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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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 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     72927      125     3431       343     219727      343     1331125     343     7291       729     491327      729     3375125     729     2197343     729     13311       1331    926127      1331    6859125     1331    4913343     1331    3375729     1331    21971       2197    1562527      2197    12167125     2197    9261343     2197    6859729     2197    49131       3375    2438927      3375    19683125     3375    156251331    2197    3375343     3375    12167729     3375    92611331    3375    68591       4913    3593727      4913    29791125     4913    24389343     4913    196832197    3375    4913729     4913    156251331    4913    121671       6859    5065327      6859    42875125     6859    359372197    4913    9261343     6859    29791729     6859    243891331    6859    196833375    4913    68592197    6859    156251       9261    6892127      9261    59319`

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

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