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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(4): computer solutioNs ....NOT YET Comment 6 of 6 |
(In reply to re(3): computer solutioNs ....NOT YET by Ady TZIDON)

Absolutely right; I had < signs instead of > signs. Embarassed.

    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:Crcrnd=int(Crc+0.5)
   75              :if abs(Crc-Crsrnd)>0.000000001 then
   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

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
17      459     2125
19      513     2375
21      567     2625
5       625     3645
23      621     2875
25      675     3125
135     625     1715
29      783     3625
31      837     3875
7       875     5103
33      891     4125
35      945     4375
3       1029    6591
37      999     4625
189     875     2401
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
53      1431    6625
55      1485    6875
57      1539    7125
13      1625    9477
59      1593    7375
297     1375    3773
61      1647    7625
5       1715    10985

  Posted by Charlie on 2013-09-30 00:28:11
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