Friday, December 3, 2010

The cubes: a(n) = n^3

0^3 = 0
1^3 = 1
2^3 = 8

3^3 = 27
4^3 = 64

5^3 = 125
6^3 = 216
7^3 = 343
8^3 = 512
9^3 = 729

10^3 = 1000 ..... 16^3 = 4096
11^3 = 1331 ..... 17^3 = 4913
12^3 = 1728 ..... 18^3 = 5832
13^3 = 2197 ..... 19^3 = 6859
14^3 = 2744 ..... 20^3 = 8000
15^3 = 3375 ..... 21^3 = 9261

22^3 = 10648 ..... 35^3 = 42875
23^3 = 12167 ..... 36^3 = 46656
24^3 = 13824 ..... 37^3 = 50653
25^3 = 15625 ..... 38^3 = 54872
26^3 = 17576 ..... 39^3 = 59319
27^3 = 19683 ..... 40^3 = 64000
28^3 = 21952 ..... 41^3 = 68921
29^3 = 24389 ..... 42^3 = 74088
30^3 = 27000 ..... 43^3 = 79507
31^3 = 29791 ..... 44^3 = 85184
32^3 = 32768 ..... 45^3 = 91125
33^3 = 35937 ..... 46^3 = 97336
34^3 = 39304 .........................


47^3 = 103823 ..... 65^3 = 274625 ..... 83^3 = 571787
48^3 = 110592 ..... 66^3 = 287496 ..... 84^3 = 592704
49^3 = 117649 ..... 67^3 = 300763 ..... 85^3 = 614125
50^3 = 125000 ..... 68^3 = 314432 ..... 86^3 = 636056
51^3 = 132651 ..... 69^3 = 328509 ..... 87^3 = 658503
52^3 = 140608 ..... 70^3 = 343000 ..... 88^3 = 681472
53^3 = 148877 ..... 71^3 = 357911 ..... 89^3 = 704969
54^3 = 157464 ..... 72^3 = 373248 ..... 90^3 = 729000
55^3 = 166375 ..... 73^3 = 389017 ..... 91^3 = 753571
56^3 = 175616 ..... 74^3 = 405224 ..... 92^3 = 778688
57^3 = 185193 ..... 75^3 = 421875 ..... 93^3 = 804357
58^3 = 195112 ..... 76^3 = 438976 ..... 94^3 = 830584
59^3 = 205379 ..... 77^3 = 456533 ..... 95^3 = 857375
60^3 = 216000 ..... 78^3 = 474552 ..... 96^3 = 884736
61^3 = 226981 ..... 79^3 = 493039 ..... 97^3 = 912673
62^3 = 238328 ..... 80^3 = 512000 ..... 98^3 = 941192
63^3 = 250047 ..... 81^3 = 531441 ..... 99^3 = 970299
64^3 = 262144 ..... 82^3 = 551368 ...........................

100^3 = 1000000


Notice that
1 + 8 = 9 = 3^2
1 + 8 + 27 = 36 = 6^2
1 + 8 + 27 + 64 = 100 = 10^2
1 + 8 + 27 + 64 + 125 = 225 = 15^2
1 + 8 + 27 + 64 + 125 + 216 = 441 = 21^2

So it seems that the sum of cubes is always a square, if you start with 1.

Could you provide a formal proof?

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