The unresolved issue of Christian Goldbach

But the truth is: no matter how many even numbers we take, it can be expressed as the sum of two primes. For example: 24 = 13 + 11. Or 100 = 83 + 17. Or 7112 = 5119 + 1993. You can take an arbitrarily large number, and the hypothesis will be true. The problem is precisely to find a general mathematical proof of Goldbach's statement. A prime number is a number that is divisible by only 1 and by itself. So, 2, 3, 5, 7 are prime numbers, but 4, 6, 9 are not. The larger the even number, the more ways to present it as two simple ones:

Goldbach's problem itself is conditionally divided into two statements. The first, directly expressed by Christian Golbach, is called a “weak problem”, and Euler’s clarification is called a “strong problem.” From the validity of the statement of a strong Goldbach problem, weak justice automatically follows. The Goldbach problem is primarily approached “on the forehead”. That is, it is checked consistently for each subsequent prime number. Thus, to date, all even numbers up to 3 * 1018 are checked, and the verification continues. But this method has a significant drawback. In this way, a hypothesis can be refuted if it is false. But the theorem cannot be proved in this way, since it cannot be guaranteed that the number that the program could verify in its next step will not be the first exception to the rule.

For a long time it was not possible to find any ways at all to study the Goldbach problem. It was only in 1923 that the English mathematicians Gottfrey Hardy and John Littlewood managed to prove that if some theorems (not proved even now) regarding the so-called Dirichlet L-series are true, then any sufficiently large odd number is the sum of three prime numbers. Hardy:

In 1930, the mathematician Lev Schnirelman published a proof of the theorem that an integer greater than one is the sum of a limited number of primes, and this number does not exceed 300, 000. This was a serious step forward. But a limited number is not the number 2 indicated in the hypothesis. Therefore, the Schnirelman proof was only a particular solution to the problem. It was published only in 1939, a year after the tragic death of a mathematician (he committed suicide). Nevertheless, the result of Schnirelman was repeatedly specified; The last refinement was made in 1995 by the French mathematician Ramaret: he showed that any even number is the sum of no more than 6 primes. Schnirelman:

In 1937, the Soviet mathematician Ivan Vinogradov took the most serious step forward in solving the Goldbach problem. He proved that any sufficiently large odd number is represented as the sum of three primes, that is, in essence, he solved the Goldbach problem for odd numbers. True, “a sufficiently large number” in the formulation of Vinogradov is 3.33 * 1043000, which today is practically not applicable in applied mathematics. In addition, he presented a proof of a particular case of the Goldbach conjecture for some bounded groups of even numbers, and also showed that there exists a finite n such that any even number can be represented as a sum of no more than n simple summands. In 1975, his proof was developed by Hugo Montgomery and Robert Vaughan. They showed that there are positive constants c and C, such that the number of even numbers, not greater than N, that cannot be represented as the sum of two primes, does not exceed CN1-C. Vinogradov:

A significant step towards proving the Goldbach problem was made in 1966 by the Chinese mathematician Chen Jingzhun. He proved that any sufficiently large even number can be represented either as the sum of two primes, or as the sum of a prime and semisimple (that is, having only 2 divisors, not counting 1 and itself). In 1997, the weak Goldbach problem was proved to be true for another special case: numbers over 1020.

Finally, in 2013, the Peruvian mathematician Harald Andres Helfgott finally proved the weak (or otherwise called ternary) Goldbach problem. That is, the statement “Every odd number greater than 5 can be represented as the sum of three primes” is true. Goldbach’s strong problem remains a stone wall for researchers.

On the Internet you can find a lot of “evidence” of Goldbach’s strong hypothesis. But usually such evidence has errors, or is not evidence at all. It is likely that this hypothesis is simply unprovable, and the observed pattern is a complex combination of mathematical coincidences. This statement is connected, in particular, with the fact that the so-called "law of prime numbers" also does not exist. The discovery of each new prime number occurs exclusively by the method of "enumeration", and recently, due to the enormous numerical "distances" between each new prime number and the one following it, such discoveries occur extremely rarely and are significant mathematical achievements. On the other hand, many even numbers can be represented using several pairs of primes, that is, there are several combinations. If you build a graph of the dependence of the number of combinations of pairs of prime numbers on the increase in even composite numbers, it turns out that with an increase in the even number, there is a tendency to increase the number of pairs of prime numbers that add up to a given number, and this increase occurs according to a certain law. This fact does not allow mathematicians to quit looking for evidence.

And Euler slyly looks from an old picture:

Characteristically, Goldbach was not the luminary of the mathematical science of his time. He was born in 1690 and graduated from the law department of the University of Koenigsberg: mathematics was just his hobby. In 1725, Goldbach came to Russia, where he received the title of academician of the St. Petersburg Academy of Sciences, and since 1742 he worked in the College of Foreign Affairs. He had friendly correspondence with Euler for 35 years, until his death in 1764 in Moscow. In 1843, 177 letters of this correspondence were published. He traveled quite a lot and was friends with many famous mathematicians, including the Bernoulli family. In his life, he has published less than a dozen medium-sized mathematical works, although two of them - on endless series - have made him well-known in the scientific community. However, in wide circles Christian Goldbach became famous thanks to several phrases in a single letter to his friend. Such are the games of fate.

Fermat's Theorem

It is worth noting that the simplest mathematical statements are often extremely difficult to prove. For example, Fermat’s Great Theorem (or, as it is called, “the last”) was proved only several hundred years after it was formulated. The theorem says that the equation an + bn = cn for any positive integer n> 2 does not have positive integers a, b, and c. The theorem was formulated in 1637 and, according to legend, written on the sidelines of the "Arithmetic" of Diophantus. Most likely, Fermat did not have evidence at all, since over the next 30 years of his life he never published it. Special cases for n = 3, 5, 7 and some limited groups of numbers were published by Dirichlet, Lame, Kummer and other mathematicians in different years, but Fermat's theorem was finally proved only in 1995 by the Anglo-American mathematician Sir Andrew John Wiles. He worked on the proof since 1986, and it took more than 130 pages.

The complete correspondence of Goldbach and Euler can be downloaded and read here.

Tim Skorenko,


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