2005-09-15, 06:46 #6 trilliwig     Oct 2004 tropical Massachusetts 10001012 Posts Hm, I can get close, but no cigar. Code: ? Nfull=2^1378+1 ? L=2^689-2^345+1 ? M=2^689+2^345+1 ? flist=factor(2^26*x^52+1) %4 = [2*x^2 - 2*x + 1 1] [2*x^2 + 2*x + 1 1] [4096*x^24 - 4096*x^23 + 2048*x^22 - 1024*x^20 + 1024*x^19 - 512*x^18 + 256*x^16 - 256*x^15 + 128*x^14 - 64*x^12 + 32*x^10 - 32*x^9 + 16*x^8 - 8*x^6 + 8*x^5 - 4*x^4 + 2*x^2 - 2*x + 1 1] [4096*x^24 + 4096*x^23 + 2048*x^22 - 1024*x^20 - 1024*x^19 - 512*x^18 + 256*x^16 + 256*x^15 + 128*x^14 - 64*x^12 + 32*x^10 + 32*x^9 + 16*x^8 - 8*x^6 - 8*x^5 - 4*x^4 + 2*x^2 + 2*x + 1 1] ? P=flist[4,1] %5 = 4096*x^24 + 4096*x^23 + 2048*x^22 - 1024*x^20 - 1024*x^19 - 512*x^18 + 256*x^16 + 256*x^15 + 128*x^14 - 64*x^12 + 32*x^10 + 32*x^9 + 16*x^8 - 8*x^6 - 8*x^5 - 4*x^4 + 2*x^2 + 2*x + 1 ? Mprim=subst(P,x,2^26) %6 = 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113 ? z=2*x+1/x %7 = (2*x^2 + 1)/x ? P/x^12-eval(Pol([1,2,-22,-44,172,344,-560,-1120,672,1344,-224,-448,-64],'z)) %8 = 0 ? Q=Pol([1,2,-22,-44,172,344,-560,-1120,672,1344,-224,-448,-64],'z) %9 = z^12 + 2*z^11 - 22*z^10 - 44*z^9 + 172*z^8 + 344*z^7 - 560*z^6 - 1120*z^5 + 672*z^4 + 1344*z^3 - 224*z^2 - 448*z - 64 ? xmod=Mod(2^26,Mprim) %10 = Mod(67108864, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? zmod=2*xmod+1/xmod %11 = Mod(285152538601387169506781365053581230592433074122088195472170561991719913693300399990037647880200513410357643430058818003293158177788323557633287625439025178669303926415798445740983284907638783, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? subst(Q,'z,zmod) %12 = Mod(0, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? z2mod=zmod^2 %13 = Mod(8498207948384856356148164994775234696058589277482298461713184792056933159713585920683296902912626823080244012032643423420301080480146077365260197802849412416499960801672859247332818950, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? neg_c0mod=64/(1+2/zmod) %14 = Mod(271942652322184754529069161754860585240769190626965451171865847338390750856691737618601477021484144551752615924320273899334633409481993696363818114000495981271054974909227003705267585088, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? neg_c0=component(neg_c0mod,2) %15 = 271942652322184754529069161754860585240769190626965451171865847338390750856691737618601477021484144551752615924320273899334633409481993696363818114000495981271054974909227003705267585088 ? mpoly=Pol([1,-22,172,-560,672,-224,-neg_c0],zsquare) %16 = zsquare^6 - 22*zsquare^5 + 172*zsquare^4 - 560*zsquare^3 + 672*zsquare^2 - 224*zsquare - 271942652322184754529069161754860585240769190626965451171865847338390750856691737618601477021484144551752615924320273899334633409481993696363818114000495981271054974909227003705267585088 ? subst(mpoly,zsquare,z2mod) %17 = Mod(0, 285152542850491175357501403430350705193444129732329634402348763225297093979008916513412511048984163942486513060254213059514241871206854248405437313525494851669503299027787512159802595435610113) ? log(neg_c0)/log(10) %18 = 185.43447732901241625166315915028769913 ? log(Mprim)/log(10) %19 = 191.45507724876353223637684615191137519 P is a 24th degree polynomial so the degree-halving trick only gets us down to degree 12. There's still some relationships among the coefficients which allows us to get down to a sextic in $\displaystyle{\LARGE z^2 = \frac{2^{106}+2^{54}+1}{2^{52}}}$, but it leaves a HUUGE constant coefficient $\LARGE\displaystyle{\frac{-64}{1+\frac{2}{z}}}$, which is 186 digits, not much smaller than the SNFS difficulty at 191.5.