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Lucas Digits Look (Posted on 2016-11-07) Difficulty: 3 of 5
L(N) denotes the Nth Lucas Number.

Find the first three digits of L(2016)

No Solution Yet Submitted by K Sengupta    
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Solution solution+ | Comment 4 of 10 |
The reference gives the value of Lucas number n, when n >=2, as the nearest integer to phi^n.
                                                             
phi ~= 1.618033988749895                                                             
2016 * log(phi) ~= 421.3190827439572   (common log)                                         
10^0.3190827439572 ~= 2.084888068563217 

The first three digits of L(2016) are 208.

In fact we should have lost only 3 or 4 significant digits in the log, so it's safe to say L(2016) begins 208488806856....

To be more exact,

    5   open "lucasdig.txt" for output as #2
   10   point 200
   20   Phi=(1+sqrt(5))/2
   30   L=2016*log(Phi)/log(10)
   50   N=10^L
   60   print N
   70   print #2,int(N+0.5)
   80   close #2

prints

 2084888068563090912458516390561325281171020479491164173354049239002708488595943349565836183183731118374225
 4888383410947905292915756002825362140440186026243514077500621846337171808675725220399882236988796256624844
 5402479607357252824391008958625008649722442717841891717457928730425527558619547497634874094593660339644192
 68550729415193388283501690725596998976916646809132118229619532708413826849176632242912696523533298094082 
 
as the exact value of all the digits. 


  Posted by Charlie on 2016-11-07 10:54:14
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