Richard Brent - Research Interests (1)

Computational number theory

• integer factorisation algorithms
  • Pollard rho method
    • complete factorisation of the Fermat number F8
  • elliptic curve method (ECM)
    • large factors found by ECM
      • the old list (1991-1998), details;
      • the lists for 2004, 2003, 2002, 2001, 2000, 1999, 1998, 1997, and 1996;
      • some graphs and extrapolations
    • complete factorisation of the Fermat numbers F10 and F11
    • use of special-purpose hardware (jointly with Harvey Dubner)
      • new 27-33 digit factors of F13, F15 and F16
        (jointly with Richard Crandall, Karl Dilcher and Chris Van Halewyn)
    • the ECMNET project
  • number field sieve (NFS) (jointly with Brian Murphy)
    • application to security and cryptography
      • factorisation of a 140-digit number RSA140 by NFS
      • factorisation of a 512-bit number RSA155 by NFS
      • some graphs and extrapolations from my recent survey
    • the NFSNET project
  • factorization on an IBM PC
    • description (Tech Report TR-CS-89-23)
    • programs and factor database (self-extracting IBM PC archive)
  • extension of the Cunningham tables to bases 13..99
    • tables (June 1992, 363 pages, and updates),
  • Scott Contini's FactorWorld page
• primality testing
  • here is a talk on the AKS, ECPP and Rabin-Miller algorithms
• perfect numbers (jointly with Graeme Cohen and Herman te Riele)
  • proof that there is no odd perfect number less than 10³⁰⁰
• distribution of primes and twin primes
  • for a recent extension by Robert Harley press here and here
  • see also the recent paper on "jumping champions" by Odlyzko, Rubinstein and Wolf
  • for my review of a paper by Thomas Nicely on Brun's constant, twin primes, the Pentium bug etc, press here
  • large gaps between successive primes
  • small gaps between successive primes
• Riemann Hypothesis (RH) and its generalisations
  • computation of zeros of L-functions
  • conditions equivalent to RH (jointly with Herman te Riele)
• other people working on number theory

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