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Cubic Minimum (Posted on 2014-09-26) Difficulty: 3 of 5
S is the minimum positive integer whose cube root is of the form T + U, where T is a positive integer and U is a positive real number less than 0.00001.

Determine the value of S.

No Solution Yet Submitted by K Sengupta    
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Hints/Tips re: generalised approach. | Comment 6 of 7 |
(In reply to generalised approach. by broll)


Accepting your terminology, let S(n) denote a qualifying number, such that a cubic root  of S(n)^3+1 is a smallest positive integer number to produce a result with n zeroes immediately after the decimal point .

So a sequence  2, 6 , 19, 58, 183, 578, 1826, 5777, 18268, 57768  …fits the above description (n - a positive integer).

If we evaluate q(n)= S(n+1)/ S(n) we get 2,  3.1667, 3.0526,  3.1551, ….,3,1622, 3.16225…

It looks like the ratio is closer to  sqrt(10),  and lim S(n+1)/ S(n) it is likely to be this number.

So the formula  SC(n+1)=int ( S(n)*SQRT(10) ) seems more appropriate than yours.

SC- denotes a candidate for S,  however IMHO as of 58 and on you get not a candidate, but the true S.

I did not recheck my numbers and if I err, please let me know.

Edited on September 27, 2014, 5:21 am
  Posted by Ady TZIDON on 2014-09-27 04:24:22

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