Squares:680s

680

The smallest squares containing k 680's :
636804 = 798²,
35368068096 = 188064²,
746808816806809 = 27327803².

680² = (2² + 4)(9² + 4)(26² + 4).

20^k + 260^k + 265^k + 680^k are squares for k = 1,2,3 (35², 775², 18725²).

Komachi equation: 680² = 98² * 7² - 6² * 5² - 43² * 2² + 10².

Page of Squares : First Upload August 8, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

681

The smallest squares containing k 681's :
1681 = 41²,
2096815681 = 45791²,
68116812330681 = 8253291².

The squares which begin with 681 and end in 681 are
6813016681 = 82541², 68142403681 = 261041², 681105234681 = 825291²,
681382560681 = 825459², 681517942681 = 825541²,...

681³ = 315821241, and 3² + 1² + 5² + 8² + 2² + 1² + 24² + 1² = 681.

1 / 681 = 0.00 1 4 6 8 4 2 8 7 8 1 2 0 4 1 1 1 6 0 0 5 8 7 3 7 1 5 1 2 4 8 ...,
where the sum of the squares of its digits is 681.

3-by-3 magic squares consisting of different squares with constant 681²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 449, 512, 476, 368, 319, 487, 356, 316),	(6, 333, 594, 477, 426, 234, 486, 414, 237),
(8, 116, 671, 479, 476, 88, 484, 473, 76),	(8, 151, 664, 256, 616, 137, 631, 248, 64),
(18, 141, 666, 426, 522, 99, 531, 414, 102),	(25, 220, 644, 356, 545, 200, 580, 344, 95),
(31, 148, 664, 424, 524, 97, 532, 409, 116),	(31, 328, 596, 392, 479, 284, 556, 356, 167),
(44, 217, 644, 448, 476, 191, 511, 436, 112),	(52, 224, 641, 316, 577, 176, 601, 284, 148),
(64, 356, 577, 449, 412, 304, 508, 409, 196),	(80, 356, 575, 385, 500, 256, 556, 295, 260),
(104, 361, 568, 407, 424, 344, 536, 392, 151),	(112, 304, 599, 424, 503, 176, 521, 344, 272),
(127, 304, 596, 416, 511, 172, 524, 332, 281),	(161, 388, 536, 424, 484, 223, 508, 281, 356)

681² = 463761, 4 + 6 + 3 + 7 + 61 = 9²,
681² = 463761, 4 + 63 + 7 + 6 + 1 = 9²,
681² = 463761, 4 + 63 + 76 + 1 = 12²,
681² = 463761, 46 + 37 + 61 = 12².

Page of Squares : First Upload August 8, 2005 ; Last Revised July 27, 2009
by Yoshio Mimura, Kobe, Japan

682

The smallest squares containing k 682's :
682276 = 826²,
13682682729 = 116973²,
1568268268268689 = 39601367².

the square root of 682 is 26. 1 1 5 1 2 9 7 14 4 0 11 9 0 3 9 1 0 3 ..., where
26² = 1² + 1² + 5² + 1² + 2² + 9² + 7² + 14² + 4² + 0² + 11² + 9² + 0² + 3² + 9² + 1² + 0² + 3².
similar for 26. 1 15 1 2 9 7 1 4 4 0 1 1 9 0 3 9 10 3 ..., 26. 11 5 1 2 9 7 1 4 4 0 1 19 ..., 26. 11 5 1 2 9 7 14 4 0 1 1 9 0 3 9 1 0 3 ...

682² = (4² + 6)(5² + 6)(26² + 6).

682^k + 5084^k + 20553^k + 20770^k are squares for k = 1,2,3 (217², 29667², 4215907²).
33^k + 205^k + 441^k + 477^k are squares for k = 1,2,3 (34², 682², 14246²).

682² = 465124, 46 + 51 + 24 = 11².

Page of Squares : First Upload August 8, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

683

The smallest squares containing k 683's :
268324 = 518²,
966836836 = 31094²,
1683416834068369 = 41029463².

683 = (1² + 2² + 3² + ... + 682²) / (1² + 2² + 3² + ... + 77²).

683² + 684² + 685² + ... + 2146² = 56486².

1 / 683 = 0.0014641..., and 14641 = 121².

the square root of 683 is 26. 1 3 4, and 26 = 1² + 3² + 4².

683² = 242² + 441² + 462² : 264² + 144² + 242² = 386².

3-by-3 magic squares consisting of different squares with constant 683²:

10^k + 44^k + 57^k + 58^k are squares for k = 1,2,3 (13², 93², 683²).

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(6, 183, 658, 233, 618, 174, 642, 226, 57),	(6, 422, 537, 462, 393, 314, 503, 366, 282),
(9, 82, 678, 282, 618, 71, 622, 279, 42),	(26, 162, 663, 393, 546, 118, 558, 377, 114),
(42, 258, 631, 442, 471, 222, 519, 422, 138),	(55, 258, 630, 390, 530, 183, 558, 345, 190),
(57, 378, 566, 422, 426, 327, 534, 377, 198),	(71, 222, 642, 462, 489, 118, 498, 422, 201),
(78, 198, 649, 231, 622, 162, 638, 201, 138),	(82, 327, 594, 414, 498, 217, 537, 334, 258),
(90, 215, 642, 370, 558, 135, 567, 330, 190)

Page of Squares : First Upload August 8, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

684

The smallest squares containing k 684's :
31684 = 178²,
14684107684 = 121178²,
3368406843684 = 1835322².

The squares which begin with 684 and end in 684 are
68475375684 = 261678², 684223443684 = 827178², 684461691684 = 827322²,
6841771399684 = 2615678², 6842524735684 = 2615822²,...

Cubic Polynomials :
(X + 209²)(X + 528²)(X + 684²) = X³ + 889²X² + 403788² + 75480768²,
(X + 448²)(X + 684²)(X + 2679²) = X³ + 2801²X² + 2211828²X + 820931328².

the square root of 684 is 26. 1 5 ..., and 26 = 1² + 5².

684² = 467856, 4 + 6 + 7 + 8 + 5 + 6 = 6².

684² = (2² + 8)(7² + 8)(26² + 8).

684² = 467856, 4 + 6 + 7 + 8 + 56 = 9²,
684² = 467856, 4 + 6 + 78 + 56 = 12²,
684² = 467856, 46 + 7 + 85 + 6 = 12².

684² = 467856 appears in the decimal expression of e:
e = 2.71828•••467856••• (from the 21561st digit).

Page of Squares : First Upload August 8, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

685

The smallest squares containing k 685's :
568516 = 754²,
16858685281 = 129841²,
636856856856529 = 25236023².

685³ - 684³ + 683³ - 682³ + .. + 1³ = 12691².

1 / 685 = 0.00 1 4 5 9 8 5 4 0 1 4 5 9 8 5 4 0 1 4 5 9 8 5 4 0 1 ...,
the sum of the squares of its digits is 685.

685² = 1³ + 40³ + 74³.

3-by-3 magic squares consisting of different squares with constant 685²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 411, 548, 440, 420, 315, 525, 352, 264),	(12, 116, 675, 291, 612, 100, 620, 285, 60),
(24, 243, 640, 460, 480, 165, 507, 424, 180),	(35, 180, 660, 420, 516, 163, 540, 413, 84),
(45, 192, 656, 240, 620, 165, 640, 219, 108),	(45, 240, 640, 368, 549, 180, 576, 332, 165),
(60, 285, 620, 480, 460, 165, 485, 420, 240),	(60, 325, 600, 480, 408, 269, 485, 444, 192),
(80, 165, 660, 264, 620, 123, 627, 240, 136),	(100, 315, 600, 357, 540, 224, 576, 280, 243),
(100, 315, 600, 453, 420, 296, 504, 440, 147)

685² = 469225, 4 + 69 + 2 + 25 = 10²,
685² = 469225, 4 + 69 + 22 + 5 = 10²,
685² = 469225, 46 + 9 + 2 + 2 + 5 = 8²,
685² = 469225, 46² + 92² + 2² + 5² = 103².

Page of Squares : First Upload August 8, 2005 ; Last Revised July 27, 2009
by Yoshio Mimura, Kobe, Japan

686

The smallest squares containing k 686's :
36864 = 192²,
26468686864 = 162692²,
526861968668676 = 22953474².

686² = 470596, a square with different digits.

686²± 3 are primes.

686² is the ninth square which is the sum of 6 seventh powers : 7⁶ + 7⁶ + 7⁶ + 7⁶.

Komachi equations:
686² = 98² * 7² + 6² - 5² - 4² + 3² - 2² */ 1² = 98² * 7² - 6² + 5² + 4² - 3² + 2² * 1².

38² + 39² + 40² + ... + 686² = 10384².

the square root of 686 is 26. 1 9 16 0 17 0 7 ...,
where 26² = 1² + 9² + 16² + 0² + 17² + 0² + 7².

The 4-by-4 magic square consisting of different squares with constant 686:

0²	1²	18²	19²
7²	24²	5²	6²
14²	3²	16²	15²
21²	10²	9²	8²

Page of Squares : First Upload August 8, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

687

The smallest squares containing k 687's :
687241 = 829²,
36874368729 = 192027²,
1687166879568769 = 41075137².

687² = 471969, a zigzag square.

687² + 688² + 689² + ... + 1984² = 49973²,
687² + 688² + 689² + ... + 215055² = 57579143².

(1³ + 2³ + ... + 263³)(264³ + 265³ + ... + 312³)(313³ + 314³ + ... + 687³) = 275628098592000².

687² = 471969, 4 + 7 + 1 + 9 + 6 + 9 = 6².

687² = 242² + 443² + 466² : 664² + 344² + 242² = 786².

3-by-3 magic squares consisting of different squares with constant 687²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 133, 674, 283, 614, 122, 626, 278, 53),	(2, 283, 626, 469, 418, 278, 502, 466, 53),
(2, 298, 619, 331, 542, 262, 602, 299, 142),	(2, 382, 571, 469, 418, 278, 502, 389, 262),
(9, 228, 648, 312, 576, 207, 612, 297, 96),	(10, 262, 635, 310, 565, 238, 613, 290, 110),
(10, 395, 562, 437, 430, 310, 530, 362, 245),	(11, 82, 682, 422, 539, 58, 542, 418, 59),
(14, 197, 658, 427, 518, 146, 538, 406, 133),	(37, 194, 658, 262, 613, 166, 634, 242, 107),
(37, 350, 590, 410, 485, 262, 550, 338, 235),	(53, 334, 598, 466, 422, 277, 502, 427, 194),
(58, 178, 661, 362, 571, 122, 581, 338, 142),	(58, 277, 626, 373, 514, 262, 574, 362, 107),
(86, 298, 613, 443, 494, 178, 518, 373, 254),	(154, 293, 602, 322, 574, 197, 587, 238, 266),
(154, 418, 523, 443, 466, 242, 502, 283, 374)

687² = 471969, 47 + 19 + 6 + 9 = 9²,
687² = 471969, 471 + 96 + 9 = 24².

Page of Squares : First Upload August 8, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

688

The smallest squares containing k 688's :
6889 = 83²,
4688688676 = 68474²,
266886688668889 = 16336667².

688² = 473344, a square with just 3 kinds of digits.

Cubic Polynomial :
(X + 576²)(X + 688²)(X + 903²) = X³ + 1273²X² + 901968²X + 357848064².

688² + 689² + 690² + ... + 2223² = 59632².

688² = 473344, 4 + 7 + 3 + 3 + 4 + 4 = 5²,
688² = 473344, 473 + 3 + 4 + 4 = 22².

688² = (1² + 7)(6² + 7)(37² + 7),

686² = 473344 appears in the decimal expression of e:
e = 2.71828•••473344••• (from the 84979th digit).

Page of Squares : First Upload August 8, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

689

The smallest squares containing k 689's :
13689 = 117²,
2268902689 = 47633²,
241689526898689 = 15546367².

The squares which begin with 689 and end in 689 are
68967688689 = 262617², 68976092689 = 262633², 689094233689 = 830117²,
689120797689 = 830133², 689509354689 = 830367²,...

Komachi equations: 689² = 9² + 8² * 76² + 54² * 3² * 2² */ 1².

689² = 474721, 4 + 7 + 4 + 7 + 2 + 1 = 5².

689² = 201² + 464² + 468² : 864² + 464² + 102² = 986².

3-by-3 magic squares consisting of different squares with constant 689²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 265, 636, 364, 540, 225, 585, 336, 140),	(9, 292, 624, 384, 516, 247, 572, 351, 156),
(23, 264, 636, 456, 471, 212, 516, 428, 159),	(36, 184, 663, 481, 468, 156, 492, 471, 104),
(49, 144, 672, 288, 616, 111, 624, 273, 104),	(76, 351, 588, 384, 468, 329, 567, 364, 144),
(96, 201, 652, 247, 624, 156, 636, 212, 159),	(104, 273, 624, 399, 536, 168, 552, 336, 239),
(156, 393, 544, 468, 464, 201, 481, 324, 372)

689² = 474721, 4 + 7 + 472 + 1 = 22²,
689² = 474721, 474 + 7 + 2 + 1 = 22².

Page of Squares : First Upload February 6, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan