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Imperfectly Cubed (Posted on 2013-09-29) Difficulty: 3 of 5
Consider three distinct odd positive integers A, B and C - none of which is a perfect cube.

Can the respective cube roots of A, B and C form three consecutive terms of an arithmetic sequence?

If so, provide an example. If not, prove it.

No Solution Yet Submitted by K Sengupta    
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Solution re(2): computer solutios | Comment 4 of 6 |
(In reply to re: computer solutios by Ady TZIDON)

Yes, rounding errors caused the cube roots to differ from an integer by a tiny amount in those instances.  The corrected program:

    5    point 5
   10    for Tot=84 to 999999 step 2
   20       for A=1 to int(Tot/2) step 2
   30          B=Tot-A
   40          Cra=A^(1/3)
   50          Crb=B^(1/3)
   55            Crarnd=int(Cra+0.5):Crbrnd=int(Crb+0.5)
   60          if abs(Cra-Crarnd)<0.000000001 and abs(Crb-Crbrnd)<0.0000000001 and A<>B then
   70              :Crc=Crb+Crb-Cra
   80              :C=Crc*Crc*Crc
   90              :Crnd=int(C+0.5)
  100              :if abs(C-Crnd)<0.000000000001 and Crnd@2<>0 then
  110                   :print A,B,Crnd:Ct=Ct+1
  115                   :if Ct@44=0 then while inkey="":wend:print
  120        next A
  130     next Tot

1       125     729
27      125     343
1       343     2197
27      343     1331
125     343     729
1       729     4913
27      729     3375
125     729     2197
343     729     1331
1       1331    9261
27      1331    6859
125     1331    4913
343     1331    3375
729     1331    2197
1       2197    15625
27      2197    12167
125     2197    9261
343     2197    6859
729     2197    4913
1       3375    24389
27      3375    19683
125     3375    15625
1331    2197    3375
343     3375    12167
729     3375    9261
1331    3375    6859
1       4913    35937
27      4913    29791
125     4913    24389
343     4913    19683
2197    3375    4913
729     4913    15625
1331    4913    12167
1       6859    50653
27      6859    42875
125     6859    35937
2197    4913    9261
343     6859    29791
729     6859    24389
1331    6859    19683
3375    4913    6859
2197    6859    15625
1       9261    68921
27      9261    59319

  Posted by Charlie on 2013-09-29 20:00:04
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