Sifting Function Partition by Integer Sort for the Goldbach Problem
首发时间:2006-04-04
Abstract:All sieve methods for the Goldbach problem sift out all the composite numbers; even though, strictly speaking, it is not necessary to do so. Furthermore, the sifting out of all composite numbers is, in general, very difficult. A new method introduced in this paper shows that the Goldbach problem can be solved by using a combination of algebraic techniques and existing results from sieve theory under sifting out only some composite numbers. In fact, in order to prove the Goldbach conjecture, it is only necessary to show that there are prime numbers left in the residual integers after the initial sifting! This idea can be implemented by using a method called “sifting function partition by integer sort”. Under this scheme, the integers are first classified according to their number of prime divisors; the sifting functions are then partitioned according to integer sort; the total number of integers in each class (containing the class of prime) is then determined by using the method of “separation of variables”; and the Goldbach conjecture is proved. This is a feasible method for solving both the Goldbach problem and the problem of twin primes. An added bonus of the above scheme is the elimination of the indeterminacy of the sifting functions brought about by their upper and lower bounds.
keywords: Sifting function partition by integer sort Selberg’s sieve Goldbach problem Problem of twin primes.
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Goldbach问题的筛函数按整数类别分割法
摘要:Goldbach问题中的所有筛法都要求筛除所有合数,虽然严格说那是不必要的,何况筛除所有合数一般是非常困难的. 本文引入的一种新的方法表明Goldbach问题可以在只筛除部分合数的情况下,利用现有筛法结果结合代数方法加以解决. 事实上,为了证明Goldbach猜想,只需证明在筛余整数中存在许多素数就足够了. 这种想法可以用一种称为“筛函数按整数类别分割法”实现. 在这一方案中,首先将整数按素因子的个数进行分类,然后将筛函数按整数类别分割;筛函数中各类整数(包括素数)的总数则可以用“分离变量法”确定,由此即可证明Goldbach猜想. 这是解决Goldbach问题和孪生素数问题的切实可行的方法. 这种方法的额外收获是:由上下限所带来的筛函数的不确定性得以消除.
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