N^2+1形式素数个数的估计  

Estimation of the Number of Prime Numbers with N^2+1 Form

作　　者：吴新生[1] 

机构地区：[1]信息产业部电子第十六研究所,合肥230061

出　　处：《安徽广播电视大学学报》2006年第1期121-125,共5页Journal of Anhui Radio & TV University

摘　　要：本文发现Landau集合L={n2+1|n∈N+}中的关键元素为4n+1形式的奇数。由于(2k)2=4k2,因此,一个偶数的平方加上1可能是一个4n+1形式的奇合数,也可能是一个4n+1形式的奇素数。这是一个随机事件。当把这些奇数当作来自奇数集G={1,3,5,…}中的随机样本时,可以证明集合L中素数有无穷多个。为了估计区间[1,x]内n2+1形式的素数个数,利用M=x2-1,P(L)=23,P(G)~ln2x而建立一个随机抽样的数学模型;πn(2+x1;=4P,1≤x)~32lnxx-1,x→∞。至此,Landau猜想已被证明是一个肯定的结果。本文同时用新的方法获得了4n+1形式的素数个数的估计式。The critical elements in Landau＇s set L={n^2＋1｜n∈N^＋} are found to be the odd integers with 4n＋1 form in the paper. Because （2k）^2 =4k^2 ,so the square of an even integer adding one is probably an odd composite integer or an odd prime number. This is a random event. When these odd integers are regarded as random samples from set G= { 1,3,5,…}, the number of prime numbers in the set L can be proved to be infi- nite. A mathematical model of random sampling is set for estimating the number of prime numbers with n^2 ＋ 1 form by M=√x-1/2,P（G）-2/lnx and P（L）=3/2,get π（x;4,1）-3√x-1/2lnx,x→∞ n^2＋1=P≤x. So,Landau＇s conjecture is proved to be an affirmative result. The estimating formula for the number of prime numbers with 4n ＋ 1 form is obtained by the new method at the same time.

关 键 词：Landau猜想 素数定理 随机抽样 数学模型 

分 类 号：O222[理学—运筹学与控制论] O156.3[理学—数学]