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:

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:

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.

## 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, nostradamvs.livejournal.com*