■x^2−92y^2=1(その13)

  an^2−5bn^2=1

から,

  an+1=161an+360bn,bn+1=72an+161bn

を逆算できるだろうか?

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 an^2−5bn^2=1

が成り立つ最小解は(a,b)=(9,4)である.

  (9+4√5)^n=an+bn√5

  (9−4√5)^n=an−bn√5

  an+1+bn+1√5=(9+4√5)(an+bn√5)

          =(9an+20bn)+(4an+9bn)√5

より

  an+1=9an+20bn,bn+1=4an+9bn

  an+2=9an+1+20bn+1=161an+360bn

  bn+2=4an+1+9bn+1=72an+161bn

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