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

34 Bui Minh Phong, Li Dongdong For each positive integer k > 1, let Ak '•= {2n > pk'. 2n — p\, In — p2

,..., 2n — pk all are composite numbers}. Since pi • • - pk G C N, therefore has a minimum element. Let S(k) min AkWe shall prove that the following conjectures are equivalent to Conjecture A. Conjecture B. For every positive integer k, we have S{k) > pk+i 4- 3. Conjecture C. For every positive integer k, the number S(k) is the sum of two odd primes. The purpose of this note is to prove the following Theorem. We have (a) Every even integer 2 n > 4 is the sum of two odd primes if and only if (1) S(k) > Pk+

1 + 3. holds for every positive integer k. (b) Every even integer 2n > 4 is the sum of two odd primes if and only if the number S(k

) is the sum of two odd primes for all positive integers k. In the other words, Conjectures A, B and C are equivalent. 2. Lemmas In the following we denote by G the set of all even positive integers which are the sums of two odd primes. Goldbach's conjecture states that G contains all even integers 2n > 6. Lemma 1. We have { 2n: 6 < 2n <p

k + 3 } C G if and only if {2«: 6 <2 n < S{k)} C G. Proof. It follows from the definition of S(k) that S(k) > pk -f 9, consequently {2n: 6 < 2n < p

k + 3} C G if {2n: 6 < 2n < S{k)} C G.