The first integer which is the sum of two non-zero squares: 2 = 12 + 12.
The smallest squares containing k 2's are :
25 = 52, 225 = 152, 22201 = 1492, 2002225 = 14152,
21022225 = 45852, 212722225 = 145852, 11222224225 = 1059352,
132922222225 = 3645852, 12222722202201 = 34961012.
Every prime number of the form 4k + 1 can be written uniquely as the sum of 2 squares.
13 + 23 = 32.
(1/1)3 + (1/2)3 = (32/23)
n2 - (n-1)2 + (n-2)2 - (n-3)2 + ... - 12 = a2, (n = 2u2, v2 - 2u2 = 1).
n2 - (n-1)2 + (n-2)2 - (n-3)2 + ... + 12 = b2, (n = v2, v2 - 2u2 = -1).
Example-1: 82 - 72 + 62 - 52 + 42 - 32 +22 - 12 = 62.
Example-2: 492 - 482 + 472 - 462 + ... + 32 - 22 + 12 = 352
In the set {2, 167, 674, 6722} the sum of any two numbers is a square :
2 + 167 = 132, 2 + 674 = 262, 2 + 6722 = 822,
167 + 674 = 292, 167 + 6722 = 832, 674 + 6722 = 862.
Also, the sets {2, 359, 482, 3362}, {2, 1022, 2114, 6722} and {2, 1442, 4487, 7394} have the same property.
Komachi Fractions: 22 = 15768/3942 = 17568/4392 = 23184/5796 = 31824/7956 = 81576/20394 = 81756/20439
Komachi equations:
22 = 1 + 2 + 3 + 4 - 5 * 6 + 7 + 8 + 9 = 1 + 2 + 3 - 4 * 5 - 6 + 7 + 8 + 9
= 1 + 2 + 3 * 4 + 5 - 6 + 7 - 8 - 9, and more 274 equations,
22 = 9 + 8 + 7 + 6 - 5 * 4 - 3 - 2 - 1 = 9 + 8 + 7 + 6 - 5 * 4 - 3 * 2 * 1
= 9 + 8 + 7 - 6 - 5 - 4 - 3 - 2 * 1, and more 303 equations,
22 = 9 + 8 + 7 + 6 - 5 - 4 + 3 - 2 * 10 = 9 + 8 + 7 - 6 + 5 + 4 - 3 - 2 * 10
= 9 + 8 + 7 - 6 + 5 - 4 - 3 - 2 - 10, and more 207 equations,
22 = 12 - 22 * 32 - 42 + 52 - 62 + 72 - 82 + 92 = 122 + 32 - 42 + 52 + 62 - 72 - 82 - 92
= 122 * 32 / 42 + 52 - 62 - 72 + 82 - 92, and more 4 equations,
22 = 92 * 82 / 72 / 62 / 542 * 32 * 212 = 982 / 72 / 62 * 542 / 32 / 212
= 92 - 82 * 72 / 62 + 52 * 42 / 32 / 22 - 12, and more 4 equations,
22 = 982 / 72 + 62 - 542 / 32 - 22 + 102 = 92 + 82 - 72 + 62 - 52 - 42 + 32 + 22 - 102
= 92 - 82 - 72 - 62 - 52 - 42 + 32 + 22 + 102, and more 5 equations,
22 = 93 - 83 + 73 - 63 - 53 - 43 * 33 / 23 + 13.
217 = 131072, 12 + 32 + 12 + 072 + 22 = 26,
218 = 262144, 22 + 62 + 22 + 142 + 42 = 28,
230 = 1073741824, 12 + 072 + 32 + 72 + 42 + 182 + 242 = 210,
235 = 34359738368, 32 + 432 + 592 + 72 + 382 + 362 + 82 = 213,
240 = 1099511627776, 102 + 92 + 92 + 52 + 12 + 12 + 62 + 22 + 72 + 72 + 72 + 62 = 29,
245 = 35184372088832, 32 + 52 + 182 + 42 + 32 + 72 + 202 + 82 + 82 + 82 + 322 = 211,
245 = 35184372088832, 32 + 52 + 182 + 42 + 372 + 22 + 0882 + 832 + 22 = 214,
246 = 70368744177664, 702 + 362 + 82 + 72 + 42 + 412 + 72 + 72 + 62 + 62 + 42 = 213,
247 = 140737488355328, 142 + 072 + 372 + 42 + 82 + 82 + 352 + 52 + 322 + 82 = 212,
251 = 2251799813685248, 22 + 22 + 52 + 12 + 72 + 92 + 92 + 812 + 32 + 62 + 852 + 22 + 482 = 214.
Page of Squares : First Upload August 17, 2003 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan