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 Imperfectly Cubed (Posted on 2013-09-29)
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 No Rating

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 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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