Taxicabs and sums of two cubes
Taxicabs and sums of two cubes
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出租车和两个立方的和
DOI：
10.1080/00029890.1993.11990409
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发表时间：
1993
影响因子：
0.5
通讯作者：
J. Silverman
中科院分区：
文献类型：
--
作者：
J. Silverman
我们的故事始于1913年，当时著名的英国数学家G·H·哈代从印度收到了一个笨重的信封，里面装满了一页又一页的方程式。每个著名的数学家都会定期收到古怪的人的来信，他们声称已经证明了最奇妙的结果。有时会包括不正确或前后不一致的证明。在其他时候，作者会索要报酬，以换取透露他的发现。这封写给哈代的信，是马德拉斯一个名叫拉马努扬的穷职员写的，里面装满了方程式，都是在没有任何证据的情况下给出的。其中一些公式是众所周知的，仅仅是练习；而其他许多公式在哈代训练有素的眼睛看来是荒谬的。如果哈代将这封信原封不动地退还给发信人，谁会责怪他呢？事实上，Ramanujan之前已经将他的结果发送给了另外两位英国数学家，他们每个人都这样做了！相反，哈代思考了这些“狂野的定理。他以前从未见过、也从未想象过的定理。”他和哈代经常合作的另一位著名数学家J·E·利特尔伍德一起，成功地证明了拉马努詹的一些令人惊叹的身份。在这一点上，哈代意识到这封信是由一位真正的数学天才写的，他决定让拉马努扬来英国继续他的数学研究。使用哈代学院提供的旅费，拉马努詹于1914年来到这里。在接下来的几年里，他继续制作和出版极具原创性的材料，并与哈代合作撰写了许多杰出的论文。1918年，30岁的拉马努扬当选为皇家学会会员和三一学院院士，这两项荣誉都是他应得的荣誉。不幸的是，在英格兰较冷的气候下，他患上了肺结核。他回到了家乡马德拉斯，于1920年去世，享年33岁。在拉马努詹的一生中，他把数字视为自己的私人朋友。为了说明这一点，哈代讲述了他有一天是如何去医院探望拉马努詹的故事。哈代无话可说，他说他是乘坐1729号出租车到达医院的。“在我看来，”他继续说道，“这是一个相当枯燥的数字。”拉马努扬回答说：“不，哈代！这是一个非常有趣的数字。它是可以用两种不同的方式表示为两个立方体之和的最小数字：”
Our story begins in 1913, when the distinguished British mathematician G. H. Hardy received a bulky envelope from India full of page after page of equations. Every famous mathematician periodically receives letters from cranks who claim to have proven the most wonderous results. Sometimes the proofs, incorrect or incoherent, are included. At other times the writer solicits a reward in return for revealing his discoveries. Now this letter to Hardy, which was from a poor clerk in Madras by the name of Ramanujan, was filled with equations, all given without any sort of proof. Some of the formulas were well-known, mere exercises; while many of the others looked preposterous to Hardy's trained eye. Who would have blamed Hardy if he had returned this missive to the sender, unread? And in fact, Ramanujan had previously sent his results to two other British mathematicians, each of whom had done just that! But instead Hardy gave some thought to these "wild theorems. Theorems such as he had never seen before, nor imagined."' And together, he and J. E. Littlewood, another eminent mathematician with whom Hardy often worked, succeeded in proving some of Ramanujan's amazing identities. At this point Hardy realized that this letter was from a true mathematical genius, and he became determined that Ramanujan should come to England to pursue his mathematical researches. Using travel money provided by Hardy's college, Ramanujan arrived in 1914. Over the next several years he continued to produce and publish highly original material, and he also collaborated with Hardy on a number of outstanding papers. In 1918, at the age of 30, Ramanujan was elected a Fellow of the Royal Society and also of Trinity College, both signal honors which he richly deserved. Unfortunately, in the colder climate of England he contracted tuberculosis. He returned to his native Madras and died, in 1920, at the age of 33. During all of Ramanujan's life, he considered numbers to be his personal friends. To illustrate, Hardy tells the story of how one day he visited Ramanujan in the hospital. At a loss for something to say, Hardy remarked that he had arrived at the hospital in taxicab number 1729. "It seemed to me," he continued, "a rather dull number." To which Ramanujan replied "No, Hardy! It is a very interesting number. It is the smallest number expressible as a sum of two cubes in two different ways:"2