The Sieve of Eratosthenes and the Theorem of Goldbach

§1. The theorem of Goldbach is well-known that one can write every even number as a sum of two prime numbers. In a letter of 1742, Euler has written: “I believe it is a completely acceptable theorem, although I cannot prove it.” This theorem has still not been proved, and it is the same about the following theorem: The sequence of the twin prime numbers¹⁾ is infinite. In an address delivered at the International Congress of Mathematics, Cambridge, 1912, E. Landau had said that he regarded these problems as “unattainable problems in modern science.”…