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670 - 679

670

The smallest squares containing k 670's :
67081 = 2592,
2867067025 = 535452,
54670467026704 = 73939482.

(12 + 22 + ... + 1752)(1762 + 1772 + ... + 5392)(5402 + 5412 + ... + 6702) = 662080419002.

Komachi equation: 6702 = 122 * 32 * 42 * 52 * 672 / 82 / 92.

Page of Squares : First Upload August 1, 2005 ; Last Revised June 25, 2010
by Yoshio Mimura, Kobe, Japan

671

The smallest squares containing k 671's :
1671849 = 12932,
6713671969 = 819372,
2671867161671449 = 516901072.

6712 + 6722 + 6732 + ... + 161282 = 11825372.

6712 = 450241, 4 + 5 + 0 + 2 + 4 + 1 = 42.

(12 + 22 + 32 + ... + 112) + (12 + 22 + 32 + ... + 1102) = 6712.

6712 = 563 + 653.

20130k + 20801k + 142252k + 267058k are squares for k = 1,2,3 (6712, 3039632, 1481292892).
58377k + 64416k + 126148k + 201300k are squares for k = 1,2,3 (6712, 2529672, 1031051892).

3-by-3 magic squares consisting of different squares with constant 6712:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(5, 150, 654, 354, 555, 130, 570, 346, 75),(14, 162, 651, 306, 581, 138, 597, 294, 86),
(14, 261, 618, 294, 558, 229, 603, 266, 126),(18, 114, 661, 219, 626, 102, 634, 213, 54),
(18, 149, 654, 446, 486, 123, 501, 438, 86),(21, 194, 642, 378, 534, 149, 554, 357, 126),
(54, 283, 606, 382, 486, 261, 549, 366, 122),(59, 138, 654, 186, 634, 117, 642, 171, 94),
(59, 282, 606, 402, 501, 194, 534, 346, 213),(94, 411, 522, 453, 346, 354, 486, 402, 229),
(114, 373, 546, 411, 474, 238, 518, 294, 309),(122, 366, 549, 459, 354, 338, 474, 437, 186)

6712 = 450241 appears in the decimal expression of e:
  e = 2.71828•••450241••• (from the 62416th digit).

Page of Squares : First Upload August 1, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

672

The smallest squares containing k 672's :
6724 = 822,
396726724 = 199182,
386724456726721 = 196653112.

6722 = (72 - 1)(972 - 1).

Cubic Polynomial :
(X + 362)(X + 4272)(X + 6722) = X3 + 7972X2 + 2883722X + 103299842.

6722 = 243 + 563 + 642.

Komachi equations:
6722 = 12 / 3 * 4 * 56 * 7 * 8 * 9,
6722 = 12 * 22 * 34562 * 72 / 82 / 92 = - 92 * 82 * 72 + 62 * 52 * 42 / 32 * 212.

6722 = 243 + 563 + 643.

6722 = 451584, 4 + 5 + 15 + 8 + 4 = 62,
6722 = 451584, 4 + 51 + 5 + 84 = 122,
6722 = 451584, 45 + 15 + 84 = 122.

Page of Squares : First Upload August 1, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

673

The smallest squares containing k 673's :
126736 = 3562,
45673673796 = 2137142,
1044673673673744 = 323214122.

6732 is the third square which is the sum of 8 sixth powers :
16 + 26 + 46 + 66 + 66 + 66 + 66 + 86.

6732 = 452929, a zigzag square.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
673 - 5365 - 7034 - 6056 - ... - 10138 - 1446 - 2312 - 673
(Note f(673) = 62 + 732 = 5365,   f(5365) = 532 + 652 = 7034, etc. See 37)

(32 - 7)(62 - 7)(82 - 7)(122 - 7) = 6732 - 7.

82k + 298k + 628k + 673k are squares for k = 1,2,3 (412, 9712, 240732).

(12 + 22 + ... + 62)(72 + 82 + ... + 1822)(1832 + 1842 + ... + 6732) = 1356515162.

3-by-3 magic squares consisting of different squares with constant 6732:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(7, 348, 576, 432, 444, 263, 516, 367, 228),(12, 81, 668, 231, 628, 72, 632, 228, 39),
(12, 336, 583, 367, 492, 276, 564, 313, 192),(16, 228, 633, 327, 556, 192, 588, 303, 124),
(47, 276, 612, 324, 528, 263, 588, 313, 96),(60, 423, 520, 452, 360, 345, 495, 380, 252),
(81, 268, 612, 452, 432, 249, 492, 441, 128),(88, 159, 648, 192, 632, 129, 639, 168, 128),
(88, 396, 537, 423, 452, 264, 516, 303, 308),(135, 348, 560, 452, 465, 180, 480, 340, 327)

6732 = 452929, 4 + 5 + 2 + 9 + 29 = 72,
6732 = 452929, 4 + 5 + 29 + 2 + 9 = 72.

Page of Squares : First Upload August 1, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

674

The smallest squares containing k 674's :
667489 = 8172,
67400467456 = 2596162,
867467486745601 = 294528012.

10k + 218k + 542k + 674k are squares for k = 1,2,3 (382, 8922, 218122).

6742 = 454276, 4 + 5 + 42 + 7 + 6 = 82,
6742 = 454276, 45 + 4 + 2 + 7 + 6 = 82,
6742 = 454276, 45 + 42 + 7 + 6 = 102,
6742 = 454276, 4 + 5 + 4 + 276 = 172.

Page of Squares : First Upload February 6, 2006 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

675

The smallest squares containing k 675's :
586756 = 7662,
6756675601 = 821992,
1067546754675556 = 326733342.

Komachi Square Sum : 6752 = 93 + 183 + 243 + 573 + 633.

6752 = 455625 : 45 + 5 + 625 = 675

3-by-3 magic squares consisting of different squares with constant 6752:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 82, 670, 418, 526, 65, 530, 415, 50),(2, 289, 610, 335, 530, 250, 586, 302, 145),
(12, 141, 660, 309, 588, 120, 600, 300, 75),(12, 384, 555, 435, 420, 300, 516, 363, 240),
(14, 305, 602, 398, 490, 239, 545, 350, 190),(22, 79, 670, 254, 622, 65, 625, 250, 50),
(34, 287, 610, 463, 434, 230, 490, 430, 175),(49, 430, 518, 470, 350, 335, 482, 385, 274),
(50, 175, 650, 250, 610, 145, 625, 230, 110),(50, 175, 650, 454, 490, 97, 497, 430, 154),
(50, 250, 625, 350, 545, 190, 575, 310, 170),(50, 250, 625, 415, 482, 226, 530, 401, 118),
(58, 394, 545, 470, 415, 250, 481, 358, 310),(65, 170, 650, 362, 559, 110, 566, 338, 145),
(70, 209, 638, 385, 538, 134, 550, 350, 175),(75, 300, 600, 456, 420, 267, 492, 435, 156),
(79, 278, 610, 470, 415, 250, 478, 454, 145),(98, 386, 545, 430, 385, 350, 511, 398, 190),
(110, 230, 625, 274, 593, 170, 607, 226, 190),(113, 266, 610, 350, 550, 175, 566, 287, 230),
(118, 335, 574, 449, 470, 182, 490, 350, 305),(120, 285, 600, 336, 552, 195, 573, 264, 240),
(166, 335, 562, 362, 530, 209, 545, 250, 310) 

6752 = 455625, 4 + 5 + 5 + 62 + 5 = 92,
6752 = 455625, 45 + 5 + 6 + 25 = 92,
6752 = 455625, 4 + 5 + 562 + 5 = 242.

Page of Squares : First Upload August 1, 2005 ; Last Revised July 21, 2009
by Yoshio Mimura, Kobe, Japan

676

The square of 26.

The smallest squares containing k 676's :
676 = 262,
169676676 = 130262,
95316676676676 = 97630262.

The squares which begin with 676 and end in 676 are
67613520676 = 2600262,   676463480676 = 8224742,   676549020676 = 8225262,
6760135200676 = 26000262,   6762465024676 = 26004742,...

Komachi equations:
6762 = 14 * 2344 * 564 / 74 / 84 / 94 = 14 * 2344 / 564 * 74 * 84 / 94.

61642 = 5812 + 5822 + 5832 + ... + 6762

(13 + 23 +... + 263)(273 + 283 + ... + 1563)(1573 + 1583 + ... + 6763) = 9817602958802.

(13 + 23 + ... + 6753)(6763) = 40099644002.

(12 + 22 + 32 + ... + 662) + (12 + 22 + 32 + ... + 1022) = 6762.

6762 = 456976, 4 + 5 + 6 + 9 + 76 = 102.

Page of Squares : First Upload August 1, 2005 ; Last Revised June 25, 2010
by Yoshio Mimura, Kobe, Japan

677

The smallest squares containing k 677's :
677329 = 8232,
8677667716 = 931542,
6773677967755449 = 823023572.

6772 = 458329, a square with different digits.

46152 = 6282 + 6292 + 6302 + 6312 + ... + 6772.

the square root of 677 is 26. 0 1 9 22 3 6 6 2 5 ...,
where 262 = 02 + 12 + 92 + 222 + 32 + 62 + 62 + 22 + 52.

3-by-3 magic squares consisting of different squares with constant 6772:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 132, 664, 232, 624, 123, 636, 227, 48),(3, 196, 648, 452, 483, 144, 504, 432, 133),
(20, 195, 648, 360, 552, 155, 573, 340, 120),(24, 228, 637, 448, 483, 156, 507, 416, 168),
(43, 228, 636, 372, 524, 213, 564, 363, 92),(48, 304, 603, 396, 477, 272, 547, 372, 144),
(84, 277, 612, 308, 564, 213, 597, 252, 196),(108, 344, 573, 384, 507, 232, 547, 288, 276)

6772 = 458329, 4 + 5 + 8 + 3 + 29 = 72,
6772 = 458329, 4 + 5 + 83 + 29 = 112.

Page of Squares : First Upload August 1, 2005 ; Last Revised July 21, 2009
by Yoshio Mimura, Kobe, Japan

678

The smallest squares containing k 678's :
467856 = 6842,
44678967876 = 2113742,
3826786785216784 = 618610282.

77482 = 5102 + 5112 + 5122 + 5132 + ... + 6782.

the square root of 678 is 26. 0 3 8 4 3 3 13 2 5 8 3 0 7 4 0 7 4 0 3 7 5 6 7 ...,
where 262 = 02 + 32 + 82 + 42 + 32 + 32 + 132 + 22 + 52 + 82 + 32 + 02 + 72 + 42 + 02 + 72 + 42 + 02 + 32 + 72 + 52 + 62 + 72.

6782 = 223 + 453 + 713.

6782 = 459684, 4 + 5 + 9 + 6 + 8 + 4 = 62.

6782 + 6792 + 6802 + ... + 22502 = 22512 + 22522 + 22532 + ... + 28222.

28137k + 48477k + 150177k + 232893k are squares for k = 1,2,3 (6782, 2827262, 1271026262).

The 4-by-4 magic square consisting of different squares with constant 678:

02 72102232
112222 32 82
142 12202 92
192122132 22

6782 = 459684, 4 + 59 + 6 + 8 + 4 = 92,
6782 = 459684, 45 + 9 + 6 + 84 = 122,
6782 = 459684, 45 + 96 + 84 = 152.

6782 = 459684 appears in the decimal expressions of π:
  π = 3.14159•••459684••• (from the 12872nd digit),
  (459684 is the seventh 6-digit square in the expression of π.)

Page of Squares : First Upload August 1, 2005 ; Last Revised September 7, 2011
by Yoshio Mimura, Kobe, Japan

679

The smallest squares containing k 679's :
1679616 = 12962,
32367967921 = 1799112,
5679679439867904 = 753636482.

6792 = 152 + 162 + 172 + ... + 1112.

6792 = (62 + 63 + 64 + 65 + 66 + 67 + 68)2 + (69 + 70 + 71 + 72 + 73 + 74 + 75)2.

1 / 679 = 0.00 1 4 7 2 7 5 4 0 5 0 0 7 3 6 3 7 7 0 2 5 0 3 6 8 1 8 8 5 1 ...,
where the sum of the squares of its digits is 679.

6792 = 461041, 4 + 6 + 1 + 0 + 4 + 1 = 42.

Komachi equation: 6792 = - 12 + 22 + 32 + 452 * 62 + 72 * 892.

3-by-3 magic squares consisting of different squares with constant 6792:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 186, 653, 221, 618, 174, 642, 211, 66),(6, 243, 634, 381, 526, 198, 562, 354, 141),
(13, 306, 606, 474, 438, 211, 486, 419, 222),(18, 131, 666, 419, 522, 114, 534, 414, 67),
(18, 274, 621, 429, 486, 202, 526, 387, 186),(22, 114, 669, 354, 573, 86, 579, 346, 78),
(45, 390, 554, 446, 435, 270, 510, 346, 285),(51, 122, 666, 258, 621, 94, 626, 246, 93),
(51, 302, 606, 346, 534, 237, 582, 291, 194),(54, 141, 662, 237, 626, 114, 634, 222, 99),
(54, 365, 570, 445, 450, 246, 510, 354, 275),(66, 387, 554, 418, 414, 339, 531, 374, 198),
(86, 246, 627, 339, 562, 174, 582, 291, 194),(94, 342, 579, 474, 381, 302, 477, 446, 186)

Page of Squares : First Upload August 1, 2005 ; Last Revised June 25, 2010
by Yoshio Mimura, Kobe, Japan