a problem

[托斯卡尼艷陽下]

Fn is Fabonacci number

Prove that 5|Fn <=> 5|n



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Post by 托斯卡尼艷陽下
Fn is Fabonacci number
Prove that 5|Fn <=> 5|n

就數列除以5的餘數 依序為


1 1 2 3 0 3 3 1 4 0 4 4 3 2 0 2 2 4 1 0

然後循環

即可得到 5|Fn <=> 5|n
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[托斯卡尼艷陽下]

※ 引述《obel (托斯卡尼艷陽下)》之銘言：
: Fn is Fabonacci number
: Prove that 5|Fn <=> 5|n

我想起來了..

F(5n)=F(5n-1)+F(5n-2)
=2xF(5n-2)+F(5n-3)
=3xF(5n-3)+2xF(5n-4)
=5xF(5n-4)+3xF(5n-5) ...(1)

Claim: 5|F(5k) for k≧1 ...(*)

(1)k=1, F(5)=5 => 5|F(5)
(2)Assume that k=m (*) is true, which is 5|F(5m) => F(5m)=5s ...(2)

When k=m+1,

F(5k)=F(5m+5)=5xF(5m+1)+3xF(5m)
=5[F(5m+1)+3s]

So k=m+1 is also true by (1),(2) and the claim is proved.


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