Goldbach's problem is one of the oldest and most hyped problems in the history of all mathematics.
This conjecture has been proven to be true for all integers less than 4 × 1018, but remains unproven despite considerable efforts by mathematicians.
Number
The Goldbach number is a positive even integer that is the sum of a pair of odd primes. Another form of the Goldbach conjecture is that all even integers greater than four are Goldbach numbers.
Separation of such numbers is called Goldbach's partition (or partition). Below are examples of similar sections for some even numbers:
6=3 + 38=3 + 510=3 + 7=5 + 512=7 + 5…100=3 + 97=11 + 89=17 + 83=29 + 71=41 + 59=47 + 53.
Discovery of the hypothesis
Goldbach had a colleague named Euler, who liked to count, write complex formulas and put forward unsolvable theories. In this they were similar to Goldbach. Euler made a similar mathematical riddle even before Goldbach, with whom heconstant correspondence. He then proposed a second suggestion in the margin of his manuscript, according to which an integer greater than 2 could be written as the sum of three primes. He considered 1 to be a prime number.
The two hypotheses are now known to be similar, but this did not seem to be a problem at the time. The modern version of Goldbach's problem states that every integer greater than 5 can be written as the sum of three primes. Euler replied in a letter dated June 30, 1742, and reminded Goldbach of an earlier conversation they had ("… so we are talking about the original (and not marginal) hypothesis arising from the following statement").
Euler-Goldbach problem
2 and its even numbers can be written as the sum of two prime numbers, which is also Goldbach's conjecture. In a letter dated June 30, 1742, Euler stated that every even integer is the result of the addition of two primes, which he considers to be a well-defined theorem, although he cannot prove it.
Third version
The third version of Goldbach's problem (equivalent to the other two versions) is the form in which the conjecture is usually given today. It is also known as the "strong", "even", or "binary" Goldbach conjecture to distinguish it from the weaker hypothesis known today as the "weak", "odd", or "ternary" Goldbach conjecture. The weak conjecture states that all odd numbers greater than 7 are the sum of three odd primes. The weak conjecture was proven in 2013. The weak hypothesis isa consequence of a strong hypothesis. The reverse corollary and the strong Goldbach conjecture remain unproven to this day.
Check
For small values of n, the Goldbach problem (and hence the Goldbach conjecture) can be verified. For example, Nils Pipping in 1938 carefully tested the hypothesis up to n ≦ 105. With the advent of the first computers, many more values of n were calculated.
Oliveira Silva performed a distributed computer search that confirmed the hypothesis for n ≦ 4 × 1018 (and double checked up to 4 × 1017) as of 2013. One entry from this search is that 3,325,581,707,333,960,528 is the smallest number that does not have a Goldbach split with a prime below 9781.
Heuristics
The version for the strong form of Goldbach's conjecture is as follows: since the quantity tends to infinity as n increases, we expect that every large even integer has more than one representation as the sum of two primes. But in fact, there are a lot of such representations. Who solved the Goldbach problem? Alas, still nobody.
This heuristic argument is actually somewhat imprecise, as it assumes that m is statistically independent of n. For example, if m is odd, then n - m is also odd, and if m is even, then n - m is even, and this is a non-trivial (complex) relation, because apart from the number 2, only odd numbers can be prime. Similarly, if n is divisible by 3 and m was already a prime other than 3, then n - m is also mutuallyprime with 3, so more likely to be a prime number as opposed to a total number. Carrying out this type of analysis more carefully, Hardy and Littlewood in 1923, as part of their famous Hardy-Littlewood simple tuple conjecture, made the above refinement of the whole theory. But it has not helped to solve the problem so far.
Strong hypothesis
The strong Goldbach conjecture is much more complicated than the weak Goldbach conjecture. Shnirelman later proved that any natural number greater than 1 can be written as the sum of at most C primes, where C is an effectively computable constant. Many mathematicians tried to solve it, counting and multiplying numbers, offering complex formulas, etc. But they never succeeded, because the hypothesis is too complicated. No formulas helped.
But it is worth moving away from the question of proving Goldbach's problem a little. The Shnirelman constant is the smallest C number with this property. Shnirelman himself got C <800 000. This result was subsequently supplemented by many authors, such as Olivier Ramaret, who showed in 1995 that every even number n ≧ 4 is actually the sum of at most six primes. The most famous result currently associated with the Goldbach theory by Harald Helfgott.
Further development
In 1924, Hardy and Littlewood assumed G. R. H. showed that the number of even numbers up to X, violating the binary Goldbach problem, is much less than for small c.
In 1973 Chen JingyunI tried to solve this problem, but it didn't work. He was also a mathematician, so he was very fond of solving riddles and proving theorems.
In 1975, two American mathematicians showed that there are positive constants c and C - those for which N is sufficiently large. In particular, the set of even integers has zero density. All this was useful for work on the solution of the ternary Goldbach problem, which will take place in the future.
In 1951, Linnik proved the existence of a constant K such that every sufficiently large even number is the result of adding one prime number and another prime number to each other. Roger Heath-Brown and Jan-Christoph Schlage-Puchta found in 2002 that K=13 works. This is very interesting for all people who like to add to each other, add up different numbers and see what happens.
Solution of the Goldbach problem
As with many well-known conjectures in mathematics, there are a number of alleged proofs of the Goldbach conjecture, none of which are accepted by the mathematical community.
Although Goldbach's conjecture implies that every positive integer greater than one can be written as the sum of at most three prime numbers, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime number at each step. The Pillai sequence keeps track of the numbers requiring the most primes in their greedy representations. Therefore, the solution to the Goldbach problemstill in question. Nevertheless, sooner or later it will most likely be solved.
There are theories similar to Goldbach's problem in which prime numbers are replaced by other specific sets of numbers, such as squares.
Christian Goldbach
Christian Goldbach was a German mathematician who also studied law. He is remembered today for the Goldbach conjecture.
He worked as a mathematician all his life - he was very fond of adding numbers, inventing new formulas. He also knew several languages, in each of which he kept his personal diary. These languages were German, French, Italian and Russian. Also, according to some sources, he spoke English and Latin. He was known as a fairly well-known mathematician during his lifetime. Goldbach was also quite closely connected with Russia, because he had many Russian colleagues and the personal favor of the royal family.
He continued to work at the newly opened St. Petersburg Academy of Sciences in 1725 as professor of mathematics and historian of the academy. In 1728, when Peter II became Tsar of Russia, Goldbach became his mentor. In 1742 he entered the Russian Foreign Ministry. That is, he actually worked in our country. At that time, many scientists, writers, philosophers and military people came to Russia, because Russia at that time was a country of opportunities like America. Many have made a career here. And our hero is no exception.
Christian Goldbach was multilingual - he wrote a diary in German and Latin, his letterswere written in German, Latin, French and Italian, and for official documents he used Russian, German and Latin.
He died on November 20, 1764 at the age of 74 in Moscow. The day when Goldbach's problem is solved will be a fitting tribute to his memory.
Conclusion
Goldbach was a great mathematician who gave us one of the greatest mysteries of this science. It is not known whether it will ever be solved or not. We only know that its supposed resolution, as in the case of Fermat's theorem, will open up new perspectives for mathematics. Mathematicians are very fond of solving and analyzing it. It is very interesting and curious from a heuristic point of view. Even math students like to solve the Goldbach problem. How else? After all, young people are constantly attracted to everything bright, ambitious and unresolved, because by overcoming difficulties one can assert oneself. Let's hope that soon this problem will be solved by young, ambitious, inquisitive minds.
