双重概率筛法与素数分布──关于Goldbach猜想、Hilbert第八问题与余新河猜想的研究  

The Dual Probability Sieve Method and Distribution of Primes──On the Goldbach conjecture，the 8-th problem of Hilbert and YU Xin-he conjecture

作　　者：宋富高[1] 

机构地区：[1]深圳大学理学院,深圳518060

出　　处：《深圳大学学报（理工版）》1999年第2期1-21,共21页Journal of Shenzhen University(Science and Engineering)

摘　　要：用双重概率筛法证明了下列定理：①随机整数x的素数概率；②随机函整数x的素数概率；身随机整数x1，…，xn-1与关联整数xn＝N＋Σεixi同时为素数的概率；④模k的一些简化剩余类中的随机函整数x1，…，xn-1与关联函整数xn＝N＋Σεixi同时为素数的概率（以上εi=±1，1≤i≤n－1）；⑤模a的简化剩余类中的随机函整数x与模b的简化剩余类中的关联函整数y同时为素数的概率由此导出了各个不同领域中素数分布的精确公式，解决了Goldbach猜想、Hilbert第八问题和余新河猜想等一系列重大问题；并预言了在等差级数中的素数分布、孪生素数分布和Goldbach猜想、余新河猜想的解组以及其它场合都存在精细结构所得结果经计算机大范围验证，与实际情况符合良好。The following fundamental theorems have been proved using the Dual Probability Sieve Method： Tyeorem 1 If x is a random integer, then the prime probability of x is Theorem 2 Suppose that f（t） is an irreducible integral valued function， is a random function integer． then the prime probablilty of x is where （p） is the solution number of the congrence f(t) = 0（mod p）；(x) is a function correlative with f(t)：c～e． Theorem3 It x1，…，xn-1 are random integers．and N is a given integer，then the probability that all of x1…，xn are simultaneously primes is Theorem 4 Suppose that N, k, ti，…， tn are given integers, x1，…, Xn-1 are random integers，satisfy then the probabilitythat all of x1,…,xn are simultaneously primes is Theorem 5 Suppose that a， a0 ；， b， b0 ；． k， which satisfy are given integers； t is a random integer，x=at＋ a0， y= b（k＋ t）＋ b0 ； （p） is the number of different solutions of congruences x=0(mod p)and y=0(mod p）； then the probability for x，y simultaneously being primes is From theorems 1 to 5，the exact formulae of prime dlstribution in various vary field can be deduced，a series of important problems，as the eighth problem of Hilbert，the Goldbach comecture and YU Xin－he conjecture，can be solved．For example，the following theorems hold：Theorem 6(prime number theorem) Theorem 7 Ignoring the event of absolute small probability， the number of representations of an even number N as the sum of two odd primes is Fig． 4 and 5 show comparisons of this theorem with fact within range Theorem 8 Suppose that b，t1，t2， are given integers， with （t1，30）＝（t2，30）＝ 1； N＝ 30b＋t1＋ t2； x＝ 30b1＋ t1 and y= 30b2＋ t2 are primes； Ignoring the event of absolute small probability, then the number of representations of integer b as b=b1+b2 is Theorem 9 Let N�

关 键 词：哥德巴赫猜想 双重概率筛法 H问题 素数分布 

分 类 号：O156.4[理学—数学]