Quote:
Originally Posted by R.D. Silverman
<snip>
Greg may indeed do R323 before he does 12^3191. I think he will. R323 might well be
done by a reciprocal octic to take advantage of the algebraic factor 10^191. Whether
the octic would be easier than the obvious sextic might be an interesting experiment.
It might also be interesting to see if a septic would be any better. I think a septic
will be slightly better in general for numbers of this size.
Let's do a "back of the envelope" look at the norms. Take (10^6, 10^6) == (a,b) as a
'typical lattice point'.
For a sextic, an algebraic norm is ~ a^6 ~ 10^36 and a
linear norm is ~ b * (10^324/6) ~ 10^60. For a septic an anorm is ~a^7 ~ 10^42
and a linear norm is b *(10^322/7) ~ 10^52. The norms are closer for the
septic and their product is slightly smaller. A septic seems slightly superior.
For the reciprocal octic an anorm is a^8 ~ 10^48 and a linear norm is b * (10^38) ~ 10^44 which seems even better still.
.

I'd like to hear ideas from others about what I wrote just above. It seems that
a degree 7 polynomial would be better (than degree 6) for Greg to use moving forward
for numbers that NFS@Home is about to undertake.