Please solve questions 7,8 and 9. A lower bound for II(N).In this question we will prove that there are infinitely many primes by examining theprime counting function II(N), whereII(N) = The number of primes between 2 and NWe say that a natural number n 2 2 is square-free if it is not divisible by a square.For example, 6 is square-free, but 8 is not, since it is divisible by 4 = 22.1. Use the fundamental theorem of arithmetic to show that every integer n > 2 canbe written as a product n = 8•r2, where s, r > 1 are integers and s is square-free.We call this the square-free factorization.2. For example, the square-free factorization of 8 = 2· (2)2 (s = 2, r = 2) and of600 = 6 · 102 (s = 6, r = 10.) Find the square-free factorization of 72 and 2366.3. DefineS(N) = The number of squares between 1 and NF(N)== The number of square-free numbers between 1 and NWhat is S(4), S(25) and S(49)? F(6) and F(50)?4. In part 1, we showed that every positive integer factors uniquely as a product ofa square and square-free number. Use this to conclude that N < S(N) · F(N).In the next parts we'll overestimate both S(N) and F(N).5. Show that S(N)< N for every N 1.6. Show that F(N) < 2(N). (Hint: every square free-number smaller than N of theform q1 · q2 · .. · qk where the q; are distinct primes smaller than N.)7. Now using 4, 5, and 6, we have N < VN - 2(N). Get II(N) by itself on theright-hand side of this inequality.8. Take the limit from the solved inequality in 6 to conclude that limN II(N)and hence that there are infinitely many primes.= 09. Using part 6, what's the smallest N guaranteed to have 500 primes smaller thanit?

Name: Please solve questions 7,8 and 9. A lower bound for II(N).In this ques
Uploaded: 2024-03-05T11:53:49.000Z
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Description: Please solve questions 7,8 and 9. A lower bound for II(N).In this question we will prove that there are infinitely many primes by examining theprime counting function II(N), whereII(N) = The number of primes between 2 and NWe say that a natural numbe