Integral Points on Conic Curves: x^2-Dy^2=±1(D \in [2,999])

[2003.08.16]x²-Dy²=±1 (D \in [2,999])の整点

• x^2-Dy^2=±1の基本解を計算するプログラム(pari/gp)

gp> read("pell.gp")
time = 42 ms.
gp> findpell(1,10000,50)
   realprecision = 57 significant digits (50 digits displayed)
d=2; [1, 1, -1]
d=2; [3, 2, 1]
d=3; [2, 1, 1]
d=5; [2, 1, -1]
d=5; [9, 4, 1]
d=6; [5, 2, 1]
...省略...
   realprecision = 105 significant digits (100 digits displayed)
d=9997; [1056881013819720204460013572819482, 10570395816504471539188482748085, -1]
d=9997; [2233994954745199219608947478540746520621707535213601737381917496649, 22343301294045950038591020734822406588061732170848426457972383940, 1]
   realprecision = 57 significant digits (50 digits displayed)
d=9998; [9999, 100, 1]
d=9999; [100, 1, 1]
time = 36,687 ms.
gp> quit
Good bye!
bash-2.05a$ 

• D=9949
  x=23551019614858223475933893515741198183163217312913587552899320396564478041197360918469501097146448985821854465768234479384482435117587576296319428592757548743265811454938493105633433315887574461850060798834186249,
  y=236113054062810988826514929828649213339688520849720684015415366388626019230322623673232286474879711003505448178417385617250641629212134427833135509077013929303770208680820795381507114806491325360400076633910900.
  
• D=9349
  x=51656307989267703540879380845975127917680122854237934497655785917557388336490256146317144854190586157776781818049349744794264045796317561017465497628741492011059434201801,
  y=534245389472923247131114573159529285543703702417501076227299174052709611618962327512263802885455987900834865331119140126590408279603786893623302902270665924551562453420
  

• x^2-Dy^2=±1, D \in [1000,1999]
• x^2-Dy^2=±1, D \in [2000,2999]
• x^2-Dy^2=±1, D \in [3000,3999]
• x^2-Dy^2=±1, D \in [4000,4999]
• x^2-Dy^2=±1, D \in [5000,5999]
• x^2-Dy^2=±1, D \in [6000,6999]
• x^2-Dy^2=±1, D \in [7000,7999]
• x^2-Dy^2=±1, D \in [8000,8999]
• x^2-Dy^2=±1, D \in [9000,9999]

[参考文献]

• [1]G.H.Hardy, E.M.Wright, "An Introduction to the Theory of Numbers 5th edition", Oxford University Press, New York, p129-153, 1979, ISBN0-19-853171-0.
  
• [2]John Cremona, "G13NUM: Number Theory Cource Notes 2000", p28-34.
  

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