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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 computer solutios | Comment 2 of 6 |

  5    point 5
 10    for Tot=2 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)
 60          if Cra<>int(Cra) and Crb<>int(Crb) 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

finds

 3       81      375
 5       135     625
 7       189     875
 9       243     1125
 11      297     1375
 13      351     1625
 3       375     2187
 15      405     1875
 81      375     1029
 125     343     729
 17      459     2125
 19      513     2375
 21      567     2625
 5       625     3645
 23      621     2875
 25      675     3125
 135     625     1715
 29      783     3625
 125     729     2197
 31      837     3875
 7       875     5103
 33      891     4125
 35      945     4375
 3       1029    6591
 37      999     4625
 189     875     2401
 343     729     1331
 39      1053    4875
 81      1029    3993
 9       1125    6561
 41      1107    5125
 43      1161    5375
 45      1215    5625
 47      1269    5875
 243     1125    3087
 49      1323    6125
 11      1375    8019
 375     1029    2187
 51      1377    6375
 125     1331    4913
 53      1431    6625
 55      1485    6875
 57      1539    7125
 13      1625    9477
 
 etc.
 
 in order of incrasing A+B.

  Posted by Charlie on 2013-09-29 12:02:18
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