The Mathematical Intelligencer

, Volume 17, Issue 3, pp 41–47 | Cite as

Discovery of a lost factoring machine

  • Jeffrey Shallit
  • Hugh C. Williams
  • François Morain


Despite the importance of the work of Eugène Carissan and the remarkable machine he built, very few people were aware of his work. Perhaps this was because both Pierre and Eugène died within 5 years of the exhibition of the machine in Paris in 1920.

Indeed, when D. H. Lehmer began to construct his first automatic sieve in 1926, he had never heard of Carissan’s sieve, and he remained ignorant of it until 1989. Lehmer’s 1926 sieve, which was in some ways much less sophisticated than Carissan’s, made use of bicycle chains driven by an electric motor. It was his first attempt of many. Each time he built a new sieve, the most advanced technology was used: photoelectric cells in 1932,16-mm movie film in 1936, and delay lines in 1965 for the DLS-127 (later DLS-157). For an account of all of this, see [18].

One could be led to think that sieves are old-fashioned and outclassed by modern computers. As a matter of fact, sieve methods remained the fastest known method for factoring integers up to 1970. Sieve devices are still much faster than current software programs. For instance, Williams,et al. (see [16] and [18]) have constructed devices that can perform up to 2 x 1012 trials per second, compared to 2,000,000 on a workstation.

Modern sieve devices are no longer used for factoring purposes, because an approach based on more sophisticated mathematical ideas and massive distribution of programs on workstations is much more powerful (see [15]). We should also point out that another special-purpose factoring machine, which was not a sieve, was constructed by Smith and Wagstaff [17] in 1983.

However, for other special problems, sieve devices still seem to be the only way. For example, Stephens and Williams [18] used their device to find pseudosquares, numbers that behave like squares modulo many small primes. This study could eventually lead to a very fast primality-proving algorithm, if some highly plausible (yet hard) conjecture in number theory were true.

Although sieve machines are not used to factor integers anymore, thebasic principle of sieving remains at the heart of the modern factoring methods such as the quadratic sieve method or the number field sieve algorithm. Thus sieving, which was known to Euclid, is still a powerful method used in everyday life of number-theorists.


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  1. 1.
    Archives Nationales, Carton F/17/25723 (Instruction Public), containing items on Pierre-Eugène-Georges Carissan; Paris, France.Google Scholar
  2. 2.
    E. Carissan, Machine à résoudre les congruences,Bull. Soc. Encouragement Ind. Nat. 132 (1920), 600–607.Google Scholar
  3. 3.
    E. Carissan, Principe mécanique et description de la machine du Ct. E. Carissan. Unpublished manuscript (1920).Google Scholar
  4. 4.
    P. Carissan, Description mécanique de la spirale logarithmique,Nouv. Ann. Math. 17 (1917), 273–277.Google Scholar
  5. 5.
    P. Carissan, Réponse 511,Sphinx-Oedipe 15 (1920), 43–46.Google Scholar
  6. 6.
    P. Carissan, Le tonneau des Danaïdes,Sphinx-Oedipe 18 (1923), 73–75.Google Scholar
  7. 7.
    M. ďOcagne, Vue ďensemble sur les machines à calculer,Bull. Sci. Math. 46 (1922), 102–144.Google Scholar
  8. 8.
    M. ďOcagne,Le Calcul Simplifié par les Procédés Mécaniques et Graphiques, Paris: Gauthier Villars (1928). [English translation in J. Howlett and M. R. Williams, eds.,Le Calcul Simplifié: Graphical and Mechanical Methods for Simplifying Calculation, Cambridge, MA: MIT Press (1986).]Google Scholar
  9. 9.
    A Gérardin, Question et réponse 355,Sphinx-Oedipe 7 (1912), 47–48, 61–64; also see p. 31.Google Scholar
  10. 10.
    A. Gérardin, Rapport sur diverses méthodes de solutions employées en théorie pour la decomposition des nombres en facteurs,Assoc. Française Avan. Sci. Comptes Rendus 41 (1912), 54–57, IIe partie.Google Scholar
  11. 11.
    A. Gérardin, Sur une nouvelle machine algébrique,Br. As- soc. Reports 82 (1912), 405–406.Google Scholar
  12. 12.
    A. Gérardin, Sur quelques nouvelles machines algébriques,Proc. 5th International Congress of Mathematicians, Vol. II (E. W. Hobson and A. E. H. Love, eds.), Cambridge UK: Cambridge University Press (1913), 572–573.Google Scholar
  13. 13.
    W. S. Jevons,The Principles of Science: A Treatise on Logic and Scientific Method, Vol. I. London: Macmillan and Co. (1874).Google Scholar
  14. 14.
    F. W. Lawrence, Factorisation of numbers.Quart. J. Pure Appi. Math. 28 (1896), 285–311. [French translation inSphinx-Oedipe 5 (1910), 98-121.]zbMATHGoogle Scholar
  15. 15.
    A. K. Lenstra and Mark S. Manasse, Factoring by electronic mail,Advances in Cryptology (J.-J. Quisquater, ed.), New York: Springer-Verlag (1990).Google Scholar
  16. 16.
    R. F. Lukes, C. D. Patterson, and H. C. Williams, Numerical sieving devices: their history and some applications,Nieuw Archief voor Wiskunde (in press).Google Scholar
  17. 17.
    J. W. Smith and S. S. Wagstaff, Jr., An extended precision operand computer,Proc. 21st Southeast Region ACMConference (1983), 209–216.Google Scholar
  18. 18.
    A. J. Stephens and H. C. Williams, An open architecture number sieve,Number theory and cryptography (J. H. Loxton, ed.), Cambridge UK: Cambridge University Press (1990), 38–75.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Jeffrey Shallit
    • 1
  • Hugh C. Williams
    • 1
  • François Morain
    • 1
  1. 1.Department of DefenseDélégation Générale pour ľArmementNew york

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