From: rgep@pmms.cam.ac.uk (Richard Pinch)
Newsgroups: sci.math
Subject: Re: Primes in the Decimal Pi
Date: 28 Feb 1995 10:16:27 GMT
Keywords: Maple; primality test; Morain; ECPP

In article <9502260032591.kevin2003.DLITE@delphi.com>, 
kevin2003@delphi.com (Kevin Brown) writes:
|> KB>  For which integers k do the first k decimal digits of pi give a 
|> KB>  prime?  All I've been able to find are k = 1, 2, 6, and 38, which
|> KB>  correspond to the primes
|> KB>
|> KB>     3    31    314159    31415926535897932384626433832795028841
|> KB>
|> KB>  but I haven't checked any further.  Does anyone know the next prime
|> KB>  in this sequence? 
|>  
|> MM = Mark McClure
|> MM>  I had Maple check primality for k = 1 to 500 and found no more 
|> MM>  primes.  
|> 
|> I suppose Maple is checking for "probable primality" rather than 
|> actually proving primality.  (As recently as 1988 Paulo Ribenboim's 
|> 'Book of Prime Number Records' said the state of the art in primality
|> proving was 213 digit numbers in about 10 minutes, so proving primality 
|> of 500 digit numbers must be fairly challenging.)

Yes, Maple V.2 uses a rather weak probabilistic test (try 2152302898747 = 
6763*10627*29947).  V.3 uses a much better one.  Morain's Elliptic Curve 
Primality Prover can now routinely prove primality of numbers well over 
1000 digits.

A review of some primality testing routines is available from
ftp://emu.pmms.cam.ac.uk/pub/rgep/Primality/latest.tex

Richard Pinch;	Queens' College, Cambridge