Every integer > 19 is the sum of 6 nonzero squares.
The smallest squares containing k 19's :
196 = 142, 19061956 = 43662, 1901919321 = 436112,
191921971921 = 4380892, 1951919319441961 = 441805312.
192 = 361, a zigzag squares.
A1 + B1 + C1, A2 + B2 + C2, A3 + B3 + C3 are squares for A = 108, B = 124, C = 129:
1081 + 1241 + 1291 = 192, 1082 + 1242 + 1292 = 2092, 1083 + 1243 + 1293 = 23052.
In general {A = 108m2, B = 124m2, C = 129m2} has the same property (m = 1,2,...).
See the second primitive solution 71.
22k + 73k + 130k + 136k are squares for k = 1,2,3 (192, 2032, 22612).
42k + 66k + 108k + 145k are squares for k = 1,2,3 (192, 1972, 21612).
192 = 361, 3 + 61 = 82.
192 = 361, and 3 * 6 + 1 = 19.
192 is the 4th square which is the sum of 3 squares in just 3 ways:
12 + 62 + 182 = 62 + 62 + 172 = 62 + 102 + 152.
192 is the first square which is the sum of 5 distinct squares in just 7 ways.
192 = 361, where 36 and 1 are squares.
192 - 1 = 3 x 5!
192 = 30 + 32 + 33 + 34 + 35.
Komachi Fractions: 5184/207936 = (3/19)2, 20736/51984 = (12/19)2,
48735/91260 = (19/26)2, 3249/15876 = 6498/31752 = 16245/79380 = (19/42)2,
6498/570312 = (19/178)2, 3249/576081 = (19/253)2
Komachi equations:
192 = 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 35 equations,
192 = 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 42 equations,
192 = 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 31 equations,
192 = 122 / 32 * 42 * 52 - 62 - 782 + 92 = - 12 + 232 + 42 - 52 + 62 - 72 - 82 - 92
= 12 - 22 * 32 + 42 * 52 - 62 + 72 + 82 - 92,
192 = - 92 + 82 + 72 - 62 + 52 * 42 - 32 * 22 + 12 = - 92 + 82 - 72 + 62 - 52 - 42 - 32 + 212,
192 = 982 / 72 / 62 - 52 * 42 / 32 + 22 * 102 = - 92 - 82 - 72 + 62 - 52 + 42 * 32 + 22 * 102
= - 92 * 82 + 762 + 52 + 42 * 32 - 22 * 102,
192 = 93 - 83 - 73 + 63 - 543 / 33 / 23 + 103 = - 93 - 83 - 73 + 63 + 543 / 33 / 23 + 103.
(192 + 3) = (22 + 3)(72 + 3), (192 + 7) = (32 + 7)(42 + 7),
(192 - 1) = (22 - 1)(112 - 1) = (42 - 1)(52 - 1),
(192 - 1) = (22 - 1)(32 - 1)(42 - 1).
(1 + 2 + ... + 18)(19) = 572,
(13 + 23 + ... + 193) = (1 + 2 + ... + 19)2 = 1902,
(products of at most 3 factors)
12 + 22 + 32 + ... + 192 = 2470, which consists of different digits.
(15 + 25)(35 + 45 + 55 + 65)(75)(85)(95)(105 + 115)(125 + 135 + ... + 195) = 54619152230522882.
197 = 893871739, 89 + 3 + 87 + 173 + 9 = 192,
198 = 16983563041, 169 + 83 + 5 + 63 + 041 = 192,
199 = 322687697779, 3 + 226 + 87 + 6 + 9 + 7 + 7 + 7 + 9
= 32 + 2 + 68 + 76 + 97 + 7 + 79 = 32 + 2 + 68 + 76 + 97 + 77 + 9 = 192.
1910 = 6131066257801, 62 + 12 + 32 + 102 + 62 + 62 + 22 + 52 + 72 + 82 + 012 = 192,
6 + 1 + 3 + 1 + 06 + 6 + 257 + 80 + 1 = 6 + 13 + 10 + 66 + 257 + 8 + 01 = 192,
1911 = 116490258898219, 1 + 1 + 6 + 4 + 9 + 02 + 5 + 8 + 8 + 98 + 219 = 192 and more 30 equations,
1912 = 2213314919066161, 2 + 2 + 1 + 3 + 31 + 49 + 190 + 66 + 16 + 1 = 192 and more 62 equations.
Every integer greater than 19 is the sum of 6 nonzero squares.
192 = 361 appears in the decimal expressions of π and e:
π = 3.14159•••361••• (from the 1028th digit), 361 is the 8th 3-digit square in the expr. of π.
e = 2.71828•••361••• (from the 503th digit), 361 is the 7th 3-digit square in the expr. of e.
Page of Squares : First Upload September 29, 2003 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan