Use the estimate in the text based on the Prime Number Theorem to give approximate values of the following. (a) The number of primes between 1 and 1030. (b) The number of primes between 1 and 1029.

a) 148.47

b) 148.34

Step-by-step explanation:

Let π (x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, π (7) = 4 because there are four prime numbers (2, 3, 5, 7) less than or equal to 7.

We can see that π (7) = π (10) because there are four prime numbers (2,3,5,7) less than or equal to 10.

A good aproximation for π (x) is x / log x, where log x is the natural logarithm of x.

a) The number of primes between 1 and 1030.

1030/log (1030) = 148.47

b) The number of primes between 1 and 1029.

1029/log (1029) = 148.34

Extra: π (1029) = π (1030) because 1030 is not a prime.