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620 - 629

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, 2011
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload June 27, 2005 ; Last Revised January 6, 2011
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, 2011
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload June 27, 2005 ; Last Revised August 17, 2013
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.

Page of Squares : First Upload June 27, 2005 ; Last Revised March 17, 2011
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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload June 27, 2005 ; Last Revised June 6, 2009
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, 2014
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload June 27, 2005 ; Last Revised December 7, 2013
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, 2011
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload June 27, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan