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Little cubes (Posted on 2010-02-27) Difficulty: 4 of 5
The difference of the cubes of two non-negative successive integers is a square of an integer. Find few possible pairs.
How about a general formula or evaluation algorithm for a(n)?

See The Solution Submitted by Ady TZIDON    
Rating: 4.0000 (2 votes)

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Some Thoughts computer exploration | Comment 1 of 5

Each line of the output from the program shows, in order, n, n^3, (n+1)^3, (n+1)^3-n^3 and sqrt((n+1)^3-n^3):

run
 0  0  1  1  1
 7  343  512  169  13
 104  1124864  1157625  32761  181
 1455  3080271375  3086626816  6355441  2521
 20272  8330859163648  8332092086417  1232922769  35113
 282359  22511524427594279  22511763608256000  239180661721  489061
 3932760  60826430604528576000  60826477004344027081  46399815451081  6811741
OK
list
   10   for I=1 to 10000000
   20     Cube=I*I*I
   30     D=Cube-Pcube:Sr=int(sqrt(D)+0.5)
   40     if Sr*Sr=D then
   50       :print I-1;Pcube;Cube;D;Sr
   90     Pcube=Cube
  100   next
OK

Unfortunately Sloane's OLEIS seems to be unavailable this morning. Otherwise it might have a formula for this.


  Posted by Charlie on 2010-02-27 14:29:40
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