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760

The smallest squares containing k 760's :
57600 = 240²,
2476057600 = 49760²,
6676027607601 = 2583801².

Komachi equation: 760² = - 9³ + 87³ + 6³ + 5³ - 43³ - 2³ - 10³.

760² + 761² + 762² + ... + 98441² = 17832269².

(1³ + 2³ + ... + 608³)(609³ + 610³ + ... + 760³) = 41127592128².

Page of Squares : First Upload October 3, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

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761

The smallest squares containing k 761's :
4761 = 69²,
4944761761 = 70319²,
57610847612761 = 7590181².

The squares which begin with 761 and end in 761 are
76137916761 = 275931²,   761135849761 = 872431²,   761376659761 = 872569²,
761572127761 = 872681²,   761813006761 = 872819²,...

761² = 579121, 5 + 7 * 9 * 12 * 1 = 761.

761² = 579121, 5 + 7 + 9 + 1 + 2 + 1 = 5²,
761² = 579121, 57 + 91 + 21 = 13².

1³ - 2³ + 3³ - 4³ + 5³ - ... - 760³ + 761³ = 14859².

Komachi Square Sum : 761² = 2² + 6² + 31² + 98² + 754².

The square root of 761 is 27.586228448267443433301195759...,
and 5² + 8² + 6² + 2² + 2² + 8² + ... + 5² + 9² = 27².

3-by-3 magic squares consisting of different squares with constant 761²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 87, 756, 399, 644, 72, 648, 396, 49),	(4, 279, 708, 384, 612, 239, 657, 356, 144),
(41, 204, 732, 276, 687, 176, 708, 256, 111),	(60, 239, 720, 489, 540, 220, 580, 480, 111),
(72, 336, 679, 441, 536, 312, 616, 423, 144),	(81, 384, 652, 428, 564, 279, 624, 337, 276),
(84, 472, 591, 516, 399, 392, 553, 444, 276),	(105, 364, 660, 500, 465, 336, 564, 480, 175),
(148, 456, 591, 489, 396, 428, 564, 463, 216)

761² = 579121 appears in the decimal expression of e:
  e = 2.71828•••579121••• (from the 138877th digit)

Page of Squares : First Upload October 3, 2005 ; Last Revised August 31, 2009
by Yoshio Mimura, Kobe, Japan

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762

The smallest squares containing k 762's :
762129 = 873²,
5762176281 = 75909²,
3643476276276225 = 60361215².

762^k + 1938^k + 2334^k + 3066^k are squares for k = 1,2,3 (90², 4380², 221940²).
210^k + 606^k + 762^k + 1338^k are squares for k = 1,2,3 (54², 1668², 55404²).

762² = 580644, 5 + 8 + 0 + 64 + 4 = 9²,
762² = 580644, 5 + 80 + 644 = 27².

Page of Squares : First Upload October 3, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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763

The smallest squares containing k 763's :
37636 = 194²,
40077637636 = 200194²,
676376376374169 = 26007237².

763² = 582169, a square with different digits.

3-by-3 magic squares consisting of different squares with constant 763²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 393, 654, 426, 542, 327, 633, 366, 218),	(6, 87, 758, 537, 538, 66, 542, 534, 57),
(9, 258, 718, 498, 542, 201, 578, 471, 162),	(38, 135, 750, 450, 610, 87, 615, 438, 110),
(38, 222, 729, 519, 542, 138, 558, 489, 178),	(56, 357, 672, 483, 504, 308, 588, 448, 189),
(78, 457, 606, 486, 498, 313, 583, 354, 342),	(87, 218, 726, 362, 654, 153, 666, 327, 178),
(102, 281, 702, 498, 558, 151, 569, 438, 258),	(137, 426, 618, 474, 438, 407, 582, 457, 186),
(138, 430, 615, 470, 537, 270, 585, 330, 362),	(146, 423, 618, 522, 394, 393, 537, 498, 214),
(162, 394, 633, 423, 582,254,614,297,342)

763² = 582169, 5 + 8 + 21 + 6 + 9 = 7²,
763² = 582169, 5 + 82 + 169 = 16²,
763² = 582169, 5 + 821 + 6 + 9 = 29².

Page of Squares : First Upload October 3, 2005 ; Last Revised August 31, 2009
by Yoshio Mimura, Kobe, Japan

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764

The smallest squares containing k 764's :
1764 = 42²,
376476409 = 19403²,
73867647647641 = 8594629².

The squares which begin with 764 and end in 764 are
7648901764 = 87458²,   76429025764 = 276458²,   76475477764 = 276542²,
764676793764 = 874458²,   764823709764 = 874542²,...

190^k + 314^k + 410^k + 530^k are squares for k = 1,2,3 (38², 764², 15988²).

764² = 583696, 5 + 8 + 36 + 9 + 6 = 8²,
764² = 583696, 5 + 83 + 696 = 28².

(1² + 2² + 3² + ... +91²) + (1² + 2² + 3² + ... + 99²) = 764².

764² = 49³ + 60³ + 63³.

Page of Squares : First Upload October 3, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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765

The smallest squares containing k 765's :
765625 = 875²,
7657650064 = 87508²,
2867657656765225 = 53550515².

765² = 585225, a square consisting of just 3 kinds of digits.

765² = (6² + 9)(114² + 9).

Komachi Square Sum : 765² = 9² + 12² + 354² + 678².

(1 + 2 + 3 + ... + 9)(10 + 11 + ... + 24)(25 + 26) = 765².

3-by-3 magic squares consisting of different squares with constant 765²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 360, 675, 459, 540, 288, 612, 405, 216),	(4, 235, 728, 272, 680, 221, 715, 260, 80),
(8, 244, 725, 419, 608, 200, 640, 395, 140),	(11, 140, 752, 400, 640, 125, 652, 395, 64),
(11, 160, 748, 448, 605, 136, 620, 440, 85),	(15, 150, 750, 330, 678, 129, 690, 321, 78),
(15, 150, 750, 510, 561, 102, 570, 498, 111),	(20, 85, 760, 139, 748, 80, 752, 136, 35),
(20, 85, 760, 344, 680, 67, 683, 340, 56),	(20, 340, 685,524, 493, 260, 557, 476, 220),
(30, 210, 735, 273, 690, 186, 714, 255, 102),	(32, 245, 724, 349, 640, 232, 680, 340, 85),
(35, 392, 656, 520, 469, 308, 560, 460, 245),	(53, 304, 700, 496, 547, 200, 580, 440, 235),
(68, 301, 700, 476, 532, 275, 595, 460, 140),	(76, 332, 685, 365, 620, 260, 668, 301, 220),
(80, 428, 629, 515, 496, 272, 560, 395, 340),	(85, 340, 680, 440, 536, 323, 620, 427, 136),
(85, 340, 680, 440, 580, 235, 620, 365, 260),	(85, 340, 680, 472, 560, 221, 596, 395, 272),
(92, 256, 715, 440, 605, 160, 619, 392, 220),	(125, 272, 704, 400, 629, 172, 640, 340, 245),
(136, 277, 700, 323, 664, 200, 680, 260, 235)

Page of Squares : First Upload October 3, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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766

The smallest squares containing k 766's :
276676 = 526²,
27667664896 = 166336²,
476609676657664 = 21831392².

766² = 586756, a zigzag square.

766² = 586756, 5 + 8 + 6 + 75 + 6 = 10².

Komachi Square Sum: 766² = 7² + 89² + 425² + 631².

766² = 586756 appears in the decimal expression of π:
  π = 3.14159•••586756••• (from the 11840th digit)
  (586756 is the fifth 6-digit square in the expression of π.)

Page of Squares : First Upload October 3, 2005 ; Last Revised September 11, 2006
by Yoshio Mimura, Kobe, Japan

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767

The smallest squares containing k 767's :
76729 = 277²,
55767767104 = 236152²,
767017677767056 = 27695084².

₇₆₇C₁ + ₇₆₇C₂ + ₇₆₇C₃ + ₇₆₇C₄ = 8672².

54457^k + 154934^k + 171808^k + 207090^k are squares for k = 1,2,3 (767², 315237², 133541603²).

3-by-3 magic squares consisting of different squares with constant 767²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 174, 747, 474, 587, 138, 603, 462, 106),	(2, 198, 741, 309, 678, 182, 702, 299, 78),
(21, 258, 722, 358, 642, 219, 678, 331, 138),	(27, 138, 754, 538, 534, 117, 546, 533, 78),
(34, 357, 678, 453, 538, 306, 618, 414, 187),	(42, 155, 750, 405, 642, 110, 650, 390, 117),
(42, 277, 714, 507, 546, 182, 574, 462, 213),	(43, 306, 702, 378, 603, 286, 666, 362, 117),
(78, 182, 741, 282, 699, 142, 709, 258, 138),	(78, 299, 702, 398, 618, 219, 651, 342, 218),
(78, 453, 614, 533, 474, 282, 546, 398, 363),	(117, 218, 726, 286, 693, 162, 702, 246, 187),
(117, 390, 650, 450, 565, 258, 610, 342, 315),	(133, 378, 654, 522, 434, 357, 546, 507, 182)

767² = 588289, 588 + 28 + 9 = 25²,
767² = 588289, 58³ + 8³ + 2³ + 89³ = 949².

Page of Squares : First Upload October 3, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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768

The smallest squares containing k 768's :
17689 = 133²,
7684976896 = 87664²,
768947686768836 = 27729906².

768² = 589824, 5 + 8 + 9 + 8 + 2 + 4 = 6²,
768² = 589824, 58 + 9 + 8 + 2 + 4 = 9².

768² is the 6th square which is the sum of 9 eighth powers.

768² is the first square which is the sum of 3 cubes in just 3 ways
768² = 24³ + 40³ + 80³ = 40³ + 62³ + 66³ = 54³ + 59³ + 61³.

Komachi equation: 768² = 9² * 8² + 765² - 4² * 3² - 21².

(1³ + 2³ + ... + 24³)(25³ + 26³ + ... + 159³)(160³ + 161³ + ... + 768³) = 1125490464000².

The square root of 768 is 27. 7 1 2 8 1 2 9 2 11 0 20 ...,
and 27² = 7² + 1² + 2² + 8² + 1² + 2² + 9² + 2² + 11² + 0² + 20².

Page of Squares : First Upload October 3, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

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769

The smallest squares containing k 769's :
12769 = 113²,
876929769 = 29613²,
9976987697769 = 3158637².

The squares which begin with 769 and end in 769 are
76930233769 = 277363²,   76943547769 = 277387²,   769327214769 = 877113²,
769369316769 = 877137²,   769765833769 = 877363²,...

769² = 591361, 5 + 9 + 1 + 3 + 6 + 1 = 5²,
769² = 591361, 59 + 136 + 1 = 14².

1 / 769 = 0.00130039011703511053315994798439531...,
and 1² + 3² + 0² + 0² + 3² + 9² + 0² + 1² + 1² + 7² + ... + 3² + 1² =769.

769² = 264² + 327² + 644² : 446² + 723² + 462² = 967².

3-by-3 magic squares consisting of different squares with constant 769²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 327, 696, 492, 536, 249, 591, 444, 212),	(24, 159, 752, 543, 536, 96, 544, 528, 129),
(24, 292, 711, 436, 591, 228, 633, 396, 184),	(39, 464, 612, 508, 444, 369, 576, 423, 284),
(49, 132, 756, 348, 679, 96, 684, 336, 103),	(63, 184, 744, 456, 609, 112, 616, 432, 159),
(72, 249, 724, 284, 684, 207, 711, 248, 156),	(84, 399, 652, 537, 436, 336, 544, 492, 231),
(132, 399, 644, 441, 572, 264, 616, 324, 327)

Page of Squares : First Upload October 3, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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