Chapter 6.3, Problem 74E

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

# A prime number is a positive integer that has no factors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17,... We denote by π ( n ) the number of primes that are less than or equal to n. For instance, π ( 15 )   =   6 because there are six primes smaller than 15.(a) Calculate the numbers π ( 25 ) ) and π (100).[Hint: To find π (100), first compile a list of the primes up to 100 using the sieve of Eratosthenes: Write the numbers from 2 to 100 and cross out all multiples of 2. Then cross out all multiples of 3. The next remaining number is 5, so cross out all remaining multiples of it, and so on.](b) By inspecting tables of prime numbers and tables of logarithms, the great mathematician K. F. Gauss made the guess in 1792 (when he was 15) that the number of primes up to n is approximately n/In n when n is large. More precisely, he conjectured that lim n → ∞ π ( n ) n / In  n = 1 This was finally proved, a hundred years later, by Jacques Hadamard and Charles de la Vallée Poussin and is called the Prime Number Theorem. Provide evidence for the truth of this theorem by computing the ratio of π ( n ) to n/In n for n = 100 , 1000, 104, 105, 106, and 107. Use the following data: π ( 1000 ) = 168 π ( 10 4 ) = 1229 ,   π ( 10 5 ) = 9592 ,   π ( 10 6 ) = 78 , 498 ,   π ( 10 7 ) = 664 , 579. (c) Use the Prime Number Theorem to estimate the number of primes up to a billion.

(a)

To determine

To find:

We have to find the total number of primes less than or equal to 25 and 100 i.e. we have to find

π25,π(100).

Explanation

Calculation:

To find π25, we have to find all prime numbers less than or equal to 25.

Prime numbers less than 25 are 2,3,5,7,11,13,17,19,23.

So, π25=9.

Now, we find prime numbers less than or equal to 100.

Prime numbers less than 100 are

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,57,59,61,67,83,87,89,91,97

(b)

To determine

To show: We have to prove the prime number theorem by computing the ratio of π(n) to nlnn for n=100,1000,10,000,100,000,1,000,000,10,000,000

(c)

To determine

To find:

We have to estimate the number of primes less than to a billion using the prime number theorem.

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