On the Connections between Goldbach Conjecture and Prime Number Theorem

Abstract

By our suggested definition, an even number L n is called the largest strong Goldbach number generated by the n -th prime P n if every even number from 4 to L n is the sum of two primes not greater than P n but L n + 2 is not such a sum. We discovered the existence of step-type distribution for L n arising from observed fact that L n ≤ L n +1 and we proved that L n ≤ L n +1 for all n > 0. Every such step is called a Goldbach step whose width is ( n 2 + 1) – n 1, where n 1 is the starting point and n 2 is the finishing point for the step. We proved that if Goldbach conjecture is true then there are infinitely many Goldbach steps. It is expected that distribution of Goldbach steps is asymptotically expressed as Q ( n ) ~ n /log n same as prime number theorem, where Q ( n ) is the number of Goldbach steps. It means Goldbach steps have like-prime nature, thus, all n 1 can be called like-primes and g i = n 1-( i + 1) – n 1- i is defined as gap between the i -th and the ( i + 1)-th like-primes. We proved that if there are infinitely many like-prime gaps whose length k is uncertain but bounded by a finite integer N > 1, then Goldbach conjecture is true. Considering k = 1 for twin like-primes, it is conjectured that there are infinitely many like-primes n 1 such that n 1 + 1 is also like-prime to imply Goldbach conjecture and it is expected that distribution of twin like-primes is asymptotically expressed as Q 2 ( n ) ~ 2 C 2 n /(log n ) 2 akin to prime number theorem and same as a special case of the first Hardy-Littlewood conjecture, where Q 2 ( n ) is the number of twin like-primes and C 2 is twin prime constant. We also studied distributions of triplet like-primes and quadruplet like-primes to imply Goldbach conjecture. We presented there are bounds of L n /2 such that n log n + n loglog n – n < L n /2 < n log n + n loglog n for n ≥ 20542, and in this paper, the bounds have been verified up to n = 4000000000. If it can be proven that bounds of prime, n log n + n loglog n – n < P n < n log n + n loglog n for n ≥ 6, can be used as bounds of L n /2 for n ≥ 20542, then Goldbach conjecture is true. Further, we proved that if there is a bounded integer k > 20541 such that bounds of prime can be used as bounds of L n /2 for n ≥ k then Goldbach conjecture is true, where bounded integer k > 20541 means value of k is uncertain but there exists upper bound N > 20542 for k .