620
The smallest squares containing k 620's :
62001 = 2492,
2620620864 = 511922,
620620962025 = 7877952.
186k + 234k + 258k + 478k are squares for k = 1,2,3 (342, 6202, 120682).
(223 / 620)2 = 0.129367845... (Komachic).
Page of Squares : First Upload January 25, 2006 ; Last Revised March 17, 2011by Yoshio Mimura, Kobe, Japan
621
The smallest squares containing k 621's :
262144 = 5122,
6216218649 = 788432,
1662162159562129 = 407696232.
6212 = 385641, a square with different digits.
6212 = 363 + 95 + 67.
Komachi Fraction : 6941538 / 72 = (621 / 2)2.
Komachi equation: 6212 = 123 - 33 - 453 - 63 + 783 + 93.
(319 / 621)2 = 0.263874951... (Komachic).
3-by-3 magic squares consisting of different squares with constant 6212:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(4, 251, 568 344, 472, 211 517, 316, 136), | (8, 211, 584, 244, 536, 197, 571, 232, 76), |
(8, 389, 484, 416, 356, 293, 461, 328, 256), | (15, 246, 570, 354, 465, 210, 510, 330, 129), |
(16, 77, 616, 308, 536, 59, 539, 304, 52), | (29, 172, 596, 356, 484, 157, 508, 349, 76), |
(29, 332, 524, 388, 419, 244, 484, 316, 227), | (42, 129, 606, 186, 582, 111, 591, 174, 78), |
(56, 133, 604, 179, 584, 112, 592, 164, 91), | (56, 212, 581, 256, 539, 172, 563, 224, 136), |
(59, 244, 568, 328, 496, 179, 524, 283, 176), | (64, 307, 536, 421, 416, 188, 452, 344, 251), |
(66, 294, 543, 426, 417, 174, 447, 354, 246), | (76, 197, 584, 284, 536, 133, 547, 244, 164), |
(76, 284, 547, 419, 428, 164, 452, 349, 244), | (88, 301, 536, 389, 392, 284, 476, 376, 133), |
(111, 354, 498, 426, 318, 321, 438, 399, 186) |
6212 = 385641, 3 + 8 + 5 + 64 + 1 = 92,
6212 = 385641, 3 + 8 + 564 + 1 = 242,
6212 = 385641, 3 + 85 + 641 = 272,
6212 = 385641, 3 + 856 + 41 = 302.
by Yoshio Mimura, Kobe, Japan
622
The smallest squares containing k 622's :
46225 = 2152,
2622976225 = 512152,
172726226226225 = 131425352.
Komachi Fraction : 6222 = 20891736 / 54.
6222 = 386884, 38 + 6 + 8 + 8 + 4 = 82.
6222 = 473 + 55 + 67.
Page of Squares : First Upload June 27, 2005 ; Last Revised January 6, 2011by Yoshio Mimura, Kobe, Japan
623
The smallest squares containing k 623's :
962361 = 9812,
62324623201 = 2496492,
3162380623562304 = 562350482.
6232 + 6242 + 6252 + 6262 + ... + 28942 = 894602.
6232 = 2222 + 4022 + 4212 : 1242 + 2042 + 2222 = 3262.
12 + 22 + 32 + ... + 6232 = 80795624, the 5th 8-digit sum consisting of different digits.
Komachi equations:
6232 = 12 * 22 - 32 + 42 + 52 - 62 + 72 * 892 = - 12 * 22 + 32 - 42 - 52 + 62 + 72 * 892.
3-by-3 magic squares consisting of different squares with constant 6232:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 178, 597, 277, 534, 162, 558, 267, 74), | (6, 362, 507, 438, 363, 254, 443, 354, 258), |
(11, 78, 618, 222, 578, 69, 582, 219, 38), | (18, 187, 594, 363, 486, 142, 506, 342, 123), |
(27, 74, 618, 142, 603, 66, 606, 138, 43), | (27, 250, 570, 430, 405, 198, 450, 402, 155), |
(30, 123, 610, 402, 470, 75, 475, 390, 102), | (38, 174, 597, 426, 443, 102, 453, 402, 14), |
(43, 318, 534, 402, 394, 267, 474, 363, 178), | (54, 318, 533, 358, 453, 234, 507, 286, 222), |
(78, 331, 522, 362, 402, 309, 501, 342, 142), | (101, 198, 582, 258, 549, 142, 558, 218, 171), |
(138, 398, 459, 421, 402, 222, 438, 261, 358) |
6232 = 388129, 3 + 8 + 8 + 1 + 29 = 72,
6232 = 388129, 3 + 8 + 81 + 29 = 112,
6232 = 388129, 3 + 88 + 1 + 29 = 112,
6232 = 388129, 388 + 1 + 2 + 9 = 202.
by Yoshio Mimura, Kobe, Japan
624
The smallest squares containing k 624's :
4624 = 682,
866242624 = 294322,
6242162458624 = 24984322.
The squares which begin with 624 and end in 624 are
62466004624 = 2499322, 624207444624 = 7900682, 624782746624 = 7904322,
624997762624 = 7905682, 6240343732624 = 24980682,...
481k + 624k + 1404k + 1716k are squares for k = 1,2,3 (652, 23532, 904152).
6242 = 65 + 85 +105 + 125, the 9 th square which is the sum of 4 fifth powers.
252 + 262 + 272 + 282 + ... + 6242 = 90102.
(12 + 22 + ... + 242)(252 + 262 + ... + 6242) = 6307002.
(15)(25)(35 + 45 + 55 + 65) = 6242.
6242 = 253 + 393 + 683 = 323 + 523 + 603.
6242 = 389376, 3 + 8 + 9 + 3 + 7 + 6 = 62,
6242 = 389376, 38 + 93 + 7 + 6 = 122,
6242 = 389376, 3893 + 76 = 632.
by Yoshio Mimura, Kobe, Japan
625
The square of 25.
The smallest squares containing k 625's :
625 = 252,
1625625 = 12752,
62562515625 = 2501252.
The squares which begin with 625 and end in 625 are
6252855625 = 790752, 62512500625 = 2500252, 62537505625 = 2500752,
62562515625 = 2501252, 62587530625 = 2501752,...
6252 = 390625, a zigzag square with diffent digits.
The sum of (13x + 1)2 is 25412, x running from 0 through 48.
6252 = 390625, a square containing itself.
6252 is the first square which is the sum of 5 seventh powers: 57 + 57 + 57 + 57 + 57 = 58.
3-by-3 magic squares consisting of different squares with constant 6252:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 175, 600, 375, 480, 140, 500, 360, 105), | (0, 220, 585, 375, 468, 176, 500, 351, 132), |
(15, 144, 608, 400, 465, 120, 480, 392, 81), | (41, 120, 612, 312, 535, 84, 540, 300, 95), |
(48, 265, 564, 311, 480, 252, 540, 300, 95), | (0, 168, 599, 300, 535, 120, 545, 276, 132), |
(60, 225, 580, 256, 540, 183, 567, 220, 144), | (60, 300, 545, 329, 480, 228, 528, 265, 204), |
(76, 300, 543, 393, 400, 276, 480, 375, 140), | (105, 248, 564, 360, 489, 148, 500, 300, 225), |
(105, 360, 500, 396, 428, 225, 472, 279, 300), | (140, 375, 480, 417, 300, 356, 444, 400, 183) |
6252 = 390625, 3 *(or /) 9 * 0 + 625 = 39 * 0 + 625 = 625.
6252 = 390625, 3 + 9 + 0 + 6 + 2 + 5 = 52,
6252 = 390625, 35 + 95 + 05 + 65 + 25 + 55 = 2652.
6252 = 390625 appears in the decimal expression of π:
π = 3.14159•••390625••• (from the 96705th digit).
by Yoshio Mimura, Kobe, Japan
626
The smallest squares containing k 626's :
4626801 = 21512,
62662605625 = 2503252,
226264862662609 = 150421032.
6262 = 391876, a square with different digits.
6262± 3 are primes.
6262 + 6272 + 6282 + 6292 + ... + 37712 = 1334192.
(13 + 23 + ... + 573)(583 + 593 + ... + 2853)(2863 + 2873 + ... + 6263) = 129221719593122.
6262 = 14 + 54 + 54 + 254.
6262 = 391876, 39 * 18 - 76 = 626.
6262 = 391876, 391 + 87 + 6 = 222.
Page of Squares : First Upload June 27, 2005 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
627
The smallest squares containing k 627's :
106276 = 3262,
34062762721 = 1845612,
1627627281762769 = 403438632.
6272 = 822 + 4332 + 4462 : 6442 + 3342 + 282 = 7262.
6272 = (22 - 1)(3622 - 1) = (72 + 8)(832 + 8).
A cubic polynomial :
(X + 2882)(X + 6042)(X + 6272) = X3 + 9172X2 + 4541882X + 1090679042.
627k + 3667k + 8037k + 10773k are squares for k = 1,2,3 (1522, 139462, 13486962).
73777k + 81092k + 107426k + 130834k are squares for k = 1,2,3 (6272, 2016852, 664388012).
Komachi Square Sum : 6272 = 322 + 782 + 952 + 6142.
Komachi equations:
6272 = 1 * 234 * 5 * 6 * 7 * 8 + 9,
6272 = 123 * 33 / 43 * 53 + 673 + 83 + 93.
3-by-3 magic squares consisting of different squares with constant 6272:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(1, 278, 562, 322, 482, 239, 538, 289, 142), | (2, 79, 622, 401, 478, 62, 482, 398, 49), |
(2, 127, 614, 223, 574, 118, 586, 218, 47), | (2, 145, 610, 310, 530, 127, 545, 302, 70), |
(2, 335, 530, 415, 398, 250, 470, 350, 223), | (14, 353, 518, 382, 406, 287, 497, 322, 206), |
(26, 202, 593, 422, 433, 166, 463, 406, 118), | (27, 216, 588, 372, 468, 189, 504, 357, 108), |
(36, 387, 492, 432, 372, 261, 453, 324, 288), | (44, 253, 572, 352, 484, 187, 517, 308, 176), |
(47, 274, 562, 338, 463, 254, 526, 322, 113), | (58, 113, 614, 146, 602, 97, 607, 134, 82), |
(58, 278, 559, 398, 449, 182, 481, 338, 218), | (62, 194, 593, 287, 538, 146, 554, 257, 142), |
(62, 271, 562, 358, 478, 191, 511, 302, 202), | (65, 202, 590, 370, 490, 127, 502, 335, 170), |
(82, 257, 566, 433, 386, 238, 446, 422, 127), | (97, 394, 478, 434, 302, 337, 442, 383, 226), |
(106, 257, 562, 337, 502, 166, 518, 274, 223), | (134, 358, 497, 383, 446, 218, 478, 257, 314) |
6272 = 393129, 33 + 93 + 33 + 13 + 23 + 93 = 392,
6272 = 393129, 3 + 9 + 3 + 12 + 9 = 62,
6272 = 393129, 39 + 31 + 2 + 9 = 92,
6272 = 393129, 3 + 9 + 3 + 129 = 122,
6272 = 393129, 3 + 93 + 129 = 152,
6272 = 393129, 32 + 92 + 32 + 122 + 92 = 182.
by Yoshio Mimura, Kobe, Japan
628
The smallest squares containing k 628's :
116281 = 3412,
846286281 = 290912,
1276287628628224 = 357251682.
A cubic polynomial :
(X + 6282)(X + 11042)(X + 35912) = X3 + 38092X2 + 46133882X + 24896833922.
6282 + 6292 + 6302 + 6312 + ... + 6772 = 46152.
The square root of 628 is 25. 0 5 9 9 2 8 17 2 2 8 3 ...,
and 252 = 02 + 52 + 92 + 92 + 22 + 82 + 172 + 22 + 22 + 82 + 32.
82k + 298k + 628k + 673k are squares for k = 1,2,3 (412, 9712, 240732).
6282 = 394384, 3 + 9 + 4 + 384 = 202.
Page of Squares : First Upload June 27, 2005 ; Last Revised March 17, 2011by Yoshio Mimura, Kobe, Japan
629
The smallest squares containing k 629's :
1062961 = 10312,
6295629025 = 793452,
629362937629449 = 250871072.
6292 = 395641, a square with different digits.
173k + 245k + 397k + 629k are squares for k = 1,2,3 (382, 8022, 182022).
21386k + 48433k + 149702k + 176120k are squares for k = 1,2,3 (6292, 2371332, 945581992).
3-by-3 magic squares consisting of different squares with constant 6292:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 204, 595, 296, 525, 180, 555, 280, 96), | (12, 324, 539, 411, 404, 252, 476, 357, 204), |
(44, 219, 588, 357, 476, 204, 516, 348, 91), | (51, 136, 612, 228, 576, 109, 584, 213, 96), |
(51, 168, 604, 408, 469, 96, 476, 384, 147), | (72, 269, 564, 424, 396, 243, 459, 408, 136), |
(109, 324, 528, 432, 424, 171, 444, 333, 296), | (136, 333, 516, 408, 444, 179, 459, 296, 312) |
6292 + 6302 + 6312 + 6322 + ... + 21642 = 574242.
6292 = 395641, 3 + 9 + 5 + 6 + 41 = 82,
6292 = 395641, 39 + 56 + 4 + 1 = 102.
by Yoshio Mimura, Kobe, Japan