## Monday, November 29, 2010

### Arithmetic Progression with squares

An arithmetic progression (AP) is

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81

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

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

The squares 1, 25 and 49 are in arithmetic progression, with a common difference of 24.

Find others.

Note: we can have only three squares in arithmetic progression, that is to say, it is not possible to have 4 squares in arithmetic progression. A proof will be needed.