440
The smallest squares containing k 440's :
14400 = 1202,
3440174409 = 586532,
124409440594404 = 111538982.
4402± 3 are primes.
The sum of (19x + 3)2 is 12622, where x runs over 0, 1, ..., 23.
62k + 73k + 266k + 440k are squares for k = 1,2,3 (292, 5232, 102292).
Komachi Square Sum : 4402 = 52 + 62 + 832 + 972 + 4212.
4402 = 602 + 612 + 622 + ... + 922.
Page of Squares : First Upload February 28, 2005 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
441
The square of 21.
The smallest squares containing k 441's :
441 = 212,
441882441 = 210212,
4417441336441 = 21017712.
The squares which begin with 441 and end in 441 are
441882441 = 210212, 4419457441 = 664792, 44108820441 = 2100212,
44196232441 = 2102292, 441200164441 = 6642292,...
441 is a reversible square, 441 = 212 and 144 = 122.
441 is an exchangeable square : (441 = 122, 144 = 212).
441 is a square whose digits are squares and the sum of the digits 4, 4 and 1 is a square (9).
1/441 = 0.0022675736961451247..., the sum of the squares of its digits is 441.
4412± 2 are primes.
4412 = (22 + 5)(42 + 5)(322 + 5).
4412 = 194481, where 1 = 12, 9 = 32, 4 = 22, and 81 = 92.
4412 = 194481, 1 - 9 + 448 + 1 = 441.
4412 = 33 + 213 + 573 = 143 + 423 + 493.
33k + 205k + 441k + 477k are squares for k = 1,2,3 (342, 6822, 142462).
Komachi equations:
4412 = 122 - 32 * 42 + 562 * 72 / 82 * 92 = 122 / 32 - 42 + 562 * 72 / 82 * 92
= 122 / 32 / 42 * 562 * 72 / 82 * 92 = - 122 + 32 * 42 + 562 * 72 / 82 * 92
= - 122 / 32 + 42 + 562 * 72 / 82 * 92,
4412 = 14 * 24 / 34 * 44 * 5674 / 84 / 94 = 14 / 24 / 34 / 44 * 5674 * 84 / 94
= 984 / 74 / 64 * 544 / 34 / 24 * 14 = 984 / 74 / 64 * 544 / 34 / 24 / 14
= 94 * 84 * 74 / 64 / 54 * 44 / 324 * 104 = 94 / 84 * 74 / 64 / 54 / 44 * 324 * 104
= 984 / 74 * 64 / 54 * 44 / 324 * 104.
3-by-3 magic squares consisting of different squares with constant 4412:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(3, 114, 426, 246, 354, 93, 366, 237, 66), | (6, 222, 381, 282, 291, 174, 339, 246, 138), |
(8, 121, 424, 256, 344, 103, 359, 248, 64), | (14, 238, 371, 301, 266, 182, 322, 259, 154), |
(16, 92, 431, 244, 361, 68, 367, 236, 64), | (16, 209, 388, 284, 292, 169, 337, 256, 124), |
(17, 64, 436, 176, 401, 52, 404, 172, 41), | (40, 284, 335, 305, 260, 184, 316, 215, 220), |
(44, 113, 424, 151, 404, 92, 412, 136, 79), | (44, 164, 407, 239, 352, 116, 368, 209, 124), |
(47, 256, 356, 304, 239, 212, 316, 268, 151), | (51, 174, 402, 258, 339, 114, 354, 222, 141), |
(52, 169, 404, 271, 332, 104, 344, 236, 143), | (79, 248, 356, 284, 244, 233, 328, 271, 116) |
4412 = 194481, 19 + 4 + 4 + 8 + 1 = 62,
4412 = 194481, 1 + 94 + 48 + 1 = 122,
4412 = 194481, 19 + 44 + 81 = 122,
4412 = 194481, 1 + 94 + 481 = 242,
4412 = 194481, 1944 + 81 = 452.
by Yoshio Mimura, Kobe, Japan
442
The smallest squares containing k 442's :
442225 = 6652,
4425442576 = 665242,
1442442344291001 = 379794992.
4422 = 195364, a square with different digits.
4422 + 4432 + 4442 + 4452 + ... + 5142 = 40882.
4422 = 195364, 19 + 5 + 36 + 4 = 82.
4422 = (42 + 1)(52 + 1)(212 + 1).
12818k + 44642k + 47294k + 90610k are squares for k = 1,2,3 (4422, 1122682, 306721482).
4422 = 195364 appears in the decimal expression of e:
e = 2.71828•••195364••• (from the 78292nd digit)
by Yoshio Mimura, Kobe, Japan
443
The smallest squares containing k 443's :
443556 = 6662,
24438443584 = 1563282,
144315244344384 = 120131282.
1 / 443 = 0.00225..., 225 = 152.
3-by-3 magic squares consisting of different squares with constant 4432:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(7, 66, 438, 186, 398, 57, 402, 183, 34), | (18, 105, 430, 295, 318, 90, 330, 290, 57), |
(38, 126, 423, 153, 402, 106, 414, 137, 78), | (39, 182, 402, 258, 318, 169, 358, 249, 78), |
(42, 263, 354, 297, 246, 218, 326, 258, 153), | (87, 214, 378, 246, 342, 137, 358, 183, 186) |
4432 + 4442 + 4452 + 4462 + ... + 10102 = 177502.
4432 = 196249, 1 + 9 + 6 + 24 + 9 = 72,
4432 = 196249, 1 + 9 + 62 + 49 = 112.
by Yoshio Mimura, Kobe, Japan
444
The smallest squares containing k 444's :
1444 = 382,
36114441444 = 1900382,
144457274445444 = 120190382.
The squares which begin with 444 and end in 444 are
444171597444 = 6664622, 444272905444 = 6665382, 444838309444 = 6669622,
444939693444 = 6670382, 4441396081444 = 21074622,...
1 / 444 = 0.00225..., 225 = 152.
4442 = 197136, a square with odd digits except the last digit 6.
If N is an integer ending in 038, 462, 538, or 962, then N2 ends in 444.
Komachi equations:
4442 = 122 * 32 * 42 + 52 / 62 * 72 * 82 * 92,
4442 = - 13 - 23 * 33 + 43 - 563 - 73 + 83 * 93.
4442 = 197136, and 19997919396 = 1414142.
4442 = 197136, 1 + 9 + 7 + 13 + 6 = 62,
4442 = 197136, 19 + 7 + 1 + 3 + 6 = 62.
Page of Squares : First Upload February 28, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan
445
The smallest squares containing k 445's :
44521 = 2112,
44594458276 = 2111742,
644544552445924 = 253878822.
4452 is the second square which is the sum of ten sixth powers.
4452 = 198025, a square with different digits.
4452 = 198025, 1 + 9 + 8 + 0 + 2 + 5 = 52.
4452 = (12 + 4)(1992 + 4).
3-by-3 magic squares consisting of different squares with constant 4452:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 195, 400, 267, 320, 156, 356, 240, 117), | (12, 184, 405, 309, 288, 140, 320, 285, 120), |
(21, 120, 428, 300, 320, 75, 328, 285, 96), | (40, 99, 432, 180, 400, 75, 405, 168, 76), |
(40, 180, 405, 216, 363, 140, 387, 184, 120), | (48, 139, 420, 264, 348, 85, 355, 240, 120), |
(75, 212, 384, 300, 309, 112, 320, 240, 195), | (2, 94, 437, 283, 338, 74, 346, 277, 58), |
(75, 212, 384, 300, 309, 112, 320, 240, 195) |
by Yoshio Mimura, Kobe, Japan
446
The smallest squares containing k 446's :
446224 = 6682,
4465446976 = 668242,
446144671446769 = 211221372.
4462 = 198916, a zigzag square.
4462 = 198916, 1 + 98 + 91 + 6 = 142.
(13 + 23 + ... + 1083)(1093 + 1103 + ... + 2183)(2193 + 2203 + ... + 4463) = 131782635784802.
Page of Squares : First Upload February 28, 2005 ; Last Revised July 21, 2006by Yoshio Mimura, Kobe, Japan
447
The smallest squares containing k 447's :
447561 = 6692,
6447447616 = 802962,
44744754479104 = 66891522.
The square root of 447 is 21.142..., 21 = 12 + 42 + 22.
4472 = 199809, 1 + 9 + 9 + 8 + 0 + 9 = 62.
4472± 2 are primes.
Komachi equations:
4472 = 984 / 74 + 64 + 54 * 44 + 34 + 24 * 14 = 984 / 74 + 64 + 54 * 44 + 34 + 24 / 14.
3-by-3 magic squares consisting of different squares with constant 4472:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 94, 437, 283, 338, 74, 346, 277, 58), | (2, 107, 434, 187, 394, 98, 406, 182, 43), |
(5, 122, 430, 178, 395, 110, 410, 170, 53), | (5, 278, 350, 310, 250, 203, 322, 245, 190), |
(11, 158, 418, 262, 341, 122, 362, 242, 101), | (12, 207, 396, 252, 324, 177, 369, 228, 108), |
(46, 242, 373, 278, 277, 214, 347, 254, 122), | (53, 226, 382, 278, 283, 206, 346, 262, 107), |
(98, 203, 386, 226, 362, 133, 373, 166, 182) |
Page of Squares : First Upload February 28, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan
448
The smallest squares containing k 448's :
4489 = 672,
3944844864 = 628082,
68244484484484 = 82610222.
4482 = 84 + 164 + 164 + 164.
4482 = (12 + 7)(72 + 7)(212 + 7).
Komachi equations:
4482 = 92 * 82 * 72 / 62 / 52 * 42 / 32 * 22 * 102 = 982 / 72 * 62 / 52 * 42 / 32 * 22 * 102.
Cubic Polynomials :
(X + 4482)(X + 6842)(X + 26792) = X3 + 28012X2 + 22118282X + 8209313282,
(X + 4482)(X + 15842)(X + 233312) = X3 + 233892X2 + 384125282X + 165564241922.
(13 + 23 + ... + 1213)(1223 + 1232 + ... + 1753)(1763 + 1772 + ... + 4483) = 99152664090482.
Page of Squares : First Upload February 28, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
449
The smallest squares containing k 449's :
11449 = 1072,
4494495681 = 670412,
7744493449449 = 27828932.
The squares which begin with 449 and end in 449 are
{
44989379449 = 2121072, 449043391449 = 6701072, 449091640449 = 6701432,
449378507449 = 6703572, 449426774449 = 6703932,...
4492 - 1 = 5 x 8!
The square root of 449 is 21.1896..., 212 = 182 + 92 + 62.
Komachi equation: 4492 = - 92 * 82 + 72 * 652 - 42 * 32 + 22 - 102.
(32 - 1)(42 - 1)(62 - 1)(72 - 1) = 4492 - 1,
(22 - 1)(52 - 1)(62 - 1)(92 - 1) = 4492 - 1.
3-by-3 magic squares consisting of different squares with constant 4492:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(28, 204, 399, 231, 336, 188, 384, 217, 84), | (33, 144, 424, 296, 312, 129, 336, 289, 72), |
(39, 116, 432, 192, 396, 89, 404, 177, 84), | (39, 208, 396, 276, 324, 143, 352, 231, 156), |
(44, 144, 423, 252, 359, 96, 369, 228, 116), | (84, 271, 348, 296, 228, 249, 327, 276, 136) |
4492 = 201601, 2 + 0 + 1 + 60 + 1 = 82.
(13 + 23 + ... + 1223)(1233 + 1243 + ... + 3693)(3703 + 3713 + ... + 4493) = 379124183037602.
Page of Squares : First Upload February 28, 2005 ; Last Revised June 8, 2010by Yoshio Mimura, Kobe, Japan