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780

The smallest squares containing k 780's :
178084 = 422²,
51780367809 = 227553²,
78037806877801 = 8833901².

780² + 781² + 782² + ... + 11363² = 699258².

A, B, C, A + B, B + C, and C + A are squares for A = 780², B = 2475², C = 2992².

The integral triangle of sides 544, 2329, 2535 (or 1409, 6596, 7995) has square area 780².

Komachi equations:
780² = 9² + 8² - 7² + 65² * 4² * 3² + 2² - 10² = - 9² - 8² + 7² + 65² * 4² * 3² - 2² + 10².

(1)(2 + 3)(4)(5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14) = 780²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 19)(20 + 21 + ... + 32) = 780².

Page of Squares : First Upload October 17, 2005 ; Last Revised September 30, 2011
by Yoshio Mimura, Kobe, Japan

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781

The smallest squares containing k 781's :
781456 = 884²,
117817816516 = 343246²,
178114781781225 = 13345965².

3-by-3 magic squares consisting of different squares with constant 781²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 124, 771, 524, 573, 84, 579, 516, 92),	(12, 261, 736, 411, 628, 216, 664, 384, 147),
(16, 141, 768, 453, 624, 124, 636, 448, 69),	(16, 237, 744, 348, 664, 219, 699, 336, 92),
(27, 84, 776, 236, 741, 72, 744, 232, 51),	(27, 376, 684, 504, 531, 272, 596, 432, 261),
(33, 286, 726, 506, 561, 198, 594, 462, 209),	(51, 408, 664, 456, 524, 357, 632, 411, 204),
(61, 324, 708, 372, 636, 259, 684, 317, 204),	(69, 240, 740, 540, 520, 219, 560, 531, 120),
(96, 408, 659, 443, 516, 384, 636, 421, 168),	(128, 324, 699, 456, 603, 196, 621, 376, 288),
(189, 412, 636, 468, 579, 236, 596, 324, 387)

781² = 609961, 609 + 9 + 6 + 1 = 25².

Page of Squares : First Upload October 17, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

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782

The smallest squares containing k 782's :
378225 = 615²,
7827825625 = 88475²,
2387820378278241 = 48865329².

(599 / 782)² = 0.586732491... (Komachic).

(1² + 2² + ... + 391²)(392² + 393² + ... + 782²) = 52862418².

The square root of 782 is 27. 9 6 4 2 6 2 9 0 8 2 1 9 1 2 5 8 10 2 5 7 7 ...,
and 27² = 9² + 6² + 4² + 2² + 6² + 2² + 9² + 0² + 8² + 2² + 1² + 9² + 1² + 2² + 5² + 8² + 10² + 2² + 5² + 7² + 7².

Komachi equation: 782² = 9² * 87² + 6² - 5² / 4² * 32² - 1².

The 4-by-4 magic square consisting of different squares with constant 782:

782² = 611524, 6 + 1 + 1 + 52 + 4 = 8²,
782² = 611524, 61 + 15 + 24 = 10².

Page of Squares : First Upload October 17, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

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783

The smallest squares containing k 783's :
783225 = 885²,
32778378304 = 181048²,
2531783783783556 = 50316834².

783² = 613089, a square with different digits.

783² = 242² + 341² + 662² : 266² + 143² + 242² = 387².

81^k + 136^k + 304^k + 704^k are squares for k = 1,2,3 (35², 783², 19495²).

Komachi equations:
783² = 9² * 87² + 6² - 5² - 4² + 3² - 2² */ 1² = 9² * 87² - 6² + 5² + 4² - 3² + 2² * 1².

3-by-3 magic squares consisting of different squares with constant 783²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(5, 142, 770, 250, 730, 133, 742, 245, 50),	(10, 283, 730, 458, 590, 235, 635, 430, 158),
(26, 122, 773, 382, 677, 94, 683, 374, 82),	(33, 276, 732, 516, 543, 228, 588, 492, 15),
(37, 158, 766, 458, 626, 107, 634, 443, 122),	(38, 278, 731, 478, 571, 242, 619, 458, 142),
(58, 166, 763, 203, 742, 146, 754, 187, 98),	(58, 290, 725, 325, 670, 242, 710, 283, 170),
(58, 373, 686, 406, 602, 293, 667, 334, 238),	(59, 502, 598, 542, 458, 331, 562, 389, 382),
(74, 347, 698, 523, 542, 214, 578, 446, 283),	(82, 322, 709, 539, 538, 182, 562, 469, 278),
(87, 228, 744, 348, 681, 168, 696, 312, 177),	(87, 264, 732, 348, 672, 201, 696, 303, 192),
(94, 373, 682, 478, 514, 347, 613, 458, 166),	(98, 341, 698, 491, 518, 322, 602, 478, 149),
(105, 408, 660, 492, 480, 375, 600, 465, 192),	(107, 226, 742, 362, 677, 154, 686, 322, 197),
(133, 434, 638, 502, 443, 406, 586, 478, 203),	(142, 437, 634, 466, 562, 283, 613, 326, 362),
(154, 427, 638, 523, 418, 406, 562, 506, 203)

255² + 256² + 257² + ... + 783² = 12443².

The square root of 783 is 27. 9 8 2 1 3 7 15 9 2 6 6 4 4 5 1 3 6 6 ...
and 27² = 9² + 8² + 2² + 1² + 3² + 7² + 15² + 9² + 2² + 6² + 6² + 4² + 4² + 5² + 1² + 3² + 6² + 6².

783² = 613089, 6 + 1 + 308 + 9 = 18²,
783² = 613089, 6 + 13 + 0 + 8 + 9 = 6²,
783² = 613089, 6 + 130 + 89 = 15²,
783² = 613089, 61 + 3 + 0 + 8 + 9 = 9².

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

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784

The smallest squares containing k 784's :
784 = 28²,
196784784 = 14028²,
110544784784784 = 10514028².

The squares which begin with 784 and end in 784 are
78415680784 = 280028²,   784060662784 = 885472²,   784159838784 = 885528²,
784946384784 = 885972²,   7840156800784 = 2800028²,...

The square of 28.

Komachi equations:
784² = 12² * 3² * 4² * 56² * 7² / 8² / 9²,
784² = 9⁴ / 8⁴ * 7⁴ * 6⁴ / 54⁴ * 32⁴ * 1/⁴ = 9⁴ * 8⁴ * 7⁴ / 6⁴ * 5⁴ * 4⁴ / 3⁴ / 2⁴ / 10⁴
 = 9⁴ * 8⁴ * 7⁴ / 6⁴ / 5⁴ / 4⁴ / 3⁴ * 2⁴ * 10⁴ = 98⁴ / 7⁴ * 6⁴ * 5⁴ * 4⁴ / 3⁴ / 2⁴ / 10⁴
 = 98⁴ / 7⁴ * 6⁴ / 5⁴ / 4⁴ / 3⁴ * 2⁴ * 10⁴ = 98⁴ / 7⁴ / 6⁴ * 5⁴ * 4⁴ * 3⁴ * 2⁴ / 10⁴
 = 98⁴ / 7⁴ / 6⁴ / 5⁴ * 4⁴ * 3⁴ / 2⁴ * 10⁴.

784² = 614656, 6 * 14 / 6 * 56 = 784.

784² = 614656, 6 + 1 + 46 + 5 + 6 = 8²,
784² = 614656, 6 + 14 + 656 = 26²,
784² = 614656, 614 + 6 + 56 = 26².

784² = 28³ + 84³ = 4³ + 56³ + 76³.

(1³ + 2³ + ... + 783³)(784³) = 6737859072².

Page of Squares : First Upload October 17, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

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785

The smallest squares containing k 785's :
467856 = 684²,
34478547856 = 185684²,
327856785657856 = 18106816².

3-by-3 magic squares consisting of different squares with constant 785²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 425, 660, 471, 528, 340, 628, 396, 255),	(9, 88, 780, 520, 585, 60, 588, 516, 65),
(12, 200, 759, 385, 660, 180, 684, 375, 88),	(16, 87, 780, 420, 660, 65, 663, 416, 60),
(24, 432, 655, 465, 520, 360, 632, 399, 240),	(47, 360, 696, 504, 520, 303, 600, 465, 200),
(60, 129, 772, 240, 740, 105, 745, 228, 96),	(60, 151, 768, 255, 732, 124, 740, 240, 105),
(60, 240, 745, 340, 681, 192, 705, 308, 156),	(60, 255, 740, 360, 668, 201, 695, 324, 168),
(60, 340, 705, 439, 600, 252, 648, 375, 236),	(60, 360, 695, 452, 585, 264, 639, 380, 252),
(65, 276, 732, 420, 632, 201, 660, 375, 200),	(105, 448, 636, 540, 420, 385, 560, 489, 252),
(119, 408,660, 492, 556, 255, 600, 375, 340),	(144, 408, 655, 495, 560, 240, 592, 369, 360)

785² = 616225, 6 + 16 + 2 + 25 = 7²,
785² = 616225, 6 + 16 + 22 + 5 = 7²,
785² = 616225, 616 + 2 + 2 + 5 = 25²,
785² = 616225, 616 + 225 = 29².

785² = 616225 appears in the decimal expression of π:
  π = 3.14159•••616225••• (from the 93794th digit)

Page of Squares : First Upload October 17, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

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786

The smallest squares containing k 786's :
786769 = 887²,
7867867401 = 88701²,
378678678670224 = 19459668².

1 / 786 = 0.0012722646..., 1² + 2² + 7² + 2² + 26² + 4² + 6² = 786.

786² = 617796, 6 + 1 + 7 + 7 + 9 + 6 = 6².

786²± 5 are primes.

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

786^k + 1614^k + 3882^k + 4122^k are squares for k = 1,2,3 (102², 5940², 365004²).
190^k + 338^k + 786^k + 802^k are squares for k = 1,2,3 (46², 1188², 32356²).

The 4-by-4 magic square consisting of different squares with constant 786:

Page of Squares : First Upload October 17, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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787

The smallest squares containing k 787's :
278784 = 528²,
78747876 = 8874²,
154787787007876 = 12441374².

787² = 14³ + 44³ + 81³.

787² + 788² + 789² + ... + 37267² = 4153677².

3-by-3 magic squares consisting of different squares with constant 787²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(15, 162, 770, 238, 735, 150, 750, 230, 63),	(22, 399, 678, 498, 518, 321, 609, 438, 238),
(33, 194, 762, 454, 618, 177, 642, 447, 86),	(33, 282, 734, 554, 513, 222, 558, 526, 177),
(54, 498, 607, 527, 474, 342, 582, 383, 366),	(63, 462, 634, 526, 447, 378, 582, 454, 273),
(78, 273, 734, 481, 582, 222, 618, 454, 177),	(90, 338, 705, 513, 510, 310, 590, 495, 162),
(113, 282, 726, 474, 607, 162, 618, 414, 257),	(114, 337, 702, 383, 642, 246, 678, 306, 257)

787² = 619369, 6 + 1 + 93 + 69 = 13²,
787² = 619369, 61 + 93 + 6 + 9 = 13².

Page of Squares : First Upload October 17, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

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788

The smallest squares containing k 788's :
27889 = 167²,
47887881889 = 218833²,
70788378897889 = 8413583².

788² = 620944, 6 + 2 + 0 + 9 + 4 + 4 = 5².

788² = 28² + 29² + 30² + ... + 123².

The square root of 788 is 28.071337695236399261038922772...,
and 28² = 0² + 7² + 1² + 3² + 3² + 7² + 6² + 9² + ... + 7² + 2².

Page of Squares : First Upload October 17, 2005 ; Last Revised September 14, 2006
by Yoshio Mimura, Kobe, Japan

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789

The smallest squares containing k 789's :
78961 = 281²,
7893789409 = 88847²,
1278947894789161 = 35762381².

(1⁹ + 2⁷ + 3⁷) + (4⁹ + 5⁷ + 6⁷) = 789².

An Exchangeable Square : 789² = 622521, and 216225 = 465².

117² + 118² + 119² + ... + 789² = 12787².

The square root of 789 is 28.0891438103762785374101157849...,
and 28² = 0² + 8² + 9² + 1² + 4² + 3² + 8² + 1² + ... + 4² + 9².

Komachi equations:
789² = 1² * 2² - 3² + 4² + 5² - 6² + 789² = - 1² * 2² + 3² - 4² - 5² + 6² + 789².

3-by-3 magic squares consisting of different squares with constant 789²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 197, 764, 251, 724, 188, 748, 244, 59),	(4, 248, 749, 493, 584, 196, 616, 469, 152),
(4, 379, 692, 517, 524, 284, 596, 452, 251),	(6, 153, 774, 423, 654, 126, 666, 414, 87),
(6, 279, 738, 342, 666, 249, 711, 318, 126),	(6, 279, 738, 423, 654, 126, 666, 342, 249),
(6, 342, 711, 423, 654, 126, 666, 279, 318),	(11, 88, 784, 304, 724, 77, 728, 301, 44),
(13, 284, 736, 356, 659, 248, 704, 328, 139),	(18, 441, 654, 519, 486, 342, 594, 438, 279),
(20, 211, 760, 461, 620, 160, 640, 440, 139),	(32, 139, 776, 524, 584, 83, 589, 512, 116),
(43, 136, 776, 424, 659, 92, 664, 412, 109),	(44, 196, 763, 532, 571, 116, 581, 508, 164),
(56, 299, 728, 403, 616, 284, 676, 392, 109),	(56, 469, 632, 556, 472, 301, 557, 424, 364),
(59, 308, 724, 452, 581, 284, 644, 436, 133),	(64, 220, 755, 395, 664, 160, 680, 365, 164),
(64, 307, 724, 472, 596, 211, 629, 416, 232),	(90, 330, 711, 486, 585, 210, 615, 414, 270),
(148, 379, 676, 524, 556, 197, 571, 412, 356),	(176, 491, 592, 524, 368, 461, 563, 496, 244)

789² = 622521, 6 + 2 + 2 + 5 + 21 = 6²,
789² = 622521, 6 + 2 + 25 + 2 + 1 = 6²,
789² = 622521, 6 + 22 + 5 + 2 + 1 = 6²,
789² = 622521, 6 + 22 + 52 + 1 = 9².

Page of Squares : First Upload October 17, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

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