User:Mykhaylo

Mathematics, theory of numbers, Wetzlar, Germany, pensioner, e-mail: michusid@meta.ua Mykhaylo Khusid Binary Goldbach-Euler problem Abstract: The Goldbach-Euler binary problem is formulated as follows: Any even number, starting from 4, can be represented as the sum of two primes. the real problem has not been solved for more than two centuries, therefore, it has a colossal history and belongs to one of the unresolved tasks in the list of age-old. The ternary Goldbach problem is formulated as follows: Every odd number greater than 7 can be represented as the sum of three odd primes, which was finally solved in 2013. The author carries out the proof by the methods of elementary number theory. Keywords: solving, problems, in ,number, theory 1. Introduction, literature review and scope of work. In 1742, the Prussian mathematician Christian Goldbach sent a letter to Leonard Euler, in which he made the following conjecture: Every odd number greater than 5 can be represented as the sum of three prime numbers. Euler became interested in the problem and put forward a stronger conjecture: Every even number greater than two can be represented as the sum of two prime numbers. The first statement is called the ternary Goldbach problem, the second the binary Goldbach problem (or Euler problem). As of July 2008, Goldbach's binary conjecture has been tested for all even numbers not exceeding 1.2×1018[2]. The binary Goldbach conjecture can be reformulated as statement about the unsolvability of a Diophantine equation of the 4th degree some special kind.[3][4] 2. Content (main part) 2.1.Lemma1. Any even number starting from 12 is representable as a sum four odd prime numbers. 1. For the first even number 12 = 3+3+3+3. We allow justice for the previous N> 5: p1 + p2 + p3 + p4 =2N (1) We will add to both parts on 1 p1 + p2 + p3 + p4 + 1 = 2N + 1 (2) where on the right the odd number also agrees p1 + p2 + p3 + p4 + 1 = p5 + p6 + p7 (3) Having added to both parts still on 1 p1 + p2 + p3 + p4 + 2 = p5 + p6 + p7 + 1 (4) We will unite p6 + p7 + 1 again we have some odd number, which according to(1)we replace with the sum of three simple and as a result we receive: p1 + p2 + p3 + p4 + 2 = p5 + p6 + p7 + p8 (5) at the left the following even number is relative (1), and on the right the sum four prime numbers. p1 + p2 + p3 + p4 = 2N (6) Thus obvious performance of an inductive mathematical method. As was to be shown. Corollary: One of the four simple odds can be set arbitrarily and it should not be more than 2N-9. This follows from the finally solved Goldbach's ternary problem. 2.2.Lemma 2. The difference between the sum of two odd prime numbers and an odd prime number as well as the difference between the odd prime number and the sum of two -any odd numbers. Proof. It is necessary to prove: p1+p2−p3=2K+1, p1−p2−p3 =2K+1 (7) where K=1,2... ∞ According to the method of mathematical induction: p1+p2−p3+2= p4+p5−p6 (8) p1+p2+ p6+2= p4+p5+p3 (9) and p1−p2−p3+2= p4−p5−p6 (10) p1+p5+p6+2=p4+ p2+p3 (11) And according to the finally solved Goldbach's ternary problem: where: 2K1+3=2K2+1 (12) K2=K1+1 (13) Note: Since in the ternary problem odd numbers start with 9, then for 3, 5, and 7 where K =1,2,3 we confirm arithmetically. Q.E.D. 2.3.Theorem. The sum of four odd prime is equivalent to the sum of two prime. Let's show that starting from 12 p1+p2+ p3+ p4=p5+p6 (14) for the first even (14) 3+3+3+3=5+7 p7+p8+ p9+ p10+2= p11+p12 (15) p7+p8− p11+2=p12−p9−p10 (16) that according to Lemma2.2. 2K1+1+2=2K2+1 (17) and based on lemma2.1. we reach(14). 2.4. Binary Goldbach-Euler problem. Any even number, starting from 4, can be represented as the sum of two primes. 2+2 =4 2+2+2=3+3=6 2+2+2+2=3+5=8 2+2+3+3=3+7=5+5=10 And then in 2.3. As result, the assumption about the existence of exceptional even numbers is incorrect and the solution to Goldbach's binary problem. 3. Conclusion. the solution of this problem opens up opportunities for solving a number of problems in number theory. Literature [1]Int[Log[10,3^(3^15)]] - Wolfram|Alpha [2] Weisstein, Eric W. Goldbach Conjecture at Wolfram MathWorld. [3] Yuri Matiyasevich. Hilbert's Tenth Problem: What was done and what is t