The Goldbach problem is one of the oldest and most promoted problems in the history of all mathematics.
It has been proven that this hypothesis is true for all integers less than 4 Ă— 1018, but remains unproven, despite the considerable efforts of 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 hypothesis is that all even integers greater than four are Goldbach numbers.
The allocation of such numbers is called the partition (or section) of Goldbach. The following 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.
Hypothesis discovery
Goldbach had a colleague by the name of Euler, who liked to count, compose complex formulas and put forward unsolvable theories. In this, he and Goldbach were similar. Euler made a similar mathematical riddle even before Goldbach, with whom he kept in constant correspondence. He then proposed a second assumption in the margins of his manuscript, according to which an integer greater than 2 can be written as the sum of three primes. He considered 1 a prime number.
It is now known that the two hypotheses are similar, but at the time this did not seem to be a problem. The current version of Goldbach’s problem states that every integer greater than 5 can be written as the sum of three prime numbers. Euler answered in a letter dated June 30, 1742 and reminded Goldbach of an earlier conversation they had ("... so we are talking about the initial (and not marginal) hypothesis arising from the following statement").
Euler-Goldbach Problem
2 and even numbers can be written as the sum of two primes, which is also the Goldbach hypothesis. In a letter dated June 30, 1742, Euler stated that every even integer is the result of adding two primes, he considers this to be a completely definite theorem, although he cannot prove it.
Third version
The third version of the Goldbach problem (equivalent to the other two versions) is the form in which the hypothesis is usually clothed today. It is also known as the “strong”, “even” or “binary” Goldbach hypothesis in order to be able to distinguish it from the weaker hypothesis, known today as the “weak”, “odd” or “triple” Goldbach Hypothesis. A weak hypothesis states that all odd numbers greater than 7 are the sum of three odd primes. A weak hypothesis was proved in 2013. A weak hypothesis is a consequence of a strong hypothesis. The converse effect and Goldbach’s strong hypothesis remain unproven to this day.
Check
For small values ​​of n, the Goldbach problem (and, therefore, the Goldbach hypothesis) can be verified. For example, Niels Pipping thoroughly tested the hypothesis in 1938 to n ≤ 105. With the advent of the first computers, many more values ​​of n were calculated.
Oliveira Silva performed a distributed computer search, which 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 partition with a prime number below 9781.
Heuristic
The version for the strong form of the Goldbach hypothesis is as follows: since the quantity tends to infinity with increasing n, we expect that each large even integer has not only one representation in the form of the sum of two primes. But in fact, a lot of such ideas. Who solved the Goldbach problem? Alas, still nobody.
This heuristic argument is actually somewhat inaccurate, since it assumes that m with respect to n is statistically independent. 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 nontrivial (complex) relation, because in addition to the number 2, only odd numbers can be prime. Similarly, if n is divisible by 3, and m was already prime other than 3, then n - m is also mutually prime with 3, so it is more likely to be a prime in contrast to the total. Carrying out this type of analysis more thoroughly, Hardy and Littlewood in 1923, as part of their famous hypothesis of simple Hardy-Littlewood tuples, introduced the above refinement into the whole theory. But this has not helped to solve the problem so far.
Strong hypothesis
The strong Goldbach hypothesis is much more complicated than the weak Goldbach hypothesis. Schnirelman later proved that any natural number greater than 1 can be written as the sum of no more than prime numbers C, where C is an effectively calculated constant. Many mathematicians tried to solve it, counting and multiplying numbers, proposing complex formulas, etc. But they never succeeded, because the hypothesis is too complicated. No formulas helped.
But it is worth slightly moving away from the question of proof of the Goldbach problem. The Shnirelman constant is the smallest number C with this property. Schnirelman himself received C <800,000. This result was subsequently supplemented by many authors, such as Olivier Ramare, who in 1995 showed that every even number n ≥ 4 is actually the sum of no more than six primes. The most famous result is currently related to the Goldbach theory by Harald Helfgott.
Further development
In 1924, Hardy and Littlewood, under the assumption of G.R.H. showed that the number of even numbers up to X, violating the Goldbach binary problem, is much smaller than for small s.
In 1973, Chen Jingyun tried to solve this problem, but he did not succeed. He was also a mathematician, and therefore very fond of solving puzzles and proving theorems.
In 1975, two American mathematicians showed that there are positive constants c and C, for which there are sufficiently large numbers N. In particular, the set of even integers has zero density. All this came in handy for solving the Goldbach ternary problem, which will take place in the future.
In 1951, Linnik proved the existence of a constant K such that each 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-Pucht in 2002 found 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.
Goldbach Solution
As with many well-known hypotheses in mathematics, there is a series of alleged evidence for the Goldbach hypothesis, none of which are accepted by the mathematical community.
Although the Goldbach hypothesis implies that each positive integer greater than one can be written as the sum of no more than three primes, it is not always possible to find such a sum using a greedy algorithm that uses the maximum possible prime at each step. The Pillai sequence tracks the numbers that require the largest number of primes in their greedy representations. Therefore, the solution to the Goldbach problem is still in question. Nevertheless, sooner or later it will most likely be resolved.
There are theories similar to the Goldbach problem in which primes 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 hypothesis.
He worked as a mathematician all his life - he loved to compose numbers, invent 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 reports, he was fluent in English and Latin. He was known as a rather famous mathematician during his lifetime. Goldbach was also quite closely connected with Russia, because he had many Russian colleagues and the personal location of the imperial family.
He continued to work in the newly opened St. Petersburg Academy of Sciences in 1725 as a professor of mathematics and a historian of the academy. In 1728, when Peter II became king of Russia, Goldbach became his mentor. In 1742, he entered the Russian Foreign Ministry. That is, he, in fact, worked in our country. At that time, many scientists, writers, philosophers, and the military came to Russia, because Russia of 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 letters were 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 the Goldbach problem is resolved will be a worthy celebration of 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. It is only known that its supposed resolution, as in the case of Fermat's theorem, will open up new perspectives for mathematics. She is very fond of solving and analyzing mathematics. She is very interesting and curious from a heuristic point of view. Even mathematics students like to solve the Goldbach problem problem. How else? After all, young people are constantly attracted to everything bright, ambitious and unresolved, since by overcoming difficulties one can assert oneself. Let's hope that soon this problem will be solved by young, ambitious, inquisitive minds.