A Study of Goldbach's Conjecture That Every Even Number Greater Than 4 Can Be Expressed as the Sum of Two Odd Primes
One of the most famous unsolved problems in number theory is Goldbach's conjecture that every even number greater than 4 can be expressed as the sum of two odd Primes. At first this seems to be a very simple problem; nevertheless no one has proved it yet. Goldbach conjectured this in a letter to Euler dating from about 1742. Euler, whose mathematical intuition was acute, was also convinced of the truth of this proposition, but he couldn't find a proof. Since Euler couldn't find an attacking point, this problem was expected to be extremely difficult. Even today, some 200 years later nobody has proved Goldbach's conjecture. It is easy to verify the conjecture for small numbers. As the numbers get larger, some of them can be expressed as the sum of two odd primes in more than one way. Although Goldbach's conjecture was verified for even numbers up to 100,000 by Nils Pipping in the 1920s, it still remains to be seen if it is actually true for all even numbers greater than 4. In fact very little progress was made on the subject until the twentieth century. There have been partial results mainly of two types: theorems saying that every sufficiently large even number is the sum of at most k primes (proving that one can take k = 2 would prove Goldbach's conjecture); and theorems saying that every sufficiently large even number is the sum of two "nearprimes", numbers with a limited number of prime factors. The best result in the first direction was obtained in 1937 by I. Vinogradov who actually proved that every sufficiently large odd number can be expressed as the sum of at most 3 Primes. The result for even numbers follows from this, since if n is even, then n-3 is odd. The best results in the other direction are related to the name of Viggo Brun who in 1919 proved that every sufficiently large even number can be written as the sum of two integers, each of which is the product of at most k = 9 primes (prime is counted as a product according to its exponent; for example 12 = 2*2*3, is considered as the product of three primes). By 1940 A. Buchstab reduced k to 4. In 1948 A. Renyi showed that there exists a number k such that every large even number is the sum of a prime and an integer having at most k prime factors. The Chinese mathematician Jing-Run Chen proved Renyi's theorem for k = 2. This proof was announced in 1966 but was not published until 1973. If 2 could be replaced by 1, Goldbach's conjecture would be proved. The following paper presents: 1) Nils Pipping's method of verifying Goldbach's conjecture for all even numbers up to 100,000; 2) Pipping's method extended for all even numbers up to 290,000 by means of using a computer program and some empirical results; 3) the so called Goldbach curve - the number of distinct solutions of the equation 2n = p + q, where p and q are primes; 4) Viggo Brun's sieve method which he used to prove that every sufficiently large even number can be written as the sum of two integers, each of which is the product of at most 9 primes.
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