History

Christian Goldbach: 1690-1764

Birth: March 18, 1690 in Königsberg, Prussia 

Death: November 20, 1764 in Moscow, Russia

About: Goldbach was raised in Königsberg and attending university there, where he mainly studied law and medicine. In 1710, he took a trip around Europe, which ultimately sparked his interest in mathematics. Even though he did not know much about math, he learned from many mathematicians such as Leibnitz, Nicolaus (I) Bernoulli, and Euler. Now, Goldbach is known as a Prussian mathematician who researched number theory. He is most known for his suggestion of the Goldbach conjecture: that every even integer greater than 2 can be expressed as the sum of 2 primes. 

Summary: Christian Goldbach first proposed the conjecture to Leonhard Euler in 1742 through a letter. The letter was originally written in German, but it can be translated to: 

"Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until all terms are units."

In the margins, he proposed the weak Goldbach conjecture:

"It seems at least, that every integer greater than 2 can be written as the sum of three primes."

Leonhard Euler: 1707-1783

Birth: April 15, 1707 in Basel, Switzerland

Death: September 18, 1783 in Saint Petersburg, Russia

About: Leonhard Euler was a Swiss mathematician and physicist. He made many significant contributions in geometry, calculus, mechanics, number theory, and astronomy. In 1727 he gained the attention of Johann Bernoulli and moved to Saint Petersburg. Here, he worked for the St. Petersburg Academy of Sciences and was able to publish lots of research. Euler experienced severe blindness later in life yet was still able to be productive with his research. 

Summary: Only 23 days after Goldbach's original letter, Euler replied stating that although he agrees with the statements and would consider it certain, he can't prove it. Euler originally dismissed the letter because it seemed like a simple observation, however, he later recognized the importance of the conjecture. Even though Euler could not prove the conjecture, he considered it true because of his own observations and calculations. 

Prime number theorem: 1896

Proven in: 1896 by Jacques Hadamard and Charles Jean de la Vallée Poussin

Description: The prime number theorem tells us how primes are distributed among other integers. For any integer x, the theorem gives us an estimation of the number of prime numbers less than or equal to x. 

Define the function π(x), which outputs the number of primes less than or equal to x. The prime number theorem approximates this to:

π(x) ∼ x/ln x

For example, take x=100. We know that the actual number of primes less than or equal to 100 is 25. Using the prime number theorem, we would get an estimation of:

100/ln(100) ≈ 100/4.605 ≈  21.07

Thus the prime number theorem would suggest there are about 21.07 prime numbers that are less than or equal to 100. 

Relevance to Goldbach conjecture: The prime number theorem does not directly prove the Goldbach conjecture, however, a heuristic argument can be made to show that it might be true. The prime number theorem shows that the larger integer you have, the more prime numbers there are beneath it. Therefore, the probability of finding 2 that sum to a given even integer seems reasonable. 

Lev Shnirelman: 1905-1938

Birth: January 2, 1905 in Gomel, Belarus

Death: September 24, 1938 in Moscow, Russia

About: Lev Shnirelman was a Soviet mathematician who made contributions in number theory, topology, and differential geometry. He was very advanced as a young child and was able to complete all the courses for school mathematics from age 11 to 12. Then, he went to the University of Moscow at age 16. In 1929, he was appointed the chair of mathematics at the Don Polytechnic Institute in Novocherkassk, and in 1930, he went back to the University of Moscow to teach. After an election in 1933, he had to work at the Mathematical Institute of the Academy. Shnirelman was only 33 when he died, yet he was able to make many contributions within the field of mathematics. 

Summary: The first breakthrough in proving the Goldbach conjecture happened in 1930, when Lev Shnirelman proved that every natural number greater than 1 could be expressed as the sum of no more than C prime numbers, with C being a definite number less than 300,000. A more specific bound of C was then reached, where C=20. Therefore, in simpler terms, every natural number greater than 1 can be written as the sum of 20 or less prime numbers. This was a major step towards proving the Goldbach conjecture.

Ivan Vinogradov: 1891-1983

Birth: September 14, 1891 in Velikie Luki, Russia

Death: March 20, 1983 in Moscow, Russia

About: Ivan Vinogradov was a Soviet mathematician who is known for his contributions to analytic number theory. In 1903, his parents decided to send him to a school where he would receive scientific rather than a classical education. He completed his masters at the University of St. Petersburg in 1915. Vinogradov was successful in his research and made many discoveries, however, many mathematicians were unaware of his work (and Vinogradov was unaware of other works). This is due to the many difficulties arising from World War I and the Russian revolution, which made communication difficult. 

Summary: Ivan Vinogradov proved what is called Vinogradov's theorem in 1937, which states that every sufficiently large odd integer can be expressed as the sum of 3 prime integers. This is a weaker form of the Goldbach conjecture because it doesn't apply to odd numbers greater than 5, only sufficiently large ones. Vinogradov's finding was a major step forward in the weak Goldbach conjecture because it shows that the conjecture holds for a large range of odd numbers. 

Chen Jingrun: 1933-1996

Birth: May 22, 1933 in Fuzhou, China

Death: March 19, 1996 in Beijing, China

About: Chen Jingrun was a Chinese mathematician who is known for his significant contributions to number theory. He grew up poor with 11 other siblings, and when Jingrun was 4 years old, the second Chino-Japanese war broke out. To avoid an unsafe area, Jingrun and his family relocated for a few years, and only a few years later they were forced to move again. His high school teacher, who joked about one of his students solving the Goldbach conjecture in the future, is what drew Jingrun to mathematics. In 1949, Jingrun graduated high school and entered the Mathematics and Physics program at Xiamen University. Jingrun had 11 papers published in 2 years, and his main work at the time was working on the Goldbach conjecture. 

Summary: In 1973, Chen Jingrun made a significant contribution to the Goldbach conjecture through his Chen's theorem. This theorem proves that every sufficiently large even number can be written as the sum of a prime and a product of at most 2 primes (semiprime). Even though this does not directly prove the Goldbach conjecture, this theorem suggests one could possibly prove it by first showing how to express even integers greater than 2 as the sum of a prime and a semiprime. 

Harald Helfgott: 1977-present

Birth: November 25, 1977 in Lima, Peru

Summary: Harald Helfgott is a Peruvian mathematician who works in number theory. He is most widely known for his submission of a proof for the weak Goldbach conjecture in 2013. Although it still remains unpublished, it is widely accepted today.

2013 and onwards

Even though Harald Helfgott's proof for the weak Goldbach conjecture is broadly accepted, the strong Goldbach conjecture is still open to be solved. Large numbers are currently being verified through computers to see if they fit the Goldbach conjecture. A publishing company even offered a 1 million dollar reward for anyone who could prove it in order to give mathematicians an incentive to work on the problem. Unfortunately, the prize was only open for 2 years and no one was able to claim it.