Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

Bui MINH PHONG and LI DONGDONG, Elementary problems which are equivalent to the Goldbach's conjecture

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33 37 ELEMENTARY PROBLEMS WHICH ARE EQUIVALENT TO THE GOLDBACH'S CONJECTURE Bui Minh Phong and Li Dongdong (Budapest, Hungary) Abstract. We denote by {pi—2, P2=3, p

3 = 5,..., p*,.--) the sequence of increasing primes, and for each positive integer k>l let S(fc):=:min{2n>j>fc: 2n-p

u 2n-p

2,..., 2n-p

k all are composite numbers}. We prove that the following conjectures are equivalent to the Goldbach's conjecture. Conjecture B. For every positive integer k, we have S(/c) > p

fe + 1 + 3. Conjecture C. For every positive integer k, the number S(k) is the sum of two odd primes. 1. Introduction Goldbach wrote a letter to Euler in 1742 suggesting that every integer n > 5 is the sum of three primes. Euler replied that this is equivalent to the following statement: Conjecture A. Every even integer 2n > 4 is the sum of two odd primes. This is now known as Goldbach's conjecture. A. Schinzel showed that Goldbach's conjecture is equivalent to every integer n > 17 is the sum of three distinct primes. It has been proven that every even integer is the sum of at most six primes [2] (Goldbach suggests two) and in 1966 Chen proved every sufficiently large even integers is the sum of a prime plus a number with no more than two prime factors. In 1993 Sinisalo [5] verified Goldbach's conjecture for all integers less than 4- 10

J

I. More recently Jean-Marc Deshouillers, Yannick Saouter and Herman te Riele [1] have verified this up to 10

1

4 with the help of a Cray C9Ü and various workstations. In July 1998, Joerg Richstein [4] completed a verification to 4 • 10

1

4 and placed a üst of champions online. See the monografof P. Ribenboim [3] for more information. In the following, we shall denote by V the set of all increasing primes, that is V ~ {

P

l = 2, p2 = 3, ps = 5,..., pk, ...}•