Explanation The Sieve of Eratosthenes is used to find all the primes not exceeding a specified positive integer and we can use the principle of inclusion-exclusion with the same reasoning to find the number of primes not exceeding the positive integer n. A composite number must be divisible by a prime number not exceeding its square root. Thus, the integers not exceeding n which are not prime must have at least one prime factor not exceeding ⌊ n ⌋ . Assume the prime numbers not exceeding ⌊ n ⌋ are p 1 , p 2 , p 3 , ...... , p k . Then the composite number not exceeding n must be divisible by at least one of them. Let P i be the property that a positive integer is divisible by p i for i = 1 , 2 , 3 , ... , k . So the number of composite integers not exceeding n is N ( P i ) = ⌊ n p i ⌋ − 1 and it is divisible by p i . Similarly, the number of composite integers in this range which is divisible by a subset of those primes is N ( P i 1 P i 2 .... P i r ) = ⌊ n ∏ 1 ≤ j ≤ r p i j ⌋ and it’s divisible by p i 1 , p i 2 ,

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Author: ROSEN

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Chapter 8, Problem 15RQ

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To explain how the inclusion-exclusion principle can be used to count the number of primes not exceeding the positive integer n.

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To explain how the inclusion-exclusion principle can be used to count the number of primes not exceeding the positive integer n .

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To explain how the inclusion-exclusion principle can be used to count the number of primes not exceeding the positive integer n.

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