The number of primes equal to or less than a number x is denoted by the function `π(x)`. There is no easy way to compute this function although there are ways to compute a list of primes. Use the SieveOfEratosthenes project to generate a file of prime numbers less than 1,000,000. Then write a function `π(x)` that searches the file for the answer. The function should return the number of primes read before a value greater than x is found. To improve efficiency for subsequent calls to `π(x)` you should only read the file the first time the function is called and store the prime values in an array. This way subsequent calls to `π(x)` don't need to re-read the file. Write a test program to print the values of `π(x)` for x that are powers of 2 up to `2^19`. Will this approach work if you want to compute values for x up to 1,000,000,000 by generating an approprite file of primes? Modify the program to use an estimate `x/(ln x - 1)` for values greater than are found in the primes data file. Test by computing `π(x)` up to `2^30`.

See [[How Many Primes Are There?]]
See SieveOfEratosthenes