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The numbers can be seen as squares or cubes however some are both, and they stay.
16 is the the square of 4.
512 is the cube of 8.
Each of the other numbers can be represented as both a square and a cube:
1 = 1*1 1 = 1*1*1
64 = 8*8 64 = 4*4*4
4096 = 64*64 4096 = 16*16*16
46656 = 216*216 46656 = 36*36*36
1000000 = 1000*1000 1000000 = 100*100*100
2985984 = 1728*1728 2985984 = 144*144*144
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Note 1:
The set was derived from a sequence of numbers which had the property of being square as well as being cube, see 3rd column in table below. Finding a square and cube which were neither of the other was easy.
Note 2:
The idea arose partly from Brian Smith's "Square Patterns" and something with which that I had been toying.
The thought basically was: "Can a given number exist as both a square and a cube?" Having some doubts (possibly created from reading "Fermat's Last Theorem") I constructed a table as below in a spreadsheet:
A A^3 (A^3)^2 (A*A)^3
1 1 1 1
2 8 64 64
3 27 729 729
4 64 4096 4096
5 125 15625 15625
6 ......
The table was my source of data.
Note 2:
Charlie made a point about 6th root in a response to one comment.
The full series can be generated as x^6; x^(2*3), that to which Charlie referred. |
