LLLGCD examples

In our paper Extended gcd and Hermite normal form algorithms via lattice basis reduction, G. Havas, B.S. Majewski, K.R. Matthews, Experimental Mathematics, Vol 7 (1998) 125-136, we show that if alpha >= 3/8, our LLLGCD algorithm delivers a multiplier vector b[3] for 3 numbers such that one of b[3]+e[1]b[1]+e[2]b[2] is a shortest multiplier, where e[i]=0,-1, or 1. A similar result also holds for four integers (proved by Sean Byrnes), but not for 5 integers - see below. However something similar seems to hold in general with alpha = 1, as examples 3-6 below reveal.
  1. gcd(4,6,9) (20th July 1997) (lllgcd step by step)
  2. gcd(113192,763836,1066557,1785102) (algorithm2 of HMM, step by step)
  3. gcd(116085838,181081878,314252913,10346840) (lllgcd step by step)
  4. gcd of 5 numbers (14th December 1998)
  5. gcd of 10 numbers (17th December 1998)
  6. gcd of 20 numbers (17th December 1998)
  7. gcd of 30,40,50,60 integers (9th December 1998)

Last modified 5th August 2007