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 :-
... 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 × 1014. 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  (see also , , , , ) 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 2n2 + 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 :-
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.
Article by: J J O'Connor and E F Robertson