Thread: Polynomial
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Old 2005-09-15, 06:46   #6
trilliwig
 
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Oct 2004
tropical Massachusetts

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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.
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