujan was on his way to developing one!
拉马努金为寻找费马大定理近似反例而写的公式，成了意料之外的未来工具。
这是纯研究的绝佳案例：“他是公式天才，我认为他的目标是构造费马大定理的近似反例。于是他发展出一套寻找这些‘差一点’例子的理论，却没意识到他正在搭建的机器、写下的公式会在未来对任何人有用。”
clusterfoo查看英文原文
What a great case for pure research: > "He was a whiz with formulas and I think [his aim was] to construct those near counter-examples to Fermat's last theorem. So he developed a theory to find these near misses, without recognizing that the machine he was building, those formulas that he was writing down, would be useful for anyone, ever, in the future."
1729 这个出租车数意外出现在最快整数乘法的渐近方法中。
一个有趣的巧合：2019 年有人发现，以 O(n log n) 时间乘两个 n 位整数的最快方法涉及 1729 维傅里叶变换。如此初等的运算，其渐近最优方法竟与拉马努金著名的出租车数 1729 相连，令人惊讶。（严格说，任意不少于 1729 维都成立，但证明在小于该维数时失效；未来工作可能降低这个界，或更好解释为何这个界是必要的。）
nneonneo查看英文原文
An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778. It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number. (Technically, it works for any number of dimensions >= 1729, but the proof fails for dimensions less than that. Future work might bring the bound down, or better explain why that bound is necessary.)
出租车数故事真正的巧合不是心算，而是哈代的出租车号码出现在拉马努金的研究里。
好文！初听出租车数故事，你首先会震惊于拉马努金天生的计算能力。有意思的是，真正的巧合其实是哈代乘坐的出租车号码，恰好出现在拉马努金对费马大定理的研究中。
v64查看英文原文
Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.
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