## [2004.05.23]x^3-2y^3=c (c \in [1..100])$B$N@0E@(B

$B"#(Bc$B$r@5@0?t$H$9$k!#BJ1_6J@~(B
Cc: x3-2y3 = c ----- (1)
$B$N@0E@$r5a$a$k!#(B $B"#(B2(1/3)$B$NM-M}?t$K$h$k6a;w(B 2(1/3)$B$OL5M}?t$G$"$k!#(B
Backer$B$NDjM}(B(1967, [1]p233, [3]p57-58 Thoerem 30)$B$K$h$k$H!"G$0U$NM-M}?t(Bx/y(x,y$B$O@0?t(B)$B$KBP$7$F!"(B |(x/y)-2(1/3)| >= 10-6/|y|2.9955 ----- (2) $B$,@.N)$9$k!#(B $B4JC1$N$?$a$K!"&B(B=2(1/3)$B$H$9$k$H!"(BThue$BJ}Dx<0(B(1)$B$NG$0U$N@0?t2r(B(x,y)$B$KBP$7$F!"(B
c = x3-2y3 = |x-$B&B(By|$B!&(B|x2+$B&B(Bxy+$B&B(B2y2| = (3/4)$B&B(B2$B!&(B|x-$B&B(By| $B$h$j!"(B
|$B&B(B-(x/y)| <= 4c/(3$B&B(B2|y|3)
|2(1/3)-(x/y)| <= 4c/(3$B!&(B2(2/3)|y|3) ------ (3) $B$rF@$k!#(B(2),(3)$B$h$j!"(B 10-6/|y|2.9955 <= 4c/(3$B!&(B2(2/3)|y|3)
|y|0.0045 <= 106$B!&(B2(4/3)c/3 |y| <= {106$B!&(B2(4/3)/3}(10000/45)c(10000/45) <= 101317c223
$B$D$^$j!"(B
|y| <= 101317c223
$B$rF@$k!#(B $B$h$C$F!"(B(1)$B$N2r(B(x,y)$B$KBP$9$k(B|y|$B$O!">e3&(B101317c223$B$r;}$D!#(B
$B$?$@$7!"$3$N>e3&$O!" $B"#(Bc=1,...,100$B$K$D$$F!"BJ1_6J@~(BCcBN@0E@r(BThueBJ}Dx<0N@0?t2rH8+J7F!"(Bpari/gpBG5aakH!"0J2<Nh&KJk!#(B  c Cc(Z) 1 [1, 0], [-1, -1] 2 [0, -1] 3 [1, -1], [-5, -4] 4 - 5 - 6 [2, 1] 7 - 8 [2, 0], [-2, -2] 9 - 10 [2, -1], [4, 3] 11 [3, 2] 12 - 13 - 14 - 15 [-1, -2] 16 [0, -2] 17 [1, -2] 18 - 19 - 20 - 21 - 22 - 23 - 24 [2, -2], [-10, -8] 25 [3, 1]  c Cc(Z) 26 - 27 [3, 0], [-3, -3] 28 - 29 [3, -1] 30 - 31 - 32 - 33 - 34 [-6, -5] 35 - 36 - 37 - 38 - 39 - 40 - 41 - 42 - 43 [3, -2], [9, 7] 44 - 45 - 46 [-2, -3] 47 [63, 50] 48 [4, 2] 49 - 50 -  c Cc(Z) 51 - 52 - 53 [-1, -3] 54 [0, -3] 55 [1, -3], [29, 23] 56 - 57 - 58 - 59 - 60 - 61 - 62 [2, -3], [4, 1], [-34, -27] 63 - 64 [4, 0], [-4, -4] 65 - 66 [4, -1] 67 - 68 - 69 - 70 - 71 [5, 3] 72 - 73 - 74 - 75 -  c Cc(Z) 76 - 77 - 78 - 79 - 80 [4, -2], [8, 6] 81 [3, -3], [-15, -12] 82 [14, 11] 83 - 84 - 85 - 16 - 87 - 88 [6, 4] 89 [-7, -6] 90 - 91 - 92 - 93 [7, 5] 94 - 95 - 96 - 97 - 98 - 99 - 100 - B33G!"(BcB,40A4N)J}?tG"kH-!"D^j!"(B c = B&A(B3 (B&A(B B":(B Z, B&A(B > 0) BG"kH-!"(B Cc(Z) = {(B&A(B,0),(-B&A(B,-B&A(B)} BG"k3HO!"MF0WKJ,+k!#(B B"#(Bc=1,...,500000BKD$$$F!"BJ1_6J@~(BCc$B$N@0E@$r(Bpari/gp$B$G5a$a$k$H!"(B$B$3$N$h$&$K(B$B$J$k!#(B $B$?$@$7!"(BCc$B$,@0E@(B(x,y)$B$r;}$?$J$$bNO0lMw+i=|/!#(B c=1,...,500000BNCfG!"(B #Cc(Z)=0BHJk(BcBO!"(B488045B8D(B(97.6%), #Cc(Z)=1BHJk(BcBO!"(B 11150B8D(B(2.23%), #Cc(Z)=2BHJk(BcBO!"(B 648B8D(B(0.130%), #Cc(Z)=3BHJk(BcBO!"(B 139B8D(B(0.0278%), #Cc(Z)=4BHJk(BcBO!"(B 18B8D(B(0.00360%), #Cc(Z)>=5BHJk(BcBO!"(B 0B8D(B(0%) BG"k!#(B B33G!"(B {c \in [1..500000] | #Cc(Z)=4 } = { 899, 7192, 7750, 24273, 39302, 57536, 62000, 68923, 82522, 112375, 194184, 209250, 236629, 304606, 308357, 314416, 460288, 496000 } BG"k!#(B CcBN@0?t2r(B(x,y)BKBP7F!"(Bmax{|y|}/cB,Hf3SE*Bg-/Jk(BcBO!"0J2<Nh&KJk!#(B gp> read("zzz.gp") ii=11955 time = 1,852 ms. %1 = [11150, 648, 139, 18, 0, 0, 0, 0, 0, 0] gp> vpr(v,30) 3.319310344827586206896551724 ;thue(th,1450)=[[-2, -9], [-6064, -4813]] 2.371555224898477944139619951 ;thue(th,32259)=[[-96389, -76504]] 1.992094861660079051383399209 ;thue(th,253)=[[5, -4], [-635, -504]] 1.333333333333333333333333333 ;thue(th,3)=[[1, -1], [-5, -4]] 1.063829787234042553191489361 ;thue(th,47)=[[63, 50]] 1.000000000000000000000000000 ;thue(th,1)=[[1, 0], [-1, -1]] 0.8298275862068965517241379310 ;thue(th,11600)=[[-4, -18], [-12128, -9626]] 0.5928888062246194860349049877 ;thue(th,258072)=[[-192778, -153008]] 0.5000000000000000000000000000 ;thue(th,2)=[[0, -1]] 0.4980237154150197628458498023 ;thue(th,2024)=[[10, -8], [-1270, -1008]] 0.4450980392156862745098039215 ;thue(th,510)=[[8, 1], [-286, -227]] 0.4354838709677419354838709677 ;thue(th,62)=[[2, -3], [4, 1], [-34, -27]] 0.4181818181818181818181818181 ;thue(th,55)=[[1, -3], [29, 23]] 0.4055636896046852122986822840 ;thue(th,683)=[[349, 277]] 0.3688122605363984674329501915 ;thue(th,39150)=[[-6, -27], [-18192, -14439]] 0.3333333333333333333333333333 ;thue(th,24)=[[2, -2], [-10, -8]] 0.2999999999999999999999999999 ;thue(th,10)=[[2, -1], [4, 3]] 0.2659574468085106382978723404 ;thue(th,376)=[[126, 100]] 0.2500000000000000000000000000 ;thue(th,8)=[[2, 0], [-2, -2]] 0.2486296232185101840632392822 ;thue(th,17331)=[[11, -20], [5429, 4309]] 0.2443597618130566529076326849 ;thue(th,293383)=[[90325, 71691], [23, -52]] 0.2213438735177865612648221343 ;thue(th,6831)=[[15, -12], [-1905, -1512]] 0.2074568965517241379310344827 ;thue(th,92800)=[[-8, -36], [-24256, -19252]] 0.1968854282536151279199110122 ;thue(th,899)=[[-5, -8], [-223, -177], [43, 34], [11, 6]] 0.1959287531806615776081424936 ;thue(th,393)=[[-97, -77]] 0.1818181818181818181818181818 ;thue(th,11)=[[3, 2]] 0.1805644067156805055634630031 ;thue(th,450349)=[[-102453, -81317]] 0.1792452830188679245283018867 ;thue(th,106)=[[24, 19]] 0.1776656530891836802887025094 ;thue(th,29927)=[[-6699, -5317]] 0.1666666666666666666666666666 ;thue(th,6)=[[2, 1]] time = 87 ms.  [2004.07.28BDI5-(B] c=500001,...,1000000BKD$$$F!"BJ1_6J@~(BCc$B$N@0E@$r(Bpari/gp$B$G5a$a$k$H!"(B$B$3$N$h$&$K(B$B$J$k!#(B $B$?$@$7!"(BCc$B$,@0E@(B(x,y)$B$r;}$?$J$b$N$O0lMw$+$i=|$/!#(B c=500001,...,1000000$B$NCf$G!"(B
#Cc(Z)=0$B$H$J$k(Bc$B$O!"(B492603$B8D(B(98.5%), #Cc(Z)=1$B$H$J$k(Bc$B$O!"(B 7057$B8D(B(1.51%),
#Cc(Z)=2$B$H$J$k(Bc$B$O!"(B 284$B8D(B(0.0568%), #Cc(Z)=3$B$H$J$k(Bc$B$O!"(B 50$B8D(B(0.0100%),
#Cc(Z)=4$B$H$J$k(Bc$B$O!"(B 6$B8D(B(0.00120%), #Cc(Z)>=5$B$H$J$k(Bc$B$O!"(B 0$B8D(B(0%)
$B$G$"$k!#(B