One of the oldest unsolved problems

The Hindu

4 hours ago

A young traveller, an able administrator
The son of a pastor, Christian Goldbach was born on March 18, 1690 in Konigsberg – the historic German and Prussian name of the city we now know as Kaliningrad, Russia. Growing up in that city and attending university there, Goldbach studied some mathematics (don't raise your eyebrows), but mainly took to law and medicine.
When he was out of his teens, he set out travelling. His journey around much of Europe began in 1710 and his lengthy travels enabled him to meet many of the leading scientists of the day. We'll get to that in a bit.
After spending nearly 15 years thus, travelling, Goldbach settled down, so as to say. He had become an established mathematician by this point. Despite initial rejections, Goldbach became a professor of mathematics and historian at the newly set up Saint Petersburg Academy of Sciences.
In 1728, when Peter II became the tsar of Russia, Goldbach was named as the new tutor of the young emperor. When Peter II moved the court from St. Petersburg to Moscow, Goldbach moved with him. From this time onwards, Goldbach grew in stature as an administrator too.
Even though there were plenty of changes in the political scene, Goldbach remained unaffected. While there was a purge of officials along with the various political moves that accompanied the replacement of one Russian ruler by another, Goldbach was never one of them.
He continued to rise in status, drew bigger salaries, and also received lands. He laid down the guidelines for the education of royal children, guidelines that remained in practice for nearly 100 years.
By 1740, the administrative work occupied so much of his time that Goldbach asked his duties at the Academy to be reduced. When he further rose to a senior position in the Ministry of Foreign Affairs, he stopped working for the Academy. Goldbach died in Moscow on November 20, 1764, aged 74.
Keeping in touch
Travelling a continent and meeting prominent scientists was one thing. But keeping in touch with them years later was quite another. Goldbach was a letter writer par excellence and he was at it for nearly his whole lifetime.
Having set off in 1710, Goldbach encountered German polymath Gottfried Leibniz in Leipzig in 1711. Goldbach moved on from Leipzig but the two continued to be in touch. Their correspondence between 1711-13 included 11 letters, with Leibniz writing five and Goldbach writing six, all in Latin.
In 1712, Goldbach met French mathematician Abraham de Moivre and Swiss mathematician Nicolaus I Bernoulli, who himself was also on European travels, in London, England. Goldbach bumped into Bernoulli again in Oxford and the latter started discussing infinite series with Goldbach.
It is worth mentioning that while Goldbach was fascinated by the mathematics that he was being exposed to this way, he had little in the form of formal knowledge in the subject. In fact, during the conversation about infinite series, Goldbach confessed his ignorance, prompting Bernoulli to loan him a book on the topic by his uncle Jacob Bernoulli. Goldbach, however, was intimidated by infinite series at that time, and gave up his attempts to understand the text after finding it too difficult.
Things, however, changed in the years that followed. After reading an article about computing the area of a circle by Leibniz in 1717, Goldbach was drawn again to the theory of infinite series. He published a number of papers on mathematics in 1720 and 1724 and became a mathematician of repute by the time he decided to settle down following his travels.
In 1721, Goldbach met Swiss mathematician Nicolaus II Bernoulli in Venice, Italy, while he was also on a tour of European countries. He suggested to Goldbach that he start a correspondence with his younger brother Daniel Bernoulli, a mathematician and physicist. Goldbach began his correspondence with Daniel in 1723 and it continued for seven years.
Most famous correspondence
For someone who made letter writing a part of himself, it is fitting that he is now best remembered for what he set out on one such letter. Swiss polymath Leonhard Euler met Goldbach in St. Petersburg in 1727 and even though Goldbach moved to Moscow the following year, they had a long lasting relationship.
The correspondence between the two spanned 35 years and the nearly 200 letters between the two were written in a number of languages – Latin, German, and French – and covered a whole gamut of topics, including, of course, mathematical subjects. In fact, Euler's interest in number theory was kindled by Goldbach. Their intimacy also meant that Goldbach was the godfather of one of Euler's children.
Most of Goldbach's important work in number theory was contained in his correspondence with Euler. While Goldbach's conjecture is the most famous remnant of their correspondence now, they also discussed Fermat numbers, Mersenne numbers, perfect numbers, the representation of natural numbers as a sum of four squares, Waring's problem, and Fermat's Last Theorem, among others.
Goldbach's conjecture
In a letter to Euler dated June 7, 1742, Goldbach expressed what we now know as Goldbach's conjecture. In his own words, he asserted that 'at least it seems that every number that is greater than 2 is the sum of three primes.'
Bear in mind that in Goldbach's time, the number 1 was considered prime, a convention that is no longer followed. An equivalent form of this conjecture stated in modern terms therefore asserts that all positive even integers >=4 can be expressed as the sum of two primes.
It's been over 275 years since Goldbach stated his conjecture, but it hasn't been proven yet. Computers have shown that it holds true for trillions of numbers, but that's not quite enough. It is one thing to show through brute force that it is valid up to a certain number, quite another to prove it for all numbers.
The hunt, naturally, has been on to find a solution and Goldbach's conjecture now holds place of prominence as one of mathematics' – number theory in particular – oldest unsolved problems. There have been numerous attempts to crack that armour, but it hasn't been achieved just yet.
There have been breakthroughs, of course. Soviet mathematician Ivan Vinogradov in 1937 proved that every sufficiently large odd number is the sum of three primes. Chinese mathematician Chen Jingrun, meanwhile, showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes in 1973.
There have also been competitions and awards encouraging and challenging mathematicians to solve the problem. The British and American publishers of Apostolos Doxiadis' novel, Uncle Petros and Goldbach's Conjecture, for instance, offered a $1 million bounty to anyone who could prove Goldbach's conjecture within two years in March 2000. The prize, naturally, went unclaimed. The conjecture, however, continues to remain open – alluringly simple and tantalising in its wording, but beyond the best mathematical brains for centuries.