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2020-06-28, 19:57   #180
R.D. Silverman

Nov 2003

26×113 Posts

Quote:
 Originally Posted by R.D. Silverman Greg may indeed do R323 before he does 12^319-1. I think he will. R323 might well be done by a reciprocal octic to take advantage of the algebraic factor 10^19-1. 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.