克里斯蒂安·哥德巴赫Christian Goldbach

克里斯蒂安·哥德巴赫的父亲是柯尼斯堡的一位新教牧师。哥德巴赫在柯尼斯堡长大，并在那里的大学就读。他似乎学过一些数学，但主要学习法律和医学。1710年，他开始了一次漫长的欧洲之旅，在旅途中结识了许多顶尖科学家。1711年在莱比锡，他遇到了哥特弗里德·威廉·莱布尼茨，在哥德巴赫继续旅行后，两人保持了通信。哥特弗里德·威廉·莱布尼茨写给哥德巴赫的五封信和哥德巴赫写给哥特弗里德·威廉·莱布尼茨的六封信，均写于1711年至1713年间，在[13]中进行了讨论。两位通信者都用拉丁文写作。

1712年，尼古拉·伯努利一世也在欧洲旅行，他正在访问英国。哥德巴赫在伦敦会见了他和亚伯拉罕·棣莫弗，并在牛津再次会见了尼古拉·伯努利一世。哥德巴赫对数学着迷，但他对这个学科了解不多。当尼古拉·伯努利一世在牛津与哥德巴赫交谈时开始讨论无穷级数，哥德巴赫承认他对这个话题一无所知。尼古拉·伯努利一世借给他一本由他叔叔雅各布·伯努利写的关于这个主题的书，但哥德巴赫当时觉得无穷级数太难，放弃了理解雅各布·伯努利文本的尝试。

哥德巴赫继续他漫长的旅行，1721年在威尼斯。在这里，他遇到了也在欧洲各国旅行的尼古拉·伯努利二世。正是在尼古拉·伯努利二世的建议下，哥德巴赫于1723年开始与他的弟弟丹尼尔·伯努利通信。两人持续通信了七年。到1724年，哥德巴赫回到了他的家乡柯尼斯堡，在那里他遇到了两个将改变他生活的人，即Georg Bernhard Bilfinger和雅各布·赫尔曼。Bilfinger是一位德国哲学家、数学家和政治家，曾在图宾根担任道德哲学和数学教授，但因被指控无神论而刚被解雇。这一指控源于他与哲学家哥德巴赫 Wolff的交往，后者随后帮助安排Bilfinger参与建立帝国科学院（后来称为圣彼得堡科学院），该院将（在哥特弗里德·威廉·莱布尼茨的建议下）按照柏林科学院的模式组织。他在前往圣彼得堡的路上遇到了哥德巴赫，而雅各布·赫尔曼也在前往参与这项激动人心的新事业的路上。

1725年7月，哥德巴赫在里加时写信给拟议中的新科学院当选院长L L Blumentrost，请求在那里获得一个职位。在最初被拒绝后，哥德巴赫被提供了圣彼得堡的数学教授和历史学家职位。人们可能会想知道哥德巴赫是如何被提供这样一个重要职位的。事实上，他作为数学家的记录此时比我们上面描绘的要好得多。我们提到哥德巴赫在1712年放弃了理解无穷级数的尝试。然而在1717年，他读到了哥特弗里德·威廉·莱布尼茨关于计算圆面积的一篇文章，这促使他重新审视无穷级数理论。哥德巴赫于1720年在Acta eruditorum上发表了Specimen methodi ad summas serierum Ⓣ（求级数和方法示例）。1724年，他发表了另一篇论文，此前他还有几部其他作品出版。因此，他是一位已确立地位的数学家，尽管必须说他的论文对数学知识并没有增加太多。

哥德巴赫是1725年12月27日举行的科学院开幕式的记录秘书，并继续担任这一角色直到1728年1月。要理解哥德巴赫在俄罗斯的生活如何发展，我们需要简要看看当时那里发生的政治事件。彼得一世大帝从1682年到1725年统治俄罗斯。他和他的妻子叶卡捷琳娜是建立科学院的推动力量，科学院设在圣彼得堡，因为那是当时的俄罗斯首都。彼得大帝去世后，他的妻子叶卡捷琳娜从1725年到1727年在圣彼得堡统治。Aleksandr Danilovich Menshikov曾是彼得大帝的顾问，但在其统治末期失宠。然而，他与叶卡捷琳娜关系密切，并在1725年成功让她被命名为女皇。Menshikov随后成为实际统治者，在1727年叶卡捷琳娜去世后，他继续担任统治者，成为彼得二世的摄政，彼得二世登基时年仅十一岁。这仅持续了几个月，彼得二世就转而反对Menshikov，并请求Dolgoruky家族协助他。Menshikov于1727年9月被捕，并被送往西伯利亚。Dolgoruky家族安排了一位新导师为年轻的彼得二世接替Menshikov任命的导师。哥德巴赫被任命到这个职位，当彼得于1728年1月将宫廷迁至莫斯科时，他也搬到了莫斯科。莱昂哈德·欧拉于1727年5月17日抵达圣彼得堡，在哥德巴赫搬到莫斯科后，他于1729年开始与莱昂哈德·欧拉通信。这一重要的通信持续了大约35年，下面将讨论。

彼得二世于1730年1月死于天花，安娜·伊万诺夫娜成为俄国女皇。哥德巴赫不再需要担任家庭教师，但他继续为安娜服务。1732年，安娜将宫廷迁回圣彼得堡，哥德巴赫也回到那里，再次在科学院活跃起来，同时深度参与俄国政府事务。他于1732年被任命为科学院通讯秘书，然后在1737年成为负责科学院行政的两人之一（另一位是J D Schuhmacher）。然而，哥德巴赫的问题在于，他不仅深度参与科学院的行政事务，还在俄国政府中升任更重要的职务。安娜·伊万诺夫娜于1740年去世，她指定其侄女安娜·利奥波多夫娜的儿子伊万为继承人，由其母亲摄政。伊万成为皇帝时只有几周大，但在接下来的一年里，皇帝彼得一世大帝的女儿伊丽莎白得以废黜伊万及其母亲，随后她统治俄国20年。值得注意的是，俄国统治者更迭的各种政治变动总是伴随着官员的清洗。然而，尽管高层发生变化，哥德巴赫似乎能够继续担任具有高度影响力的职位。哥德巴赫有[1]：-

……对拉丁文风格有极佳的掌握，德语和法语也同样流利。哥德巴赫优雅的举止和国际化的朋友圈和熟人圈，确保了他在一个努力效仿其西方邻国的精英社会中的成功。

1740年，哥德巴赫请求减少他在科学院的工作，当他被任命为外交部的高级职位时，他停止了所有为科学院所做的工作。他继续随着薪水的巨幅增加而提升地位，并获得了土地。1760年，他成为枢密顾问，并被要求为王室子女的教育制定指导方针。哥德巴赫制定的这些指导方针成为此后100年公认的做法。

哥德巴赫在数论中做了重要工作，其中大部分是通过与莱昂哈德·欧拉的通信完成的。他最令人铭记的是他在1742年给莱昂哈德·欧拉的信中提出的猜想（至今仍是一个未解决的问题），即每个大于2的偶数都可以表示为两个primes之和。计算机已验证该猜想对至少达到$4 \times 10^{14}$的数成立。哥德巴赫还猜想每个奇数都是三个素数之和。伊万·维诺格拉多夫在1937年对第二个猜想取得了进展。同样在莱昂哈德·欧拉-哥德巴赫通信中，如[3]所述（另见[6]、[7]、[8]、[11]、[14]），他们讨论了皮埃尔·德·费马数、马兰·梅森数、完全数、自然数表示为四个平方数之和、爱德华·华林问题（莱昂哈德·欧拉在爱德华·华林之前解决了它）、表示众多素数的多项式、皮埃尔·德·费马最后定理，以及任何奇数表示为$2n^{2} + p$的形式，其中$p$是素数。

此处一段未译出，以下为英文原文The last conjecture was made by Goldbach in a letter written to Euler on 18 November 1752. Euler replied on 16 December, saying he had checked 哥德巴赫猜想 up to 1000. In a letter of 3 April 1753, Euler reported to Goldbach that he had checked it up to 2500. In fact the conjecture is false. In 1856 Moritz A Stern, a professor of mathematics at Göttingen, found two numbers which could not be written as twice a square plus a prime, namely 5777 and 5993. No other examples of numbers failing to satisfy this conjecture of Goldbach seem to be known. It is interesting to ponder that Goldbach could, with some hard work, have tested this conjecture to 2500 as Euler did. However he did seemed to treat mathematics as a recreation, rather than a one where hard effort should be employed. Again, however, we should note his remarkable mathematical intuition [1]:-

与莱昂哈德·欧拉的通信整体上标志着哥德巴赫是他那个时代少数理解皮埃尔·德·费马对该学科新方法含义的人之一。

哥德巴赫除了我们上面提到的作品外，还发表了许多其他作品，但他在信件中展现的洞察力被证明是他最重要的数学贡献。然而，我们应该提到他另外两篇关于无穷级数的论文De transformatione serierum Ⓣ（论级数的变换）（1729）和De terminis generalibus serierum Ⓣ（论级数的一般项）（1732）。第一篇引入了一种将一个级数变换为另一个级数而级数和保持不变的方法。第二篇扩展了上面提到的他1720年论文中开始的工作。他还研究了方程，并在与莱昂哈德·欧拉的通信中研究出如何快速检验一个代数方程是否有有理的根。

Christian Goldbach's father was a Protestant Church minister in Königsberg. Goldbach was brought up in Königsberg and attended the university there. He seems to have studied some mathematics, but he mainly studied law and medicine. In 1710 he set off on a lengthy journey around Europe, meeting many of the leading scientists on his travels. In Leipzig in 1711 he met Leibniz and after Goldbach moved on the two carried on a correspondence. Five letters from Leibniz to Goldbach and six letters from Goldbach to Leibniz, all written in the years 1711 to 1713, are discussed in [13]. Both correspondents wrote in Latin.

In 1712 Nicolaus (I) Bernoulli was also on European travels and he was visiting England. Goldbach met him and also de Moivre in London, and he met Nicolaus (I) Bernoulli again in Oxford. Goldbach was fascinated by mathematics but he did not have much knowledge of the subject. When Bernoulli started to discuss infinite series with Goldbach as they talked in Oxford, Goldbach confessed that he knew nothing about the topic. Bernoulli gave him a loan of a book on the topic by his uncle Jacob Bernoulli but Goldbach found infinite series too difficult at that time, and gave up his attempts to understand Jacob Bernoulli's text.

Goldbach continued his lengthy tour and was in Venice in 1721. Here he met Nicolaus (II) Bernoulli who was also on a tour of European countries. It was at Nicolaus (II) Bernoulli's suggestion that Goldbach began a correspondence with his younger brother Daniel Bernoulli in 1723. The two continued this correspondence for seven years. By 1724 Goldbach was back in his home town of Königsberg and there he met two people who would change his life, namely Georg Bernhard Bilfinger and Jakob Hermann. Bilfinger was a German philosopher, mathematician, and statesman, who had been professor of moral philosophy and mathematics at Tübingen but had just been sacked over a charge of atheism. The charge arose through his association with the philosopher Christian Wolff, who had then helped arrange that Bilfinger should be involved in setting up the Imperial Academy of Sciences (later called the St Petersburg Academy of Sciences) which was to be organised (at Leibniz's suggestion) along the lines of the Berlin Academy of Sciences. He was on his way to St Petersburg when he met Goldbach, and Jakob Hermann was also on his way to take part of this new exciting venture.

When he was in Riga in July 1725, Goldbach wrote to L L Blumentrost, the President elect of the proposed new Academy, asking for a position there. After an initial rejection, Goldbach was offered the positions of professor of mathematics and historian at St Petersburg. One may wonder how Goldbach was offered such an important position. In fact his record as a mathematician was by this time rather better than the picture we painted above. We mentioned that Goldbach gave up his attempts to understand infinite series in 1712. However in 1717 he read an article by Leibniz on computing the area of a circle and this led him to look again at the theory of infinite series. Goldbach published Specimen methodi ad summas serierum Ⓣ in Acta eruditorum in 1720. In 1724 he published another paper and earlier he had a couple of other works published. He was, therefore, an established mathematician although it has to be said that his papers do not add a great deal to mathematical knowledge.

Goldbach was recording secretary for the opening ceremony of the Academy which was held on 27 December 1725, and continued to act in this role until January 1728. To understand how Goldbach's life progressed in Russia we need to look briefly at the political events which were taking place there. Peter I the Great ruled Russia from 1682 to 1725. He and his wife Catherine were the driving force behind setting up the Academy and it was set up in St Petersburg because that was the Russian capital at this time. After Peter the Great died, his wife Catherine ruled from St Petersburg from 1725 to 1727. Aleksandr Danilovich Menshikov had been an advisor to Peter the Great but had fallen out of favour towards the end of his reign. However, he was close to Catherine and succeeded in having her named empress in 1725. Menshikov was then the effective ruler and on Catherine's death in 1727 he continued to be the ruler becoming regent to Peter II, who was eleven years old when he came to the throne. This lasted only a few months before Peter II turned against Menshikov, and asked the Dolgoruky family to assist him. Menshikov was arrested in September 1727, and sent to Siberia. The Dolgoruky family arranged for a new tutor for the young Peter II to take over from the one appointed by Menshikov. Goldbach was appointed to the position and he moved to Moscow when Peter moved the court there in January 1728. Euler had arrived in St Petersburg on 17 May 1727 and after Goldbach moved to Moscow he began a correspondence with Euler in 1729. This important correspondence, which continued for around 35 years, is discussed below.

Peter II died of smallpox in January 1730 and Anna Ivanovna became empress of Russia. Goldbach was no longer required as a tutor, but he continued to serve Anna. In 1732 Anna moved the court back to St Petersburg and Goldbach returned there and again became active in the Academy as well as being heavily involved with the Russian government. He was appointed as corresponding secretary to the Academy in 1732 and then in 1737 he became one of two people responsible for the administration of the Academy (the other was J D Schuhmacher). Goldbach's problem, however, was that as well as being heavily involved with the administration of the Academy, he was also rising to more responsible roles in the government of Russia. Anna Ivanovna died in 1740, having named Ivan, the son of her niece Anna Leopoldovna, as her successor with his mother as regent. Ivan was only a few weeks old when he became emperor, but in the following year, Elizabeth, the daughter of Emperor Peter I the Great, was able to remove Ivan and his mother and she then ruled Russia for the next 20 years. It is worth noting that the various political moves which replaced one Russian ruler by another always were accompanied by a purge of officials. Goldbach, however, seemed able to continue to hold positions of high influence despite the changes at the top. Goldbach had [1]:-

... a superb command of Latin style and equal fluency in German and French. Goldbach's polished manners and cosmopolitan circle of friends and acquaintances assured his success in an elite society struggling to emulate its western neighbours.

In 1740 Goldbach requested that his duties at the Academy be reduced, and when he was appointed to a senior position in the Ministry of Foreign Affairs, he ceased all his work for the Academy. He continued to raise in status with large increases in salary and he received lands. In 1760 he became a privy councillor, and was asked to lay down guidance for the education of royal children. The guidelines Goldbach drew up became the accepted practice for the next 100 years.

Goldbach did important work in number theory, much of it in correspondence with Euler. He is best remembered for his conjecture, made in 1742 in a letter to Euler (and still an open question), that every even integer greater than 2 can be represented as the sum of two primes. It has been checked by computer for numbers up to at least $4 \times 10^{14}$. Goldbach also conjectured that every odd number is the sum of three primes. Vinogradov made progress on this second conjecture in 1937. Also in the Euler-Goldbach correspondence, described in [3] (see also [6], [7], [8], [11], [14]) they discuss Fermat numbers, Mersenne numbers, perfect numbers, the representation of natural numbers as a sum of four squares, Waring's problem (which Euler solved before Waring), polynomials representing numerous primes, Fermat's Last Theorem, and the representation of any odd numbers in the form $2n^{2} + p$ where $p$ is prime.

The last conjecture was made by Goldbach in a letter written to Euler on 18 November 1752. Euler replied on 16 December, saying he had checked Goldbach's conjecture up to 1000. In a letter of 3 April 1753, Euler reported to Goldbach that he had checked it up to 2500. In fact the conjecture is false. In 1856 Moritz A Stern, a professor of mathematics at Göttingen, found two numbers which could not be written as twice a square plus a prime, namely 5777 and 5993. No other examples of numbers failing to satisfy this conjecture of Goldbach seem to be known. It is interesting to ponder that Goldbach could, with some hard work, have tested this conjecture to 2500 as Euler did. However he did seemed to treat mathematics as a recreation, rather than a one where hard effort should be employed. Again, however, we should note his remarkable mathematical intuition [1]:-

The correspondence with Euler as a whole marks Goldbach as one of the few men of his day who understood the implications of Fermat's new approach to the subject.

Although Goldbach published a number of works other than the ones we have mentioned above, it is the insight which he showed in his letters which have proved by far his most important mathematical contribution. We should, however, mention his another two of his papers on infinite series De transformatione serierum Ⓣ (1729) and De terminis generalibus serierum Ⓣ (1732). The first of these introduced a method of transforming one series into another while the sum of the series remains fixed. The second extends the work begun in his 1720 paper mentioned above. He also studied equations and worked out in his correspondence with Euler how to provide a quick test for whether an algebraic equation has a rational root.