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1、Primenumber with theproduct inequalityLianzhongLi(CampHill Middle School Sichuan CampHill637700)Abstract:Positiveintegern is between twoadjacent primenumbers square, the well-established numberof product even distributionof the formula (S)andformula (L),According to the primenumbertheorem, Mert
2、ens,Theor- em 3 launchofprimenumbers with theproductof inequality and Corollary 1,2.Keywords: Numbertheory; primenumber; inequalityCLCnumber: 015 Documentcode: ArticleID:Theorem: primenumber with theproductof inequality:Proband lemma.Lemma1:If,，……, , consecutive primes, and
3、 n, thenthe n-≠ o (m
4、od)Thenumberofvalues。Prove: I.The i= 1,∵=2,
5、n∴The conclusion wasestablished.II ,Suppose i=k,theconclusion was founded,namely:Wasestablished.When i = k+1,∵
6、n，
7、n，…，
8、n，According to theinductionhypothesis∴∵
9、n∴ n ≠ o(mod) thenumber is.They is forof 1,2,3,..., Thisa m
10、ultiples. Thenumberof multiples of the remove ,accordingto theinductionhypothesis also remaining∴∴ i = k+1,theconclusionWasestablished.ByI,II available, the conclusions aretrue when iis anypositiveinteger.Lemma1 isproved.Lemma2：if，so≤≤0.75Prove：let∴∴=Accordingto Mertens Theorem 3∴===∴ is to red
11、uce fluctuations,volatility isalsoreduced.∵==∴≤≤=0.75Thatis≤≤0.75Lemma2 isproved.Lemma3(Primenumber with theproduct distributionof the theorem):If ,，……,, forthesecondconsecutive primes ，，the numberofprimenumbers formula for(S) π(n)=（） Or (L), π (n)=（）Prove：∵n＝3+(8－3)+(24－8)+(48－24)+…++
12、…+∴ Accordingto Lemma1, the interval[), the numberofprimenumbers canbeapproximately expressedas-Betweenthe numberof remove ,，……,multiplesof thefewremainingnumber greaterthan . Thisisnotbecause p ∤ n causedby, but becausewhen =>, the numberof- no inmultiples,so getridof the ,，…… multiplesof ther
13、emaining numberof , the number of multiplesof notBut Thisisnot pis divisibleby n,. This isann by limit, - didnot reach thenumber of ... thescope of the multiples. Multiples of the front toproveLemma 1,removethe
