Proof of the Goldbach binary hypothesis

Goldbach's binary hypothesis is formulated as follows: “ any even integer number more than two can be represented as the sum of two primes ” [1].



Immediately, we note that in 1938, Niels Pepping “manually” checked the binary Goldbach hypothesis for all even numbers up to 100,000 [1].



Immediately after the "two" comes the even number 4, which is represented as the sum of two simple ones:

4 = 2 + 2 , i.e. for the "four" hypothesis is performed. Go ahead…

')

We construct a table of pairwise sums of odd numbers, in which simple and odd composite numbers are highlighted with their own colors, as follows:





As you can see, all even numbers from 6 to 62 are “sitting” in the table. At the same time, all numbers except the “six” (and the “eight”, which is represented by two copies only because of the permutation of the terms “3” and “5”) , are represented by several copies and they are located strictly parallel to the large diagonal, going from the bottom left corner to the top right and highlighted in green.



And now, we will slightly modify the table, namely, we select even numbers formed by the sum of only primes. Here is the new table:



So, despite the fact that even numbers, obtained by summing with the participation of composite odd numbers, “cut out” good “pieces” from our table, we can still type in a sequence from 6 to 62, which satisfies the Hypothesis.



It is clear that with the continuation of a sequence of odd numbers, the amounts with the participation of composite numbers will “cut out” larger “pieces” that do not satisfy the Hypothesis. As a result, in order to prove the Hypothesis, we need to refute the possibility of the situation depicted in the new table.





And the following is depicted here: “after a prime number P _{n the} next prime number is the number P _{n + 1} > 2 • P _n (n is the number of a prime number in the sequence of primes starting from two)” ...



Getting down to rebuttal. So, we need to prove that for any prime number P _n in the range from P _n to 2 • P _n there is at least the following prime number P _{n + 1} .



And to prove, it turns out, nothing is needed anymore, because this statement is Bertrand's proven postulate.

Literature

1. Stuart I. The greatest math problems / Ian Stewart; Per. from English - M .: Alpina non-fiction, 2015.— 460s.
2. Vygodsky M.Ya. Handbook of elementary mathematics. - Ed. 27th, Rev.- M .: Science. The main editors of physical and mathematical literature, 1986. - 320s.