Factoring Poetry

• Below is a poem due to John M. Pollard, called A Tale of Three Sieves, which he e-mailed to me. Everything that follows (i.e. including the notes) is a direct quote from that e-mail.

  Notes. The first verse is not original - I don`t know the author. The title was suggested by that of Pomerance: "A Tale of Two Sieves", Notices of the AMS, December 1996, 1473-1485.

  "Sieve the twos and sieve the threes -
  The sieve of Eratosthenes!
  When the multiples sublime
  The numbers which remain are prime!
  

  Add log p as you advance -
  Such is the sieve of Pomerance!
  The largest sums which you have found
  Give numbers which are smooth and round!
  

  Mark out a gigantic square.
  Add logs at random everywhere.
  This brief account you must forgive
  Of Pollard`s Number Field Sieve!"
  

  More notes for beginners:
  

  • The word "sublime" here means to "evaporate" (strictly, to go from solid to gas), i.e. to go away.
  • The numbers we call "smooth" today were sometimes "round" in older books.
  • Many people have improved the NFS but, as in my mnemonic, "space here precludes a detailed account." You don`t really add the logs at random - that`s poetic licence. I was thinking of the "lattice sieve" variation.

• "I've factored large numbers,
  I've factored small,
  Sooner or later,
  I'll factor them all."
  -- Scott Contini