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1、Primes with product distributionLianzhongLi (CampHill MiddleSchool inSichuan Camp Hill 637700) Abstract:The positiveintegern between twoadjacent primenumberssquare,number-ssquare, the well-established numberof even product distribut
2、ion formula (S)and formula (L)(S)π(n)=（）（L）π(n)=（）Keywords: numbertheory; primenumber; formulaCLCnumber: 015 Documentcode: ArticleID:Primes with product distribution theorem:If,，……,, forthesecondconsecutive primes ,, isnot greaterthan n numberofprimenumbersfor
3、mula for（S）π(n)=（）or（L）π(n)=（）Proband lemma.Lemma1:If,，……, , consecutive primes, and
4、 n, thenthe n-≠ o (mod)Thenumberofvalues。Prove: I.The i= 1,∵=2,
5、n∴The conclusion wasestablished.II ,Suppose i=k,theconclusion was founded,namely:Wasestablished.11When i = k+1,
6、∵
7、n，
8、n，…，
9、n，According to theinductionhypothesis∴∵
10、n∴ n ≠ o(mod) thenumber is.They is forof 1,2,3,..., Thisa multiples. Thenumberof multiples of the remove ,accordingto theinductionhypothesis also remaining∴∴ i = k+1,theconclus
11、ionWasestablished.ByI,II available, the conclusions aretrue when iis anypositiveinteger.Lemma1 isproved.Lemma2：if，so≤≤0.75Prove：let∴11∴=Accordingto Mertens Theorem 3∴===∴ is to reduce fluctuations,volatility isalsoreduced.∵==∴≤≤=0.75Thatis≤≤0.75Lemma2 isproved.
12、 Thefollowing proofof the primenumber with theproduct distributiontheoremPrimenumber with theproduct distributionof the theorem: If ,，……,, forthesecondconsecutive primes ，，the numberofprimenumbers formula for(S) π(n)=（） Or (L), π (n)=（）Prove：∵n＝3+(
13、8－3)+(24－8)+(48－24)+…++…+11∴ Accordingto Lemma1, the interval[), the numberofprimenumbers canbeapproximately expressedas-Betweenthe numberof remove ,，……,multiplesof thefewremainingnumber greaterthan . Thisisnotbecause p ∤ n causedby, but becausewhen =>, the num
14、berof- no inmultiples,so getridof the ,，…… multiplesof theremaining numberof , the number of multiplesof notBut Thisisnot pis divisibleby n,. This isann by lim
