Monday, November 29, 2010

Digits

Here I'll be looking at each square number and I'll add this number to its digits in order to find a square number.

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16, ..... 16 + 1 + 6 = 23
5^2 = 25, ..... 25 + 2 + 5 = 32
6^2 = 36, ..... 36 + 3 + 6 = 45
7^2 = 49, ..... 49 + 4 + 9 = 62
8^2 = 64, ..... 64 + 6 + 4 = 74
9^2 = 81, ..... 81 + 8 + 1 = 90

So far, the results did not give me any square number.
So, I'll continue with the next square numbers.

10^2 = 100 ..... 21^2 = 441
11^2 = 121 ..... 22^2 = 484
12^2 = 144 ..... 23^2 = 529
13^2 = 169 ..... 24^2 = 576
14^2 = 196 ..... 25^2 = 625
15^2 = 225 ..... 26^2 = 676
16^2 = 256 ..... 27^2 = 729
17^2 = 289 ..... 28^2 = 784
18^2 = 324 ..... 29^2 = 841
19^2 = 361 ..... 30^2 = 900
20^2 = 400 ..... 31^2 = 961

100 + 1 + 0 + 0 = 101
121 + 1 + 2 + 1 = 125
144 + 1 + 4 + 4 = 153
169 + 1 + 6 + 9 = 185
196 + 1 + 9 + 6 = 212
225 + 2 + 2 + 5 = 234
256 + 2 + 5 + 6 = 269
289 + 2 + 8 + 9 = 308
324 + 3 + 2 + 4 = 333
361 + 3 + 6 + 1 = 371
400 + 4 + 0 + 0 = 404
441 + 4 + 4 + 1 = 450
484 + 4 + 8 + 4 = 500
529 + 5 + 2 + 9 = 545
576 + 5 + 7 + 6 = 594
625 + 6 + 2 + 5 = 638
676 + 6 + 7 + 6 = 695
729 + 7 + 2 + 9 = 747
784 + 7 + 8 + 4 = 803
841 + 8 + 4 + 1 = 854
900 + 9 + 0 + 0 = 909
961 + 9 + 6 + 1 = 977

No squares found.

32^2 = 1024 ..... 49^2 = 2401 ..... 66^2 = 4356 ..... 83^2 = 6889
33^2 = 1089 ..... 50^2 = 2500 ..... 67^2 = 4489 ..... 84^2 = 7056
34^2 = 1156 ..... 51^2 = 2601 ..... 68^2 = 4624 ..... 85^2 = 7225
35^2 = 1225 ..... 52^2 = 2704 ..... 69^2 = 4761 ..... 86^2 = 7396
36^2 = 1296 ..... 53^2 = 2809 ..... 70^2 = 4900 ..... 87^2 = 7569
37^2 = 1369 ..... 54^2 = 2916 ..... 71^2 = 5041 ..... 88^2 = 7744
38^2 = 1444 ..... 55^2 = 3025 ..... 72^2 = 5184 ..... 89^2 = 7921
39^2 = 1521 ..... 56^2 = 3136 ..... 73^2 = 5329 ..... 90^2 = 8100
40^2 = 1600 ..... 57^2 = 3249 ..... 74^2 = 5476 ..... 91^2 = 8281
41^2 = 1681 ..... 58^2 = 3364 ..... 75^2 = 5625 ..... 92^2 = 8464
42^2 = 1764 ..... 59^2 = 3481 ..... 76^2 = 5776 ..... 93^2 = 8649
43^2 = 1849 ..... 60^2 = 3600 ..... 77^2 = 5929 ..... 94^2 = 8836
44^2 = 1936 ..... 61^2 = 3721 ..... 78^2 = 6084 ..... 95^2 = 9025
45^2 = 2025 ..... 62^2 = 3844 ..... 79^2 = 6241 ..... 96^2 = 9216
46^2 = 2116 ..... 63^2 = 3969 ..... 80^2 = 6400 ..... 97^2 = 9409
47^2 = 2209 ..... 64^2 = 4096 ..... 81^2 = 6561 ..... 98^2 = 9604
48^2 = 2304 ..... 65^2 = 4225 ..... 82^2 = 6724 ..... 99^2 = 9801

1024 + 1 + 0 + 2 + 4 = 1031 .......... 4356 + 4 + 3 + 5 + 6 = 4374
1089 + 1 + 0 + 8 + 9 = 1107 .......... 4489 + 4 + 4 + 8 + 9 = 4514
1156 + 1 + 1 + 5 + 6 = 1169 .... ..... 4624 + 4 + 6 + 2 + 4 = 4640
1225 + 1 + 2 + 2 + 5 = 1235 .......... 4761 + 4 + 7 + 6 + 1 = 4779
1296 + 1 + 2 + 9 + 6 = 1314 .......... 4900 + 4 + 9 + 0 + 0 = 4913
1369 + 1 + 3 + 6 + 9 = 1388 .......... 5041 + 5 + 0 + 4 + 1 = 5051
1444 + 1 + 4 + 4 + 4 = 1457 .......... 5184 + 5 + 1 + 8 + 4 = 5202
1521 + 1 + 5 + 2 + 1 = 1530 .......... 5329 + 5 + 3 + 2 + 9 = 5348
1600 + 1 + 6 + 0 + 0 = 1607 .......... 5476 + 5 + 4 + 7 + 6 = 5498
1681 + 1 + 6 + 8 + 1 = 1697 .......... 5625 + 5 + 6 + 2 + 5 = 5643
1764 + 1 + 7 + 6 + 4 = 1782 .......... 5776 + 5 + 7 + 7 + 6 = 5801
1849 + 1 + 8 + 4 + 9 = 1871 .......... 5929 + 5 + 9 + 2 + 9 = 5954
1936 + 1 + 9 + 3 + 6 = 1955 .......... 6084 + 6 + 0 + 8 + 4 = 6102
2025 + 2 + 0 + 2 + 5 = 2034 .......... 6241 + 6 + 2 + 4 + 1 = 6254
2116 + 2 + 1 + 1 + 6 = 2126 .......... 6400 + 6 + 4 + 0 + 0 = 6410
2209 + 2 + 2 + 0 + 9 = 2222 .......... 6561 + 6 + 5 + 6 + 1 = 6579
2304 + 2 + 3 + 0 + 4 = 2313 .......... 6724 + 6 + 7 + 2 + 4 = 6743
2401 + 2 + 4 + 0 + 1 = 2408 .......... 6889 + 6 + 8 + 8 + 9 = 6920
2500 + 2 + 5 + 0 + 0 = 2507 .......... 7056 + 7 + 0 + 5 + 6 = 7074
2601 + 2 + 6 + 0 + 1 = 2610 .......... 7225 + 7 + 2 + 2 + 5 = 7241
2704 + 2 + 7 + 0 + 4 = 2717 .......... 7396 + 7 + 3 + 9 + 6 = 7421
2809 + 2 + 8 + 0 + 9 = 2828 .......... 7569 + 7 + 5 + 6 + 9 = 7596
2916 + 2 + 9 + 1 + 6 = 2934 .......... 7744 + 7 + 7 + 4 + 4 = 7766
3025 + 3 + 0 + 2 + 5 = 3035 .......... 7921 + 7 + 9 + 2 + 1 = 7940
3136 + 3 + 1 + 3 + 6 = 3149 .......... 8100 + 8 + 1 + 0 + 0 = 8109
3249 + 3 + 2 + 4 + 9 = 3267 .......... 8281 + 8 + 2 + 8 + 1 = 8300
3364 + 3 + 3 + 6 + 4 = 3380 .......... 8464 + 8 + 4 + 6 + 4 = 8486
3481 + 3 + 4 + 8 + 1 = 3497 .......... 8649 + 8 + 6 + 4 + 9 = 8676
3600 + 3 + 6 + 0 + 0 = 3609 .......... 8836 + 8 + 8 + 3 + 6 = 8861
3721 + 3 + 7 + 2 + 1 = 3734 .......... 9025 + 9 + 0 + 2 + 5 = 9041
3844 + 3 + 8 + 4 + 4 = 3863 .......... 9216 + 9 + 2 + 1 + 6 = 9234
3969 + 3 + 9 + 6 + 9 = 3996 .......... 9409 + 9 + 4 + 0 + 9 = 9431
4096 + 4 + 0 + 9 + 6 = 4115 .......... 9604 + 9 + 6 + 0 + 4 = 9623
4225 + 4 + 2 + 2 + 5 = 4238 .......... 9801 + 9 + 8 + 0 + 1 = 9819

No squares found here.

In fact, we cannot find. A formal proof will be required.