Explain why we may assume a and b are both positive in the third proof of Proposition XXXXXXXXXXProve that log 2 3 is irrational. 1. Sieve of Erasothenes. Here is a method for finding many prime...

Explain why we may assume a and b are both positive in the third proof of Proposition 39.6. 39.22. Prove that log₂
3 is irrational.

1. Sieve of Erasothenes. Here is a method for finding many prime numbers. Write down all the numbers from 2 to, say, 1000. Notice that the smallest number on this list (2) is a prime. Cross off all multiples of 2 (except 2). The next smallest number on the list is a prime (3). Cross off all multiples of 3 (except 3 itself). The next number on the list is 4, but it’s crossed off. The next smallest number on the list that isn’t crossed off is 5. Cross off all multiples of 5 (except 5 itself).

a. Prove that this algorithm crosses off all composite numbers on the list but retains all the primes.

b. Implement this algorithm on a computer.

c. Let
 denote the number of primes that are less than or equal to n. For example, with the number