From: Fred W. Helenius Subject: Re: Prime numbers Date: Tue, 17 Oct 2000 17:58:19 -0400 Newsgroups: sci.math Summary: Largest prime p for which pi(p) is known J wrote: >What is the largest known prime number whose 'rank' is also known? I guess by "rank" you mean the index of the prime in the sequence of primes; i.e., the rank of the nth prime would be n. It would be hard to be sure of the present answer to your question, since it is easy to extend such a computation a little farther to set a new record. You might be able to find an out-of-date answer in Ribenboim's _The New Book of Prime Number Records_. >And are all primes lower than this number also known? To this the answer is definitely "no". There are methods for ascertaining the exact number of primes up to a given limit that do not require computing all the individual primes. The best known technique is the Meissel-Lehmer formula (with recent improvements by Lagarias, Miller and Odlyzko), which you can read about in Riesel's _Prime Numbers and Computer Methods for Factorization_. In particular, Deleglise and Rivat have found that there are exactly 2220819602560918840 primes up to 10^20, so the rank of the next prime, 10^20 + 39, is one larger. But not all the smaller primes are known; indeed, it would take billions of gigabytes to store them all. A general online source of information about primes is Chris Caldwell's Prime Pages at http://www.utm.edu/research/primes/ . For a web page that computes the nth prime for n up to 10^12, or the number of primes up to n for n up to 3*10^13, see http://www.math.princeton.edu/~arbooker/nthprime.html . -- Fred W. Helenius