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770

The smallest squares containing k 770's :
770884 = 878²,
17701770304 = 133048²,
1770687770770681 = 42079541².

770² is the third square which is the sum of 10 seventh powers.

770²± 3 are primes.

770² = (1² + 6)(2² + 6)(92² + 6) = (1² + 6)(4² + 6)(62² + 6) = (1² + 6)(4² + 6)(7² + 6)(8² + 6)
= (2² + 6)(8² + 6)(29² + 6) = (4² + 6)(7² + 6)(22² + 6) = (8² + 6)(92² + 6).

Komachi Square Sum : 770² = 8² + 94² + 136² + 752².

The 4-by-4 magic squares consisting of different squares with constant 770:

770² = 592900 appears in the decimal expression of π:
  π = 3.14159•••592900••• (from the 74418th digit)

Page of Squares : First Upload October 11, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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771

The smallest squares containing k 771's :
427716 = 654²,
7710771721 = 87811²,
77187716207716 = 8785654².

84810^k + 113337^k + 135696^k + 260598^k are squares for k = 1,2,3 (771², 326133², 149204691²).
78642^k + 98688^k + 119505^k + 297606^k are squares for k = 1,2,3 (771², 344637², 171793449²).

3-by-3 magic squares consisting of different squares with constant 771²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 161, 754, 346, 674, 143, 689, 338, 74),	(14, 209, 742, 454, 602, 161, 623, 434, 134),
(14, 226, 737, 278, 689, 206, 719, 262, 94),	(17, 86, 766, 194, 742, 79, 746, 191, 38),
(17, 374, 674, 466, 542, 289, 614, 401, 238),	(26, 193, 746, 401, 634, 178, 658, 394, 79),
(31, 82, 766, 158, 751, 74, 754, 154, 47),	(31, 326, 698, 394, 607, 266, 662, 346, 191),
(36, 147, 756, 336, 684, 117, 693, 324, 96),	(62, 266, 721, 511, 526, 238, 574, 497, 134),
(70, 415, 646, 446, 550, 305, 625, 346, 290),	(74, 214, 737, 319, 682, 166, 698, 289, 154),
(74, 271, 718, 367, 646, 206, 674, 322, 191),	(74, 422, 641, 506, 511, 278, 577, 394, 326),
(79, 290, 710, 490, 530, 271, 590, 479, 130),	(82, 266, 719, 431, 614, 178, 634, 383, 214),
(86, 446, 623, 479, 458, 394, 598, 431, 226),	(94, 418, 641, 466, 481, 382, 607, 434, 194),
(108, 444, 621, 501, 432, 396, 576, 459, 228)

771² = 594441, 5 + 94 + 4 + 41 = 12²,
771² = 594441, 5 + 94 + 44 + 1 = 12²,
771² = 594441, 59 + 44 + 41 = 12².

771⁶ = 210051732678908121,
and 2² + 1² + 0² + 0² + 5² + 17² + 3² + 2² + 6² + 7² + 8² + 9² + 0² + 8² + 12² + 1² = 771.

Page of Squares : First Upload October 11, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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772

The smallest squares containing k 772's :
77284 = 278²,
23477287729 = 153223²,
277281677257729 = 16651777².

772² = 595984, 5 + 9 + 5 + 98 + 4 = 11²,
772² = 595984, 5 + 95 + 9 + 8 + 4 = 11².

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

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773

The smallest squares containing k 773's :
677329 = 823²,
37773477316 = 194354²,
773750773773376 = 27816376².

773² = 597529, 5 + 9 + 75 + 2 + 9 = 10²,
773² = 597529, 59 + 7 + 5 + 29 = 10².

773² + 774² + 775² + ... + 1636² = 36156².

773² = 252² + 332² + 651² : 156² + 233² + 252² = 377².

773² = 9³ + 16³ + 84³.

3-by-3 magic squares consisting of different squares with constant 773²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 512, 579, 541, 408, 372, 552, 411, 352),	(27, 208, 744, 408, 636, 163, 656, 387, 132),
(28, 309, 708, 501, 532, 252, 588, 468, 181),	(35, 420, 648, 540, 477, 280, 552, 440, 315),
(48, 189, 748, 539, 528, 168, 552, 532, 99),	(72, 384, 667, 477, 548, 264, 604, 387, 288),
(84, 413, 648, 512, 456, 357, 573, 468, 224),	(93, 296, 708, 516, 552, 163, 568, 453, 264),
(136, 357, 672, 483, 568, 204, 588, 384, 323)

Page of Squares : First Upload October 11, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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774

The smallest squares containing k 774's :
7744 = 88²,
121774477444 = 348962²,
110777477407744 = 10525088².

(443 / 774)² = 0.327586149... (Komachic).

774² = (1² + 5)(9² + 5)(34² + 5).

774² = 599076, 5 + 9 + 9 + 0 + 7 + 6 = 6²,
774² = 599076, 59 + 9 + 0 + 7 + 6 = 9²,
774² = 599076, 59 + 9 + 0 + 76 = 12²,
774² = 599076, 59 + 90 + 76 = 15².

Komachi Square Sum : 774² = 2³ + 4³ + 8³ + 16³ + 59³ + 73³.

(1³ + 2³ + ... + 224³)(225³ + 226³ + ... + 279³)(280³ + 281³ + ... + 774³) = 223640607960000².

The 4-by-4 magic square consisting of different squares with constant 774:

Page of Squares : First Upload October 11, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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775

The smallest squares containing k 775's :
1577536 = 1256²,
75775775076 = 275274²,
2775775397754384 = 52685628².

775² = 600625, 600 + 625 = 35².

775² + 776² + 777² + ... + 1013² = 13862².

(1³ + 2³ + ... + 99³)(100³ + 101³ + ... + 164³)(165³ + 166³ + ... + 775³) = 18723765060000².

The square root of 775 is 27. 8 3 8 8 2 1 8 1 4 15 0 10 9 6 ...,
and 27² = 8² + 3² + 8² + 8² + 2² + 1² + 8² + 1² + 4² + 15² + 0² + 10² + 9² + 6².

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

3-by-3 magic squares consisting of different squares with constant 775²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 210, 746, 546, 530, 147, 550, 525, 150),	(10, 141, 762, 195, 738, 134, 750, 190, 45),
(10, 195, 750, 525, 550, 150, 570, 510, 125),	(10, 294, 717, 525, 530, 210, 570, 483, 206),
(10, 483, 606, 525, 450, 350, 570, 406, 333),	(18, 174, 755, 330, 685, 150, 701, 318, 90),
(35, 234, 738, 450, 595, 210, 630, 438, 109),	(45, 190, 750, 350, 675, 150, 690, 330, 125),
(45, 298, 714, 350, 630, 285, 690, 339, 98),	(45, 350, 690, 486, 525, 298, 602, 450, 189),
(66, 125, 762, 162, 750, 109, 755, 150, 90),	(78, 179, 750, 354, 678, 125, 685, 330, 150),
(90, 370, 675, 514, 477, 330, 573, 486, 190),	(125, 330, 690, 510, 557, 174, 570, 426, 307),
(150, 350, 675, 381, 630, 242, 658, 285, 294),	(195, 458, 594, 510, 531, 242, 550, 330, 435)

Page of Squares : First Upload October 11, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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776

The smallest squares containing k 776's :
5776 = 76²,
6776417761 = 82319²,
277624776629776 = 16662076².

The squares which begin with 776 and end in 776 are
77604587776 = 278576²,   776027093776 = 880924²,   776294917776 = 881076²,
776908267776 = 881424²,   7761372533776 = 2785924²,...

776² = 602176, a zigzag square.

776² = 602176, 6³ + 0³ + 2³ + 1³ + 7³ + 6³ = 28²,
776² = 602176, 602 + 17 + 6 = 25².

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

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777

The smallest squares containing k 777's :
277729 = 527²,
39077777761 = 197681²,
777777077709025 = 27888655².

777² = 603729, a square with different digits.

777² = 603729, 60 + 3 + 7 + 2 + 9 = 9²,
777² = 603729, 60 + 3 + 72 + 9 = 12²,
777² = 603729, 60 + 372 + 9 = 21².

731² + 732² + 733² + ... + 777² = 5170².

1 / 777 = 0.00128700128700128700128700128700128700128...,
and the sum of the squares of its digits is 777.

81585^k + 119658^k + 133644^k + 268842^k are squares for k = 1,2,3 (777², 333333², 155158353²).

3-by-3 magic squares consisting of different squares with constant 777²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(1, 88, 772, 452, 628, 71, 632, 449, 52),	(6, 243, 738, 333, 666, 222, 702, 318, 99),
(8, 124, 767, 383, 668, 104, 676, 377, 68),	(8, 311, 712, 361, 632, 272, 688, 328, 151),
(14, 322, 707, 413, 602, 266, 658, 371, 182),	(16, 353, 692, 388, 596, 313, 673, 352, 164),
(16, 472, 617, 503, 464, 368, 592, 407, 296),	(23, 328, 704, 512, 536, 233, 584, 457, 232),
(31, 208, 748, 352, 671, 172, 692, 332, 121),	(32, 244, 737, 484, 583, 172, 607, 452, 176),
(44, 328, 703, 428, 577, 296, 647, 404, 148),	(54, 237, 738, 522, 558, 141, 573, 486, 198),
(65, 148, 760, 200, 740, 127, 748, 185, 100),	(68, 391, 668, 479, 548, 272, 608, 388, 289),
(76, 353, 688, 508, 544, 223, 583, 428, 284),	(80, 223, 740, 415, 640, 148, 652, 380, 185),
(80, 380, 673, 545, 452, 320, 548, 505, 220),	(88, 233, 736, 464, 608, 137, 617, 424, 208),
(102, 387, 666, 531, 522, 222, 558, 426, 333),	(124, 368, 673, 412, 607, 256, 647, 316, 292),
(127, 284, 712, 376, 652, 193, 668, 313, 244),	(127, 472, 604, 536, 383, 412, 548, 484, 263),
(148, 263, 716, 296, 692, 193, 703, 236, 232),	(148, 481, 592, 508, 512, 289, 569, 332, 412),
(152, 319, 692, 412, 628, 199, 641, 328, 292),	(162, 387, 654, 438, 594, 243, 621, 318, 342)

The 4-by-4 magic square consisting of different squares with constant 777:

777² = 603729 appears in the decimal expression of e:
  e = 2.71828•••603729••• (from the 5335th digit)
  (603729 is the third 6-digit square in the expression of e.)

Page of Squares : First Upload October 11, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

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778

The smallest squares containing k 778's :
77841 = 279²,
14778778624 = 121568²,
1947789477877824 = 44133768².

778² = 605284, a zigzag square with different digits.

778² = 605284, 6 + 0 + 5 + 2 + 8 + 4 = 5²,
778² = 605284, 60 + 52 + 84 = 14².

778² = 9³ + 32³ + 83³.

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

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779

The smallest squares containing k 779's :
617796 = 786²,
4327797796 = 65786²,
226977907797796 = 15065786².

3-by-3 magic squares consisting of different squares with constant 779²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 454, 633, 498, 489, 346, 599, 402, 294),	(14, 87, 774, 249, 734, 78, 738, 246, 41),
(14, 318, 711, 423, 594, 274, 654, 391, 162),	(14, 441, 642, 534, 462, 329, 567, 446, 294),
(18, 274, 729, 311, 666, 258, 714, 297, 94),	(22, 174, 759, 489, 594, 122, 606, 473, 126),
(39, 258, 734, 462, 599, 186, 626, 426, 183),	(41, 246, 738, 402, 626, 231, 666, 393, 94),
(54, 231, 742, 546, 518, 201, 553, 534, 126),	(102, 391, 666, 441, 522, 374, 634, 426, 153),
(105, 346, 690, 410, 615, 246, 654, 330, 265),	(150, 454, 615, 535, 510, 246, 546, 375, 410),
(183, 414, 634, 454, 582, 249, 606, 311, 378)

779² = 606841, 6 + 0 + 6 + 8 + 4 + 1 = 5²,
779² = 606841, 60 + 68 + 41 = 13².

779² = 17³ + 66³ + 68³.

Page of Squares : First Upload October 11, 2005 ; Last Revised August 31, 2009
by Yoshio Mimura, Kobe, Japan

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