Integral Points on Curves: x^4-1785y^4=\pm{1}

[2003.08.24]x^4-1785y^4=±1の整点

x^4-1785y^4=1の整点を計算するプログラム(pari/gp)

■Thue方程式
     x⁴-1785y⁴ = ±1 ----- (1)
の整数解(x,y)を求める。

■Cohnの定理によると、
     x⁴-1785y⁴ = 1 ----- (2)
の正整数解(X,Y)は、
     X²=169 または X²=2*169²-1
を満たすので、(13,2)に限ることが直ちに分かる。

また、3|1785, Legendre(-1/3)=-1なので、
     x⁴-1785y⁴ = -1 ----- (3)
は、Z/3Zで解を持たない。よって、整数解を持たない。

gp>  kronecker(-1,3)
time = 2 ms.
%1 = -1

以上より、(1)の整数解は、(±1,0),(±13,±2)に限る。

■Thue方程式(1)をpari/gpを使って解く。

gp> th1=thueinit(x^4-1785)
time = 1,650 ms.
%1 = [x^4 - 1785, [[4, 2, 1, 1, 3, 0; 0, 4, 0, 2, 2, 0; 0, 0, 2, 0, 0, 0; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 0, 0, 0, 0, 0, 2], [1, 2, 1, 2, 3, 0, 2, 2, 3, 1, 1, 0, 0, 1, 1, 3, 2, 3, 1, 3, 3, 3, 0, 3, 0, 0, 1, 2, 3, 0, 0, 3, 2, 2, 1, 1, 2, 0, 0, 1, 3, 3, 0, 0, 0; 0, 3, 1, 3, 0, 1, 2, 1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 3, 1, 2, 3, 3, 2, 0, 1, 3, 0, 3, 2, 2, 2, 0, 1, 1, 0, 0, 3, 2, 0, 0, 2, 2, 1, 0, 0; 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0; 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1; 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0], [3.258092161356981401565175173 - 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12, 28, 31, 30, 32, 35, 34, 44, 49, 48], [x^4 - 1785, [2, 1], -22749646500, 8, [[1, -6.499943103486000196566558808, 21.62463017427760793034754630, -59.46711779622669612961557399; 1, 6.499943103486000196566558808, 21.62463017427760793034754630, 81.09174797050430405996312029; 1, 0.E-119 + 6.499943103486000196566558808*I, -20.62463017427760793034754630 + 0.E-118*I, -10.31231508713880396517377315 - 67.02946133162249999650606774*I], [1, 1, 2; -6.499943103486000196566558808, 6.499943103486000196566558808, 0.E-119 - 12.99988620697200039313311761*I; 21.62463017427760793034754630, 21.62463017427760793034754630, -41.24926034855521586069509260 + 0.E-118*I; -59.46711779622669612961557399, 81.09174797050430405996312029, -20.62463017427760793034754630 + 134.0589226632449999930121354*I], [4, 0.E-114, 2.000000000000000000000000000, 1.000000000000000000000000000; 0.E-114, 168.9970413942208634427803704, 0.E-113, 42.24926034855521586069509260; 2.000000000000000000000000000, 0.E-113, 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1], [4031, 143, 1868, 395; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [58, 27, 24, 36; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], [132, 42, 90, 63; 0, 6, 0, 5; 0, 0, 3, 1; 0, 0, 0, 1]]], 71.85174489553825643560031179, 0.9585709178499225557, [2, -1], [2*x - 13, 2*x + 13], 236], [[-2, 1, 1, 4, 2, 1; 0, -1, 0, -1, 1, 0; 2, 0, -1, 1, -1, 0; 0, 0, 0, 1, 1, 0], [[0, 0, 0], [0.9162679730095741500121150093 + 3.141592653589793238462643383*I, 2.351369838433177365669132361, 4.064731394486310785569481125 + 2.038274874369576940845928413*I], [0, 0, 0], [-29.04878397232543004063018055 - 3.141592653589793238462643383*I, -58.94528873545035807031592979 - 18.84955592153875943077586030*I, 64.43664797488945105602392880 - 17.42093032777233105293177720*I], [-19.78184271240233970564438973 - 9.424777960769379715387930150*I, -31.13496415698265519153949578 - 6.283185307179586476925286766*I, 12.69464372464053323976024698 - 10.04899197977096163236503645*I], [0, 0, 0]], [[49.45528762923264035596575178 + 0.E-75*I, 112.1185008863297028466512849 + 37.69911184307751886155172059*I, -161.5737885155623432026170367 + 26.35203978992464303370483770*I], [23.36991834521813519521281845 + 12.56637061435917295385057353*I, 48.94636496522597259831706526 + 12.56637061435917295385057353*I, -42.98680648672785858852696973 + 11.60816309392190419257135620*I], [22.99919419262116845690171906 + 0.E-76*I, 54.33080082116969970224448565 + 18.84955592153875943077586029*I, -77.32999501379086815914620471 + 13.17601989496232151685241885*I], [6.262697410900511726315139329 + 18.84955592153875943077586029*I, -11.82430778910417719894870047 - 6.283185307179586476925286766*I, 67.34119825583446418497938121 + 6.135104891653070242803529532*I]]], 0], [-6.499943103486000196566558808 + 0.E-67*I, 6.499943103486000196566558808 + 0.E-67*I, 0.E-71 - 6.499943103486000196566558808*I, 0.E-71 + 6.499943103486000196566558808*I]~, [4.540564651765077085632685016, 4.540564651765077085632685016]~, [-25.99988620697200039313311761 + 0.E-67*I, 0.0001137930279996068668823833893 + 0.E-67*I; -0.0001137930279996068668823833893 + 0.E-67*I, 25.99988620697200039313311761 + 0.E-67*I; -13.00000000000000000000000000 - 12.99988620697200039313311761*I, 13.00000000000000000000000000 - 12.99988620697200039313311761*I; -13.00000000000000000000000000 + 12.99988620697200039313311761*I, 13.00000000000000000000000000 + 12.99988620697200039313311761*I], [-0.04534464912569294608712693384, -0.1263870392672128513658321422; -0.1263870392672128513658321422, -0.04534464912569294608712693384], [0.007282849416398873397103819415, 9.192307690684136775362065852, 1.871793423553434862837722982, 0.1829565422449520403476238487, 1.219908897496501045 E-67, 9]]
gp> thue(th1,1)
time = 62 ms.
%2 = [[13, -2], [-13, 2], [-13, -2], [13, 2], [1, 0], [-1, 0]]
gp> thue(th1,-1)
time = 12 ms.
%3 = []

よって、(1)の整数解は、(±1,0),(±13,±2)に限る。

■以下では、参考文献[1]Chap.VIIに記述されている方法により、Thue方程式(1)の整数解を求める。
２項４次形式F(X,Y)を
     F(X,Y) = X⁴-1785Y⁴
とする。
F(X,1)の根を具体的に求めると、
     θ⁽¹⁾≒-6.499943103486000196566558808,
     θ⁽²⁾≒6.499943103486000196566558808,
     θ⁽³⁾≒-6.499943103486000196566558808*sqrt(-1),
     θ⁽⁴⁾≒6.499943103486000196566558808*sqrt(-1)
の4個である。θ⁽¹⁾,θ⁽²⁾は実数、θ⁽³⁾=conj(θ⁽⁴⁾)は純虚数である。

gp> polroots(x^4-1785)
time = 26 ms.
%12 = [-6.499943103486000196566558808 + 0.E-28*I, 6.499943103486000196566558808 + 0.E-28*I, 0.E-32 - 6.499943103486000196566558808*I, 0.E-32 + 6.499943103486000196566558808*I]~

■F(X,1)の根の１つをθ、K=Q(θ)とする。
Kのfundamental unitsは、
η₁ = 2θ-13,
η₂ = 2θ+13,
であり、Kの1の根は±1である。
また、Gal(K/Q) = D₄である。

gp> nf=bnfinit(x^4-1785)
time = 2,006 ms.
%1 = [[4, 2, 1, 1, 3, 0; 0, 4, 0, 2, 2, 0; 0, 0, 2, 0, 0, 0; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 0, 0, 0, 0, 0, 2], [3, 1, 2, 1, 2, 2, 3, 0, 2, 3, 1, 1, 0, 0, 1, 1, 3, 2, 3, 1, 3, 3, 0, 3, 1, 0, 0, 2, 3, 0, 0, 3, 2, 1, 3, 2, 1, 2, 0, 0, 1, 3, 0, 0, 0; 2, 0, 3, 1, 3, 1, 0, 1, 1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 3, 3, 2, 2, 2, 0, 1, 0, 3, 2, 0, 3, 2, 0, 0, 2, 1, 0, 0; 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0; 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1; 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0], [3.258092161356981401565175173 - 9.424777960769379715387930150*I, -9.081129303530154171265370033 - 18.84955592153875943077586030*I; -9.081129303530154171265370033 - 9.424777960769379715387930150*I, 3.258092161356981401565175173 - 18.84955592153875943077586030*I; 5.823037142173172769700194859 - 7.853972880626326644139942621*I, 5.823037142173172769700194859 - 10.99558304091243278663591767*I], [2.366137898182869327537881201 + 0.E-38*I, -2.528689402731842907962834218 + 3.141592653589793238462643383*I, 0, -0.9177711110955264372663393652 + 9.424777960769379715387930149*I, -2.646220733090678158347496194 + 9.424777960769379715387930149*I, 1.064321213232209566306125970 + 6.283185307179586476925286766*I, 29.42715361886033583131041625 + 9.424777960769379715387930149*I, -48.09252477390407789516791782 + 6.283185307179586476925286766*I, 0.6917760550973287849688908249 + 9.424777960769379715387930149*I, 6.819910872168702743492073891 + 6.283185307179586476925286766*I, -0.2674709713703914801895380768 + 3.141592653589793238462643383*I, 0.1251794324389926462933830173 + 6.283185307179586476925286766*I, -3.096088615273058539840320078 + 9.424777960769379715387930149*I, -0.9168243284726914253005671149 + 3.141592653589793238462643383*I, -3.799948012083058432280977598 + 9.424777960769379715387930149*I, 9.056929681897470367453006524 + 3.141592653589793238462643383*I, -0.5590033748747628464707736074 + 0.E-38*I, 4.357840102150048060079663924 - 4.70197740 E-38*I, 14.66400442784022887356680851 + 12.56637061435917295385057353*I, 6.479683207154275437909840160 + 3.141592653589793238462643383*I, 13.20324704194592387702116363 + 3.141592653589793238462643383*I, 2.028076970016468369394483118 + 3.141592653589793238462643383*I, 4.753101422871463613046660025 + 6.283185307179586476925286766*I, -11.50215019020444750414465750 + 3.141592653589793238462643383*I, -7.118470940675848486662762913 + 6.283185307179586476925286766*I, -1.010055745946174226356522240 + 0.E-38*I, 4.465781692731227555642181174 + 0.E-38*I, 8.367115439776008564150996041 + 9.424777960769379715387930149*I, -4.966874142995249965400844025 + 9.424777960769379715387930149*I, -16.60835421536300239987612047 + 9.40395480 E-38*I, -14.43886525943549380093484518 + 6.283185307179586476925286766*I, -8.230166278015385561529742985 + 4.70197740 E-38*I, 3.388512676380829143324952688 + 3.141592653589793238462643383*I, 3.458798948339562336641587134 + 0.E-37*I, -7.374070084363049171051262468 + 2.35098870 E-38*I, -9.929479465854498541462052706 + 9.424777960769379715387930149*I, -3.220465457829171692931725043 + 6.283185307179586476925286766*I, -5.860027112719729120687310364 + 3.141592653589793238462643383*I, 0.3062195159368716592131252560 + 0.E-38*I, 48.16338632608204721878033757 + 6.283185307179586476925286766*I, 13.53838235388755412533279569 + 6.283185307179586476925286766*I, -20.12816096231422243396121026 + 6.283185307179586476925286766*I, 64.98187720222856900349816523 - 3.76158192 E-37*I, 4.380391705195371988196198825 + 9.424777960769379715387930149*I, 18.29831510230120724611812197 + 6.283185307179586476925286766*I, -33.97455862627185448177873333 + 6.283185307179586476925286766*I, -11.94811908225262727525829092 + 3.141592653589793238462643383*I, -29.61386005091161735037616976 + 3.141592653589793238462643383*I, -2.702187920878259902575274805 + 9.424777960769379715387930149*I, -2.241351602431580738303223525 + 3.141592653589793238462643383*I, 0.9776334702080583351329674549 + 3.141592653589793238462643383*I; 2.366137898182869327537881201 + 0.E-38*I, 8.966878067503263796903754979 + 9.424777960769379715387930149*I, 0, -7.509208840865108657663589973 + 3.141592653589793238462643383*I, -9.237658462860260378744746802 + 3.141592653589793238462643383*I, 4.390903062619339616146964059 + 6.283185307179586476925286766*I, -95.12881638295371730120552445 + 6.283185307179586476925286766*I, 88.63745043711985864209093493 + 9.424777960769379715387930149*I, -5.899661674672253435428359783 + 3.141592653589793238462643383*I, 3.524192007283911633293448587 + 0.E-38*I, 2.184593898862370762045131217 + 0.E-38*I, 15.17275925811238889154524001 + 9.424777960769379715387930149*I, -11.68306425489290189038750522 + 3.141592653589793238462643383*I, 0.5182775369509117903564502376 + 0.E-38*I, -12.09190344490228455123530332 + 3.141592653589793238462643383*I, 14.08402923433424431585091973 + 3.141592653589793238462643383*I, 0.3689740567244669164508831330 + 0.E-38*I, -27.84001798899303506890033383 - 4.70197740 E-38*I, 16.05558295741323958995773479 + 2.35098870 E-38*I, 1.823248797198812550962501362 + 3.141592653589793238462643383*I, 3.360713901009779245692135239 + 3.141592653589793238462643383*I, 2.014316500800578609769848384 + 9.424777960769379715387930149*I, 9.677545921097646354185865133 + 3.141592653589793238462643383*I, -13.10001283904350019544302452 + 6.283185307179586476925286766*I, 10.05785984014680989754194060 + 3.141592653589793238462643383*I, -3.364237207146595855853798475 + 3.141592653589793238462643383*I, 4.465781692731227555642181174 + 0.E-38*I, 23.57540990563863538592150687 + 9.424777960769379715387930149*I, 20.29220919776091105618036077 + 4.70197740 E-38*I, -2.366845306722655521088807568 + 3.141592653589793238462643383*I, 10.23957767033877734472624523 + 6.283185307179586476925286766*I, 11.93048111062889285618311064 + 9.424777960769379715387930149*I, 22.72845753713305862598061675 + 4.70197740 E-38*I, -7.685704179967820210622414422 + 6.283185307179586476925286766*I, -8.715744990663484164010327224 + 6.283185307179586476925286766*I, -18.44409816970609793945665107 + 9.424777960769379715387930149*I, 14.86653974217551723233211476 + 6.283185307179586476925286766*I, 2.726948526900114229859874780 + 9.424777960769379715387930149*I, 3.345644890268164339225108060 + 3.141592653589793238462643383*I, -89.60496469913102067359872759 + 3.141592653589793238462643383*I, 32.91359707204759120690697392 + 6.283185307179586476925286766*I, -2.041155762309533508697370455 + 6.283185307179586476925286766*I, -118.0695298596238879977169449 + 9.424777960769379715387930149*I, 46.75969468873394199205804549 + 3.141592653589793238462643383*I, 15.66945393865852902474036982 + 9.424777960769379715387930149*I, -59.28899627278589957661848412 + 9.424777960769379715387930149*I, 11.73178475025619863427994401 + 6.283185307179586476925286766*I, 57.08079156158900019568417931 + 6.283185307179586476925286766*I, 13.94971541370282580703154764 + 6.283185307179586476925286766*I, 0.9776334702080583351329674549 + 0.E-38*I, -2.241351602431580738303223525 + 0.E-38*I; -4.732275796365738655075762402 + 0.E-37*I, -6.438188664771420888940920761 + 3.437770991339138610739489526*I, 0, 8.426979951960635094929929338 + 9.572875882992208853543019058*I, 11.88387919595093853709224299 + 9.572875882992208853543019058*I, -5.455224275851549182453090029 + 5.017222374034420591211553570*I, 65.70166276409338146989510820 + 10.89963000746586735538011913*I, -40.54492566321578074692301710 + 0.7058778277207349777818420426*I, 5.207885619574924650459468958 + 3.289690575812622376617732291*I, -10.34410287945261437678552247 + 11.06962324867569090369679629*I, -1.917122927491979281855593140 + 0.2221381299860872552159675256*I, -15.29793869055138153783862303 + 8.329643885243307799479455002*I, 14.77915287016596043022782530 + 10.14298302857810945916965079*I, 0.3985467915217796349441168772 + 2.038274874369576940845928413*I, 15.89185145698534298351628092 + 7.271592822044790957116319432*I, -23.14095891623171468330392626 + 1.035320127802252010712966429*I, 0.1900293181502959300198904743 + 0.8178092710065017578227170780*I, 23.48217788684298700882066991 + 5.613491465092226761290990270*I, -30.71958738525346846352454331 + 7.945970426501037224328795634*I, -8.302932004353087988872341523 + 0.3609425233460057435004902273*I, -16.56396094295570312271329887 + 6.118850380393098009026556544*I, -4.042393470817046979164331502 + 5.477753145266567335420461609*I, -14.43064734396910996723252515 + 10.56257331880683494163821437*I, 24.60216302924794769958768202 + 0.7378343624071721264986259622*I, -2.939388899470961410879177694 + 2.127045832090664413330335896*I, 4.374292953092770082210320716 + 6.327947533282894788620707200*I, -8.931563385462455111284362350 + 6.283185307179586476925286766*I, -31.94252534541464395007250291 + 10.11963018767648723240337069*I, -15.32533505476566109077951674 + 8.680306453105890188502195744*I, 18.97519952208565792096492803 + 2.916262059161149347878159061*I, 4.199287589096716456208599947 + 9.424760454073066811354598476*I, -3.700314832613507294653367658 + 2.749903653121134297629152736*I, -26.11697021351388776930556944 + 2.563479551966318761825915303*I, 4.226905231628257873980827287 + 7.059514710546822750208100758*I, 16.08981507502653333506158969 + 5.080731885624598230176540785*I, 28.37357763556059648091870378 + 6.515871640035067114270477369*I, -11.64607428434634553940038971 + 0.1480804155265162341217572345*I, 3.133078585819614890827435583 + 2.423387585781063494680922740*I, -3.651864406205035998438233316 + 10.66799245182196786541816253*I, 41.44157837304897345481839001 + 6.023884980192188319232704006*I, -46.45197942593514533223976962 + 10.96835453781894530528596139*I, 22.16931672462375594265858071 + 0.1480804155265162341217572345*I, 53.08765265739531899421877973 + 9.684305874808845625513824670*I, -51.14008639392931398025424432 + 1.625018740117068538098741344*I, -33.96776904095973627085849180 + 10.85150281944554030690538708*I, 93.26355489905775405839721746 + 6.880205597867195280345160601*I, 0.2163343319964286409783469103 + 7.147990012727714242158110535*I, -27.46693151067738284530800955 + 11.38773191880465649420189404*I, -11.24752749282456590445627284 + 4.392990848336525770201115588*I, 1.263718132223522403170256070 + 1.650744975656238627536781289*I, 1.263718132223522403170256070 + 4.632440331523347849388505477*I], [[2, [0, 0, 1, 1]~, 2, 1, [1, 1, 0, 0]~], [2, [3, 0, 1, 0]~, 1, 2, [2, 2, 1, 0]~], [3, [0, 1, 0, 0]~, 4, 1, [0, -1, 1, 1]~], [5, [0, 1, 0, 0]~, 4, 1, [0, -1, -2, -1]~], [7, [0, 1, 0, 0]~, 4, 1, [0, -1, -2, -3]~], [11, [-4, 1, 0, 0]~, 1, 1, [5, 4, -5, 4]~], [11, [4, 1, 0, 0]~, 1, 1, [-5, 4, 1, 4]~], [11, [4, 0, 2, 0]~, 1, 2, [5, 0, 2, 0]~], [13, [-3, 0, 2, 0]~, 1, 2, [1, 0, 2, 0]~], [13, [1, 0, 2, 0]~, 1, 2, [-3, 0, 2, 0]~], [17, [0, 1, 0, 0]~, 4, 1, [0, -1, -2, 4]~], [29, [5, 1, 0, 0]~, 1, 1, [-4, -5, -12, 4]~], [29, [-5, 1, 0, 0]~, 1, 1, [4, -5, 8, 4]~], [29, [-2, 1, 0, 0]~, 1, 1, [6, 3, 2, 4]~], [29, [2, 1, 0, 0]~, 1, 1, [-6, 3, -6, 4]~], [31, [-10, 1, 0, 0]~, 1, 1, [-2, 6, -13, 4]~], [31, [10, 1, 0, 0]~, 1, 1, [2, 6, 9, 4]~], [37, [16, 1, 0, 0]~, 1, 1, [-10, -4, 3, 4]~], [37, [-16, 1, 0, 0]~, 1, 1, [10, -4, -7, 4]~], [37, [-15, 1, 0, 0]~, 1, 1, [-7, 2, -9, 4]~], [37, [15, 1, 0, 0]~, 1, 1, [7, 2, 5, 4]~], [53, [18, 1, 0, 0]~, 1, 1, [16, 5, 15, 4]~], [53, [-18, 1, 0, 0]~, 1, 1, [-16, 5, -19, 4]~], [53, [-10, 1, 0, 0]~, 1, 1, [-17, -7, 18, 4]~], [53, [10, 1, 0, 0]~, 1, 1, [17, -7, -22, 4]~], [59, [-18, 1, 0, 0]~, 1, 1, [-27, 28, -25, 4]~], [59, [18, 1, 0, 0]~, 1, 1, [27, 28, 21, 4]~], [61, [22, 1, 0, 0]~, 1, 1, [-12, -5, 15, 4]~], [61, [-22, 1, 0, 0]~, 1, 1, [12, -5, -19, 4]~], [61, [-2, 1, 0, 0]~, 1, 1, [6, 3, 2, 4]~], [61, [2, 1, 0, 0]~, 1, 1, [-6, 3, -6, 4]~], [71, [-3, 1, 0, 0]~, 1, 1, [24, 8, 4, 4]~], [71, [3, 1, 0, 0]~, 1, 1, [-24, 8, -8, 4]~], [101, [47, 1, 0, 0]~, 1, 1, [-49, -14, 5, 4]~], [101, [-47, 1, 0, 0]~, 1, 1, [49, -14, -9, 4]~], [101, [-35, 1, 0, 0]~, 1, 1, [16, 12, -33, 4]~], [101, [35, 1, 0, 0]~, 1, 1, [-16, 12, 29, 4]~], [103, [-22, 1, 0, 0]~, 1, 1, [17, -32, 42, 4]~], [103, [22, 1, 0, 0]~, 1, 1, [-17, -32, -46, 4]~], [137, [51, 1, 0, 0]~, 1, 1, [16, -3, 33, 4]~], [137, [-51, 1, 0, 0]~, 1, 1, [-16, -3, -37, 4]~], [137, [-31, 1, 0, 0]~, 1, 1, [31, 1, 60, 4]~], [137, [31, 1, 0, 0]~, 1, 1, [-31, 1, -64, 4]~], [139, [-4, 1, 0, 0]~, 1, 1, [60, 15, 6, 4]~], [139, [4, 1, 0, 0]~, 1, 1, [-60, 15, -10, 4]~], [151, [-73, 1, 0, 0]~, 1, 1, [-32, 43, -7, 4]~], [151, [73, 1, 0, 0]~, 1, 1, [32, 43, 3, 4]~], [163, [-6, 1, 0, 0]~, 1, 1, [47, 35, 10, 4]~], [163, [6, 1, 0, 0]~, 1, 1, [-47, 35, -14, 4]~], [167, [-44, 1, 0, 0]~, 1, 1, [-30, -69, -81, 4]~], [167, [44, 1, 0, 0]~, 1, 1, [30, -69, 77, 4]~]]~, [2, 7, 1, 3, 4, 14, 35, 33, 6, 5, 11, 40, 43, 45, 17, 16, 21, 20, 19, 18, 24, 25, 23, 22, 27, 26, 10, 29, 47, 46, 9, 37, 36, 51, 39, 38, 8, 42, 41, 13, 50, 15, 12, 28, 31, 30, 32, 34, 44, 49, 48], [x^4 - 1785, [2, 1], -22749646500, 8, [[1, -6.499943103486000196566558808, 21.62463017427760793034754630, -59.46711779622669612961557399; 1, 6.499943103486000196566558808, 21.62463017427760793034754630, 81.09174797050430405996312029; 1, 0.E-109 + 6.499943103486000196566558808*I, -20.62463017427760793034754630 + 0.E-109*I, -10.31231508713880396517377315 - 67.02946133162249999650606774*I], [1, 1, 2; -6.499943103486000196566558808, 6.499943103486000196566558808, 0.E-109 - 12.99988620697200039313311761*I; 21.62463017427760793034754630, 21.62463017427760793034754630, -41.24926034855521586069509260 + 0.E-108*I; -59.46711779622669612961557399, 81.09174797050430405996312029, -20.62463017427760793034754630 + 134.0589226632449999930121354*I], [4, 0.E-105, 2.000000000000000000000000000, 1.000000000000000000000000000; 0.E-105, 168.9970413942208634427803704, 0.E-103, 42.24926034855521586069509260; 2.000000000000000000000000000, 0.E-103, 1786.000000000000000000000000, 893.0000000000000000000000000; 1.000000000000000000000000000, 42.24926034855521586069509260, 893.0000000000000000000000000, 19310.79474562990388180035884], [4, 0, 2, 1; 0, 0, 0, 1785; 2, 0, 1786, 893; 1, 1785, 893, 1339], [-3570, 0, 0, -1785; 0, -3570, 0, -2231; 0, 0, -1785, -892; 0, 0, 0, -1], [-5690597850, 0, 6372450, 0; 0, 6372450, 6372450, -12744900; 6372450, 6372450, -12744900, 0; 0, -12744900, 0, 0], [22749646500, [0, -6372450, -6372450, 12744900]~]], [-6.499943103486000196566558808, 6.499943103486000196566558808, 0.E-109 + 6.499943103486000196566558808*I], [1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 + 1/4*x + 1/4], [1, 0, -1, 0; 0, 1, 0, -1; 0, 0, 2, -2; 0, 0, 0, 4], [1, 0, 0, 0, 0, -1, 0, 446, 0, 0, 446, 223, 0, 446, 223, 223; 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 223, 0, 0, 223, 223; 0, 0, 1, 0, 0, 2, -1, 0, 1, -1, 1, 0, 0, 0, 0, 223; 0, 0, 0, 1, 0, 0, 2, 1, 0, 2, 0, 1, 1, 1, 1, 1]], [[256, [8, 4, 4, 2], [[29, 27, 12, 18; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [4031, 143, 1868, 395; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [58, 27, 24, 36; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], [132, 42, 90, 63; 0, 6, 0, 5; 0, 0, 3, 1; 0, 0, 0, 1]]], 71.85174489553825643560031179, 0.9585709178499225557, [2, -1], [2*x - 13, 2*x + 13], 109], [[-2, 1, 1, 4, 2, 1; 0, -1, 0, -1, 1, 0; 2, 0, -1, 1, -1, 0; 0, 0, 0, 1, 1, 0], [[0, 0, 0], [0.9162679730095741500121150093 + 3.141592653589793238462643383*I, 2.351369838433177365669132361, 4.064731394486310785569481125 + 2.038274874369576940845928413*I], [0, 0, 0], [-11.57967254580591173152959597 - 1.17549435 E-38*I, -41.47617730893083976121534521 - 15.70796326794896619231321691*I, 29.49842512185041443782275964 + 1.428625593766428377844083095*I], [-13.95880557022916693594419487 - 6.283185307179586476925286766*I, -25.31192701480948242183930092 - 3.141592653589793238462643383*I, 1.048569440294187700359857263 - 3.765806672591375155439749692*I], [0, 0, 0]], [[8.694027634020430968064387764 - 6.283185307179586476925286766*I, 71.35724089111749345874992094 + 31.41592653589793238462643383*I, -80.05126852513792442681430871 - 11.34707205315287582784688289*I], [11.72384406087178965581242873 + 6.283185307179586476925286766*I, 37.30029068087962705891667554 + 6.283185307179586476925286766*I, -19.69465791803516750972619029 - 0.9582075204372687612792173233*I], [5.530082766101650147801134483 - 3.141592653589793238462643383*I, 36.86168939465018139314390107 + 15.70796326794896619231321691*I, -42.39177216075183154094503555 - 5.673536026576437913923441449*I], [17.90877169524685726571552904 + 18.84955592153875943077586029*I, -0.1782335047578316595483107564 - 6.283185307179586476925286766*I, 44.04904968714177310617860177 + 18.70147550601224319665410306*I]]], 0]
gp> nf.clgp
time = 0 ms.
%2 = [256, [8, 4, 4, 2], [[29, 27, 12, 18; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [4031, 143, 1868, 395; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1], [58, 27, 24, 36; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], [132, 42, 90, 63; 0, 6, 0, 5; 0, 0, 3, 1; 0, 0, 0, 1]]]
gp> nf.zk
time = 0 ms.
%3 = [1, x, 1/2*x^2 + 1/2, 1/4*x^3 + 1/4*x^2 + 1/4*x + 1/4]
gp> nf.fu
time = 0 ms.
%4 = [2*x - 13, 2*x + 13]
gp> polgalois(x^4-1785)
time = 134 ms.
%5 = [8, -1, 1]
gp> nfrootsof1(nf)
time = 2 ms.
%6 = [2, [-1, 0, 0, 0]~]

また、F(X,Y)=±1の整数解X,Yに対して、
     β = X-θY
とすると、βは単数なので、未知の整数a₁,a₂に対して、
     β = ±η₁^a₁η₂^a₂
と表すことができる。
i=1,2,3,4に対して、
     β⁽ⁱ⁾ = X-θ⁽ⁱ⁾Y = μ⁽ⁱ⁾ε⁽ⁱ⁾,
     μ⁽ⁱ⁾ = ±1,
     ε⁽ⁱ⁾ = (η₁⁽ⁱ⁾)^a₁(η₂⁽ⁱ⁾)^a₂

とする。

■互いに異なる添字i,j,k∈{1,2,3,4}を、iは以下を満たすもの、j,kは任意とする。
     |β⁽ⁱ⁾| = min_{1 <=l <= n}{|β^(l)|}
とする。もちろん、iの値はa prioriに知ることはないので、可能なiの値の全てについて、それぞれ議論する。

I={i₁,i₂ : i₁≠i₂} ⊂ { 1,2,3 }に対して、2×2行列U_Iを
     U_I = [ log|η₁^(i₁)|, log|η₁^(i₂)| ;
             log|η₁^(i₂)|, log|η₂^(i₂)| ]
とする。

■定数c₁,c₂,c₃,c₄を以下のように定義する。
     c₁ = 2*min_l≠m{ |θ^(l)-θ^(m)| } ≒ 18.38461538320673508904480521 ,
     c₂ = max_{l₁≠l₂≠l₃≠l₁}{ |θ^(l₂)-θ^(l₃)|/|θ^(l₃)-θ^(l₁)| } ≒ 1.414213562373095048801688724,
     c₃ = c₁c₂ ≒ 25.99977241394400078626623523,
     c₄ = 1/max { ||U_I^-1||_∞ : det(U_I) != 0 } ≒ 2.911518571086586384850097429.
ここで、||・||_∞は行列のrow sum norm、つまり、n×n行列A=(a_i,j)に対して、
     || A ||_∞ = max_1<=i<=n{ Σ_j=1ⁿ |a_i,j| }
である。

gp> read("de1785-2.gp")
time = 353 ms.
gp> c1=cc1(r)
time = 6 ms.
%1 = 18.38461538320673508904480521
gp> c2=cc2(r)
time = 17 ms.
%2 = 1.414213562373095048801688724
gp> c3=cc3(c1,c2)
time = 0 ms.
%3 = 25.99977241394400078626623523
gp> c4=cc4([r1,r2,r3])
time = 19 ms.
%4 = 2.911518571086586384850097429

このとき、
     |log|ε^(t)|| = max_{1 <= l <= 3} |log|ε^(l)||
となるt ∈ Iが存在するならば、
     |log|ε^(t)|| >= c₄A
が成立する。

■定数c₅,c₆,c₇を以下を満たすように決める。
     c₅ < c₄/3
     c₆ = max_{1 <= l <= 4}|μ^(l)|^-1
     c₇ = min_{1 <= l <= 4}|μ^(l)|^-1
例えば、
     c₅ = c₄/(3+1e-25) ≒ 0.9705061903621954616166991108
とする。

gp> c5=cc5(c4)
time = 3 ms.
%5 = 0.9705061903621954616166991108
gp> c6=1
time = 0 ms.
%6 = 1
gp> c7=1
time = 0 ms.
%7 = 1

Case A: |β⁽ⁱ⁾| > e^-c₅A かつ |ε^(t)| >= e^-c₄Aの場合
     A <= log(c₆)/(c₄-3*c₅) = A₁
ここで、c₆=1より、A₁=0を得る。

Case B: |β⁽ⁱ⁾| > e^-c₅A かつ |ε^(t)| >= e^-c₄Aの場合
     A <= log(c₇)/(c₄-c₅) = A₂
ここで、c₇=1より、A₂=0を得る。

Case C: |β⁽ⁱ⁾| <= e^-c₅Aの場合
     A >= log(2*c₃)/c₅ = A₃
ならば、
     |e^Λ-1| = |α₁τ₁| <= 1/2
が成立する。
よって、
     |Λ| <= 2c₃e^-c₅A
となる。
ここで、A₃ ≒ 0.6632096826953253872131185437である。

gp> A1=0
time = 0 ms.
%8 = 0
gp> A2=0
time = 0 ms.
%9 = 0
gp>  A3=AA3(c3,c5)
time = 1 ms.
%10 = 4.071313510899564189923247937

■定数c₉を以下を満たすように決める。
c₉ = max_{1 <= i <= n} { 2³/|(∂F/∂X)(θ⁽ⁱ⁾,1)| } ≒ 0.007282849415670588455536760570
このとき、
|β⁽ⁱ⁾| <= c₉|Y|^-3
となる。

gp> c9=cc9(r)
time = 1 ms.
%11 = 0.007282849415670588455536760570

■定数Y₁を以下を満たすように決める。
Y₁ = (c₉/min_{2< l <=4}|Imag(θ^(l))|)^1/3 if t >= 1
Y₁ = 1 if t=0
このとき、|Y|>= Y₁ならば、添字iは、{1,2}に所属する。

gp> Y1=YY1(c9,[r3,r4])
time = 1 ms.
%12 = 0.1038637323815497425813302640

よって、|Y|>=1ならば、i=1または2である。

■i=1の場合
ある整数a₀に対して、対数の線形形式
     Λ = log(-α₂)+Σ_l=1²a_l*log(η_l⁽²⁾/η_l⁽³⁾)+a₀*2π*sqrt(-1)
の全ての値を考察する。
     A = max{ |a₁|, |a₂| }
とする。
ここで、
     α₂ = ±((θ⁽¹⁾-θ⁽³⁾)/(θ⁽²⁾-θ⁽¹⁾)) = ±(1-sqrt(-1))/2
である。α₂の最小多項式は、2x²±2x+1である。
さらに、
     (η₁⁽²⁾/η₁⁽³⁾) = ((2θ⁽²⁾+13)/(-2θ⁽²⁾*sqrt(-1)+13))
     (η₂⁽²⁾/η₂⁽³⁾) = ((2θ⁽²⁾-13)/(-2θ⁽²⁾*sqrt(-1)-13))
の最小多項式は、どちらも、
     x⁸ - 228488x⁷ + 26103154588x⁶ - 78309006776x⁵ + 117462938950x⁴ - 78309006776x³ + 26103154588x² - 228488x + 1
である。

gp> ff=(13*(x-1))^4-1785*(2*(1+I*x))^4
time = 4 ms.
%13 = x^4 + (-114244 + 114240*I)*x^3 + 342726*x^2 + (-114244 - 114240*I)*x + 1
gp> ff*conj(ff)
time = 3 ms.
%14 = x^8 - 228488*x^7 + 26103154588*x^6 - 78309006776*x^5 + 117462938950*x^4 - 78309006776*x^3 + 26103154588*x^2 - 228488*x + 1
gp> factor(%14)
time = 123 ms.
%15 = 
[x^8 - 228488*x^7 + 26103154588*x^6 - 78309006776*x^5 + 117462938950*x^4 - 78309006776*x^3 + 26103154588*x^2 - 228488*x + 1 1]

よって、それぞれのhightを計算すると、
     h(α₂) ≒ 0.3465735902799726547086160607,
     h(η₁⁽²⁾/η₁⁽³⁾) ≒ 3.084805366221783893207636301,
     h(η₂⁽²⁾/η₂⁽³⁾) ≒ 3.084805366221783893207636301
となる。

gp> ff2=%14
time = 0 ms.
%16 = x^8 - 228488*x^7 + 26103154588*x^6 - 78309006776*x^5 + 117462938950*x^4 - 78309006776*x^3 + 26103154588*x^2 - 228488*x + 1
gp> h(2*x^2-2*x+1)
time = 6 ms.
%17 = 0.3465735902799726547086160607
gp> h(ff2)
time = 118 ms.
%18 = 3.084805366221783893207636301

これらに対するmodified height h_m(α)=max{h(α), |log α|/d, 1/d}を計算すると、
     h_m(α₂) ≒ 0.3465735902799726547086160607,
     h_m(η₁⁽²⁾/η₁⁽³⁾) ≒ 3.084805366221783893207636301,
     h_m(η₂⁽²⁾/η₂⁽³⁾) ≒ 3.084805366221783893207636301
となる。

gp>  hm(2*x^2-2*x+1,alpha2,8)
time = 21 ms.
%19 = 0.3465735902799726547086160607
gp>  hm(ff2,et1(r2,r3),8)
time = 129 ms.
%20 = 3.084805366221783893207636301
gp>  hm(ff2,et2(r2,r3),8)
time = 118 ms.
%21 = 3.084805366221783893207636301

■定数c₈,A₄を以下のように定義する。
     c₈ = 18*5!*4⁵*(32*16)⁶*log(2*4*16) ≒ 193328986715464153576004.0961
     A₄ = (2/c₅)*(log(2c₃)+c₈*log(3/2)+c₈*log(c₈/c₅)) ≒ 4309338042648278199094117.649
このとき、A >= 3ならば、
     log|Λ| >= -c₈log(3A/2)
が成立する。
よって、A >= A₃ならば、
     A <= A₄ ≒ 21535609307640119588420770.63
が成立する。

gp> c8=cc8(4,16)
time = 1 ms.
%22 = 193328986715464153576004.0961
gp> A4=AA4(c3,c5,c8)
time = 2 ms.
%23 = 21535609307640119588420770.63
gp> log(A4)/log(10)
time = 2 ms.
%24 = 25.33315716377925108734902449

■LLL-algorithmにより、Aの上限(21535609307640119588420770.63)を下げる。
C=10^90とする。以下の行列
    [1, 0, 0;
    -11992647874616740556115467462949005415058101818646280163237317187178294376619009013883550827, 346573590270395016715077743825121478622116366791996309659470488561666575202449480165655410, 0;
    -785393786723370083607327927360549482923046757416549693543286723359466882860388374766840315, 785393786723370083607327927360549482923046757416549693543286723359466882860388374766840315, 6283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650]
に対して、LLL-reduced matrixを求めると、
    [-356104526893471918914171570735436638247487428252891729088507, 826365694336527353428223849797038483933388340716710737229175, 132511593889989669532949203156806777389073722201418342668944;
    -12322451327749660624915631811550905732119358720054218535250690, 28595118226144018569868100740315369828396410800419256544934746, 4585360597115904356739736633027700600074075177487826604028531;
    1495785014717043202767961602372680669918625824990820704755044, -3471074723615038799537319778128263864666331125527196429786819, -556603023685264473003244289573605805425913917783722989969195]
となる。
y=[0,[C/2],-[C/2]]^tに対して、
    l(L,y) >= 2.891623*10⁶¹
を得る。
これより、Aの新しい上限を求めると、
   A < 71.78149419826996521818850487
となる。

gp> default(realprecision,200)
   realprecision = 202 significant digits (200 digits displayed)
time = 0 ms.
gp> read("de1785-2.gp")
time = 58 ms.
gp> aa=aaa(10^90,r2,r3)
time = 32 ms.
%25 = 
[1 0 0]

[-11992647874616740556115467462949005415058101818646280163237317187178294376619009013883550827 346573590270395016715077743825121478622116366791996309659470488561666575202449480165655410 0]

[-785393786723370083607327927360549482923046757416549693543286723359466882860388374766840315 785393786723370083607327927360549482923046757416549693543286723359466882860388374766840315 6283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650]

gp> bb=qflll(aa,1)
time = 79 ms.
%26 = 
[-356104526893471918914171570735436638247487428252891729088507 826365694336527353428223849797038483933388340716710737229175 132511593889989669532949203156806777389073722201418342668944]

[-12322451327749660624915631811550905732119358720054218535250690 28595118226144018569868100740315369828396410800419256544934746 4585360597115904356739736633027700600074075177487826604028531]

[1495785014717043202767961602372680669918625824990820704755044 -3471074723615038799537319778128263864666331125527196429786819 -556603023685264473003244289573605805425913917783722989969195]

gp> lb1=lb(bb,10^90)
time = 39 ms.
%27 = 28916237315961383130332152431179608107946199062187937934824059.781536505011113058991762513307735179709964730856054930041232471052492646793081759395170965999422479898144479702484646655055969641513525873
gp> log(lb1)/log(10)
time = 6 ms.
%28 = 61.461141780350404383282629088574523251748764397181692803925993898100537904090883194052643130412805526503444904995385683018330141317093426593132499583060483095219127203131216110710850503372182997776234
gp> A42=HH(10^90,A4,2*c3,c5,lb1,1)
time = 4 ms.
%29 = 71.78149419826996521818850487

■再度、LLL-algorithmにより、Aの上限(71.78149419826996521818850487)を下げる。
C=10^6とする。行列
    [1, 0, 0;
    -11992647, 346573, 0;
    -785393, 785393, 6283185]
に対して、LLL-reduced matrixを求めると、
    [-5, -9196, 22763;
    -173, -318214, 787680;
    21, 38627, -95614]
となる。
y=[0,[C/2],-[C/2]]^tに対して、
    l(L,y) >= 794430.972593
を得る。
これより、Aの新しい上限を求めると、
   A < 4.308634784134307691579327940
となる。

gp> aa2=aaa(10^6,r2,r3)
time = 18 ms.
%30 = 
[1 0 0]

[-11992647 346573 0]

[-785393 785393 6283185]

gp> bb2=qflll(aa2,1)
time = 2 ms.
%31 = 
[-5 -9196 22763]

[-173 -318214 787680]

[21 38627 -95614]

gp> lb2=lb(bb2,10^6)
time = 7 ms.
%32 = 794430.97259364842652949275308107329913858110054677712825256252155999306056409000914717210465313692784560564571254054098353966748290173244085377913889394418523537547643510049403586325990596567307240840
gp> A43=HH(10^6,A42,2*c3,c5,lb2,1)
time = 2 ms.
%33 = 4.308634784134307691579327940

■さらにもう一度、LLL-algorithmにより、Aの上限(4.308634784134307691579327940)を下げる。
C=10^3とする。行列
    [1, 0, 0;
    -11992, 346, 0;
    -785, 785, 6283]
に対して、LLL-reduced matrixを求めると、
    [-5, 68, 117;
    -173, 2357, 4055;
    21, -286, -492]
となる。
y=[0,[C/2],-[C/2]]^tに対して、
    l(L,y) >= 4089.8389501
を得る。
これより、Aの新しい上限を求めると、
   A < 2.622487683455404397043773797
となる。

gp> aa3=aaa(10^3,r2,r3)
time = 24 ms.
%34 = 
[1 0 0]

[-11992 346 0]

[-785 785 6283]

gp> bb3=qflll(aa3,1)
time = 1 ms.
%35 = 
[-5 68 117]

[-173 2357 4055]

[21 -286 -492]

gp> lb3=lb(bb3,10^3)
time = 9 ms.
%36 = 4089.8389501539689523866064630305232202170033015797954880703249211269561332407704366089893550442518445244687752775143737211648252065805543337836940297067467990570453182924914490278703235992797450771225
gp> A44=HH(10^3,A43,2*c3,c5,lb3,1)
time = 1 ms.
%37 = 2.622487683455404397043773797

■i=2の場合。j=3,k=1を選択する。
ある整数a₀に対して、対数の線形形式
Λ = log(-α₂)+Σ_l=1²a_l*log(η_l⁽¹⁾/η_l⁽³⁾)+a₀*2π*sqrt(-1)
の全ての値を考察する。

i=1の場合と同様にして、
A <= 2.622487683455404397043773797
を得る。

■最後に、整数a₁,a₂ ∈ [-2,2]に対して、X-θY=±(η₁)^a₁(η₂)^a₂が成立するかどうか、調べる。
これを満たす±(X-θY)は、

a₁	a₂	X	Y
0	0	1	0
0	1	2	13
1	0	2	-13

に限る。
よって、(1)の整数解(X,Y)は、±(1,0),±(2,13),±(-2,13)に限る。

gp> read("de1785-2.gp")
time = 337 ms.
gp> check(2)
[-2, -2]:Mod(1352*x^2 + 57121, x^4 - 1785)
[-2, -1]:Mod(2704*x^3 - 17576*x^2 + 114242*x - 742573, x^4 - 1785)
[-2, 0]:Mod(-70304*x^3 + 456972*x^2 - 2970292*x + 19306729, x^4 - 1785)
[-2, 1]:Mod(1827896*x^3 - 11881220*x^2 + 77227254*x - 501972757, x^4 - 1785)
[-2, 2]:Mod(-47525088*x^3 + 308910368*x^2 - 2007899816*x + 13051234561, x^4 - 1785)
[-1, -2]:Mod(2704*x^3 + 17576*x^2 + 114242*x + 742573, x^4 - 1785)
[-1, -1]:Mod(-4*x^2 - 169, x^4 - 1785)
[-1, 0]:Mod(-8*x^3 + 52*x^2 - 338*x + 2197, x^4 - 1785)
[-1, 1]:Mod(208*x^3 - 1352*x^2 + 8788*x - 57121, x^4 - 1785)
[-1, 2]:Mod(-5408*x^3 + 35152*x^2 - 228486*x + 1485133, x^4 - 1785)
[0, -2]:Mod(70304*x^3 + 456972*x^2 + 2970292*x + 19306729, x^4 - 1785)
[0, -1]:Mod(-8*x^3 - 52*x^2 - 338*x - 2197, x^4 - 1785)
[0, 0]:Mod(1, x^4 - 1785)
[0, 1]:Mod(2*x - 13, x^4 - 1785)
[0, 2]:Mod(4*x^2 - 52*x + 169, x^4 - 1785)
[1, -2]:Mod(1827896*x^3 + 11881220*x^2 + 77227254*x + 501972757, x^4 - 1785)
[1, -1]:Mod(-208*x^3 - 1352*x^2 - 8788*x - 57121, x^4 - 1785)
[1, 0]:Mod(2*x + 13, x^4 - 1785)
[1, 1]:Mod(4*x^2 - 169, x^4 - 1785)
[1, 2]:Mod(8*x^3 - 52*x^2 - 338*x + 2197, x^4 - 1785)
[2, -2]:Mod(47525088*x^3 + 308910368*x^2 + 2007899816*x + 13051234561, x^4 - 1785)
[2, -1]:Mod(-5408*x^3 - 35152*x^2 - 228486*x - 1485133, x^4 - 1785)
[2, 0]:Mod(4*x^2 + 52*x + 169, x^4 - 1785)
[2, 1]:Mod(8*x^3 + 52*x^2 - 338*x - 2197, x^4 - 1785)
[2, 2]:Mod(-1352*x^2 + 57121, x^4 - 1785)
time = 130 ms.

[参考文献]

[1]Nigel P. Smart, "The Algorithmic Resolution of Diophantine Equations", LMSST 41, Cambridge University Press, 1998, ISBN0-521-64633-2.
[2]Henri Cohen, "A Course in Computational Algebraic Number Theory", GTM 138, Springer-Verlag New York Inc., 1996, ISBN-387-55640-0.
[3]Takaaki Kagawa, "Elliptic curves with everywhere goood reduction over real quadraric fields", March 1998, p1-60.

Last Update: 2005.06.12

H.Nakao