Sifting Function Partition by Intervals for the Goldbach Problem
首发时间:2006-04-05
Abstract:The term “Proposition {1, b}” means that every large even number N can be expressed as the sum of a prime and the product of at most b primes. 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. This paper shows that Proposition {1, 1} will be solved if only Proposition {1, b} is proved. In fact, there are a large number of primes in the residual integers of Proposition {1, b} after sifting, of which all those integers not exceeding N^(2/(1+b)) are primes. Therefore, as long as these primes that do not exceed N^(2/(1+b)) can be separated from the residual integers and that their total number can be proved to be greater than zero, Proposition {1, 1} will be true. This idea can be implemented by using a method called “the sifting function partition by intervals”. 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 intervals Selberg’s sieve Goldbach problem Problem of twin primes.
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Goldbach问题的筛函数按区间分割法
摘要:术语“命题{1, b}”意指每一个大偶数N 都可以表示为一个素数与一个至多b个素数的乘积之和. Goldbach问题中的所有筛法都要求筛除所有合数,虽然严格说来这是不必要的,何况筛除所有合数一般都是非常困难的. 本文表明,只要证明了任意命题{1, b},则命题{1, 1}同时得到证明. 事实上,在任意命题{1, b}的筛余整数中都存在大量素数,其中不大于N^(2/(1+b))的整数全部都是素数. 因此,只要能将这些不大于N^(2/(1+b))的素数从筛余整数中分离出来,并证明它们的数目大于零,则就证明了命题{1, 1}. 这一想法可以用“筛函数按区间分割法”实现;这是解决Goldbach问题和孪生素数问题的切实可行的方法. 这种方法的一个额外的收获是由筛函数的上下界所带来的不确定性可以一举消除.
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No.6070210581144200****
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