Divisor Goldbach Conjecture and its Partition Number

Abstract

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on the divisible factor of this integer, that is: For any positive integer $n~(n>2)$, if there exists an integer $m$, such that $m|n~( 1 < m < n )$, then $n=\sum_{i=1}^m p_i $, where $ p_i~(i=1,2 ,3...m)$ is prime number. In addition, for more prime summands, the combinatorial counting is also discussed. For some special cases, some brief proofs are given. By the use of computer, the preliminary numerical verification was given, there is no an anti-example to be found.