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

36 Bui Minh Phong, Li Dongdong 3. Proof of the theorem Proof of (a). Assume that every even integer In > 4 is the sum of two odd primes. In this case we infer from Lemma 2 that S(k

) > pk+i +3. Thus, Conjecture A implies Conjecture B. Now we assume that Conjecture B is true, that is (1) holds for every positive integer k. Hence, Lemma 2 shows that (2) {2

n: 6 < 2

n < S{k + 1) } C G holds for all positive integers k. Finally, let 2n > 4 be any even integer. It is clear to see from the definition of S{k) that S(k) > pk. Hence S(k) —» oo as k —

> oo. Consequently, S(£) > 2n is true for some positive integer t, and so we get from (2) that 2n E G. The proof of the the part (a) of the theorem is completed. Proof of (b). It is obvious that Conjecture C is a consequence of Conjecture A. Assume now that the conjecture C is true, that is, for each positive integer k, we have S(k) = p + q for for some primes p and q. Since the numbers Sk — p and S(k) — q are primes, we also have p > Pk and q > p^. Consequently, S(k) = p + q > 2p

k > pk +

1 + 1, and so Conjecture B is true. This with (a) completes the proof of (b). The assertion (b) is proved. The proof of the theorem is finished. References [1] DESHOULLIERS, J. M., TE RIELE, II. J. J., SAOUTER, Y., New experimental results concerning the Goldbach conjecture, Proc. 3rd Int. S'ymp. on Algorithmic Number Theory, LNCS 1423 (1998), 204-215. [2] RAMAR, U, O., On Schnirelman

's constant, Ann. Scuola Norm. Sup. Pisa CI. Sei. 22:4 (1995) 645-706. [3] RIBENBOIM, P., The New Book of Prime Number Records, SpringerVerlag, New York, 1995. [4] RICHSTEIN, J., Verifying the Goldbach Conjecture up to 4 • 10

1

4, to appear in Mathematics of Computation.