Counting Primes

January 7, 2011

We celebrate our 200th exercise by implementing the prime-counting function, which has long been of interest to mathematicians. Generally denoted by the letter π, the prime-counting function π(n) returns the number of primes less than or equal to n; for instance, π(100) = 25. A related function, generally denoted as Pn, returns the nth prime number; thus, π(Pn) = n.

There are various analytic methods to compute the prime-counting function; the method invented by Ernst Meissel, improved by Derrick Henry Lehmer, and automated by Andrew Odlyzko is the best-known of them. But the obvious brute-force method of generating and counting the primes is hard to beat if you pre-compute a list of starting points, then use a segmented sieve to interpolate.

Your task is to write the prime-counting function and the nth-prime function. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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