🚀 go-pugleaf

          I asked:
:>         How small do the coefficients have to be?  I have been led to
:>believe that the speed of the GNFS overall does not depend much on the
:>efficiency of the search for a good polynomial.  If this is the case it
:>would seem to be unwise to devote a lot of effort to finding a good poly-
:>nomial.  What is the point of this contest?
          Robert D. Silverman replied:
>This is NOT the case. The performance is VERY sensitive to the size
>of the coefficients and it is worthwhile spending a significant amount
>of time trying to find small coefficients.
          and also:
>I estimate that to be competitive with the Quadratic sieve they need
>to find coefficients of 'about' 14 digits.  I am not sure it is
>possible.
          Well, let us estimate how many numbers between 10**128 and
10**129 can be so represented.  m will be about 10**23 (so that c5*
m**5 is 10**129).  There are about 10**23 choices for m, 10**14 for
each of the ci.  This gives 10**23*(10**14)**6 or 10**107 representa-
tions.  Hence the chance that a random number between 10**128 and
10**129 has such a representation is below 10**-20.  It would appear
to me that running ECM would be a better bet than this.
                          James B. Shearer