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'Two' many roots (Posted on 2019-03-14) Difficulty: 3 of 5
Find the largest positive integer k possible in the following expression, n >= 0

((√2)1+(√2)2)*((√2)3+(√2)4)*...*((√2)2019+(√2)2020) = 2k + n

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 1 of 2
It would appear to be 511334, which is the truncated (i.e., integer part) of the base-2 logarithm of the given product on the LHS.

    4   kill "2manyrts.txt":open "2manyrts.txt" for output as #2
    5   point 20
   10   Sr2=sqrt(2):L2=log(2)
   20   for First=1 to 2019 step 2
   30    Fctr=Sr2^First+Sr2^(First+1)
   40    Tot=Tot+log(Fctr)/L2
   50    print First,Tot
   60   next
   70   print #2,Tot
   80   close #2

The file output is the base-2 log of the LHS:

 511334.26883619524809237004849354997890418828482963785991627526969
 5108191698252338906400265749178427037 

Running with higher precision shows 

 511334.26883619524809237004849354997890418828482963785991627526969
 510819169825233890640026574917853667276291555925708390581132661578
 5932879169868971308 

indicating a loss of 6 digits of precision (the last six digits of the first given value are wrong).

Edited on March 14, 2019, 3:56 pm
  Posted by Charlie on 2019-03-14 15:54:56

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