Squares:710s

710

The smallest squares containing k 710's :
710649 = 843²,
27105671044 = 164638²,
531710471087104 = 23058848².

Komachi equation: 710² = 12³ + 3³ + 4³ + 5³ * 6³ + 78³ + 9³.

172² + 173² + 174² + ... + 710² = 10857².

Page of Squares : First Upload August 29, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan

711

The smallest squares containing k 711's :
2471184 = 1572²,
7117115769 = 84363²,
4711711671164281 = 68641909².

3-by-3 magic squares consisting of different squares with constant 711²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 204, 681, 489, 492, 156, 516, 471, 132),	(13, 154, 694, 226, 659, 142, 674, 218, 61),
(14, 83, 706, 398, 586, 61, 589, 394, 58),	(14, 118, 701, 454, 541, 82, 547, 446, 86),
(14, 190, 685, 365, 586, 170, 610, 355, 86),	(14, 269, 658, 434,518, 221, 563, 406, 154),
(19, 82, 706, 134, 694, 77, 698, 131, 34),	(19, 358, 614,478, 461, 254, 526, 406, 253),
(22, 166, 691, 266, 643, 146, 659, 254, 82),	(22, 419, 574, 499, 418, 286, 506, 394, 307),
(24, 231, 672, 399, 552, 204, 588, 384, 111),	(34, 314, 637, 362, 541, 286, 611, 338, 134),
(50, 355, 614, 386, 530, 275, 595, 314, 230),	(54, 243, 666, 342, 594, 189, 621, 306, 162),
(61, 230, 670, 470, 490, 211, 530, 461, 110),	(74, 371, 602, 413, 514, 266, 574, 322, 269),
(82, 211, 674, 349, 602, 146, 614, 314, 173),	(96, 303, 636, 456, 516, 177, 537, 384, 264),
(98, 386, 589, 419, 446, 362, 566, 397, 166),	(131, 446, 538, 482, 454, 259, 506, 317, 386),
(134,434,547, 478, 349, 394, 509, 442, 226),	(173, 394, 566, 446, 509, 218, 526, 302, 371)

711² = 505521, 5 + 0 + 5 + 5 + 21 = 6²,
711² = 505521, 5 + 0 + 55 + 21 = 9²,
711² = 505521, 50 + 5 + 5 + 21 = 9²,
711² = 505521, 50 + 5 + 521 = 24².

711² = 505521 appears in the decimal expressions of π and e:
π = 3.14159•••505521••• (from the 102073rd digit),
e = 2.71828•••505521••• (from the 102778th digit)

Page of Squares : First Upload February 13, 2006 ; Last Revised August 4, 2009
by Yoshio Mimura, Kobe, Japan

712

The smallest squares containing k 712's :
57121 = 239²,
777127129 = 27877²,
17127571271209 = 4138547².

712² = (9 + 10 + 11 + 12)² + (13 + 14 + 15 + 16)² + ... + (69 + 70 + 71 + 72)².

Komachi equations:
712² = 12² - 3² * 4² + 56² / 7² * 89² = 12² / 3² - 4² + 56² / 7² * 89²
= 12² / 3² / 4² * 56² / 7² * 89² = - 12² + 3² * 4² + 56² / 7² * 89²
= - 12² / 3² + 4² + 56² / 7² * 89².

712² = 2⁴ + 14⁴ + 22⁴ + 22⁴.

712² = 506944, 5 + 0 + 6 + 9 + 44 = 8².

Page of Squares : First Upload February 13, 2006 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan

713

The smallest squares containing k 713's :
171396 = 414²,
871371361 = 29519²,
1713671371352196 = 41396514².

713² = 508369, a square with different digits.

(613 / 713)² = 0.739165842... (Komachic).

713² + 714² + 715² + ... + 99016² = 17988736².

1/ 713 = 0.001402524544179523141654978962131...,
and the sum of the squares of the digits is 713.

Komachi equation: 713² = 123² + 4² * 5² + 6² + 78² * 9².

713^k + 5359^k + 10511^k + 17273^k are squares for k = 1,2,3 (184², 20930², 2543432²).
138^k + 713^k + 5152^k + 7222^k are squares for k = 1,2,3 (115², 8901², 716795²).

3-by-3 magic squares consisting of different squares with constant 713²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(7, 84, 708, 492, 513, 56, 516, 488, 63),	(12, 135, 700, 385, 588, 120, 600, 380, 63),
(12, 249, 668, 479, 492, 192, 528, 452, 159),	(33, 236, 672, 336, 588, 223, 628, 327, 84),
(52, 324, 633, 417, 528, 236, 576, 353, 228),	(72, 308, 639, 492, 441, 268, 511, 468, 168),
(96, 353, 612, 452, 444, 327, 543, 432, 164),	(128, 417, 564, 444, 492, 263, 543, 304, 348),
(129, 268, 648, 312, 612, 191, 628, 249, 228)

Page of Squares : First Upload August 29, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

714

The smallest squares containing k 714's :
271441 = 521²,
171440714916 = 414054²,
5714371497145444 = 75593462².

714² = 509796, a zigzag square.

714² = 11 x 12 x 13 + 13 x 14 x 15 + 15 x 16 x 17 + ... + 43 x 44 x 45.

The integral triangle of sides 377, 4570, 4879 has square area 714².

Komachi fraction : 450 / 9176328 = (5 / 714)².

714² + 715² + 716² + ... + 1987² = 49959².

(1³ + 2³ + ... + 69³)(70³ + 71³ + ... + 77³)(78³ + 79³ + ... + 714³) = 1100194575480².

(1² + 2² + 3² + ... + 461²) + (1² + 2² + 2³ + ... + 643²) = (1² + 2² + 3³ + ... + 714²).

714² = 32³ + 61³ + 63³.

The 4-by-4 magic squares consisting of different squares with constant 714:

0²	4²	13²	23²
8²	24²	5²	7²
17²	1²	18²	10²
19²	11²	14²	6²

1²	2²	15²	22²
4²	19²	16²	9²
11²	18²	13²	10²
24²	5²	8²	7²

2²	6²	7²	25²
9²	13²	20²	8²
10²	22²	11²	3²
23²	5²	12²	4²

714² = 509796, 5 + 0 + 9 + 7 + 9 + 6 = 6²,
714² = 509796, 50 + 9 + 7 + 9 + 6 = 9²,
714² = 509796, 50 + 9 + 79 + 6 = 12²,
714² = 509796, 5 + 0 + 9796 = 99².

Page of Squares : First Upload August 29, 2005 ; Last Revised September 30, 2011
by Yoshio Mimura, Kobe, Japan

715

The smallest squares containing k 715's :
217156 = 466²,
1371517156 = 37034²,
71563715497156 = 8459534².

715² = 511225, a square with just 3 kinds of digits.

715² = 91² + 93² + 95² + 97² + 99² + 101² + ... + 155².

715716 = 846².

3-by-3 magic squares consisting of different squares with constant 715²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 275, 660, 429, 528, 220, 572, 396, 165),	(9, 270, 662, 338, 585, 234, 630, 310, 135),
(18, 249, 670, 345, 590, 210, 626, 318, 135),	(30, 185, 690, 410, 570, 135, 585, 390, 130),
(30, 266, 663, 410, 537, 234, 585, 390, 130),	(30, 410, 585, 441, 450, 338, 562, 375, 234),
(39, 130, 702, 402, 585, 86, 590, 390, 105),	(39, 150, 698, 390, 590, 105, 598, 375, 114),
(58, 210, 681, 375, 590, 150, 606, 345, 158),	(78, 329, 630, 455, 510, 210, 546, 378, 265),
(86, 327, 630, 423, 486, 310, 570, 410, 135),	(87, 166, 690, 234, 663, 130, 670, 210, 135),
(103, 270, 654, 450, 535, 150, 546, 390, 247),	(130, 306, 633, 390, 567, 194, 585, 310, 270),
(130, 390, 585, 471, 490, 222, 522, 345, 346),	(135, 410, 570, 474, 375, 382, 518, 450, 201)

715² = 511225, 5 + 1 + 1 + 2 + 2 + 5 = 4²,
715² = 511225, 5 + 1 + 12 + 2 + 5 = 5²,
715² = 511225, 5 + 11 + 2 + 2 + 5 = 5².

Page of Squares : First Upload August 29, 2005 ; Last Revised August 4, 2009
by Yoshio Mimura, Kobe, Japan

716

The smallest squares containing k 716's :
17161 = 131²,
1871687169 = 43263²,
67166007167169 = 8195487².

The squares which begin with 716 and end in 716 are
7166299716 = 84654², 71638663716 = 267654², 716301551716 = 846346²,
716822995716 = 846654², 7160151815716 = 2675846²,...

716² = 512656, 5 * 12 + 656 = 716.

716² = 512656, 5 + 1 + 2 + 6 + 5 + 6 = 5²,
716² = 512656, 512 + 6 + 5 + 6 = 23².

715716 = 846².

716² = 512656 appears in the decimal expression of e:
e = 2.71828•••512656••• (from the 61312nd digit).

Page of Squares : First Upload February 13, 2006 ; Last Revised August 31, 2006
by Yoshio Mimura, Kobe, Japan

717

The smallest squares containing k 717's :
717409 = 847²,
7170871761 = 84681²,
1327177177176249 = 36430443².

717² = 514089, a square with different digits.

717² + 718² + 719² + ... + 2230² = 59803²,
717² + 718² + 719² + ... + 224445² = 61391143².

717² = 514089, 5 * 140 + 8 + 9 = 717.

(1² + 2² + 3² + ... + 145²) + (1² + 2² + 3² + ... + 715²) = 1² + 2² + 3² + ... + 717².

3-by-3 magic squares consisting of different squares with constant 717²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(8, 155, 700, 400, 580, 133, 595, 392, 80),	(8, 236, 677, 467, 512, 184, 544, 443, 148),
(15, 342, 630, 450, 495, 258, 558, 390, 225),	(16, 83, 712, 187, 688, 76, 692, 184, 37),
(28, 116, 707, 316, 637, 92, 643, 308, 76),	(28, 133, 704, 496, 512, 77, 517, 484, 112),
(28, 379, 608, 472, 448, 301, 539, 412, 232),	(37, 268, 664, 488, 496, 173, 524, 443, 208),
(37, 344, 628, 436, 488, 293, 568, 397, 184),	(44, 197, 688, 232, 656, 173, 677, 212, 104),
(54, 198, 687, 282, 639, 162, 657, 258, 126),	(54, 447, 558, 498, 378, 351, 513, 414, 282),
(56, 293, 652, 448, 524, 197, 557, 392, 224),	(61, 292, 652, 428, 509, 268, 572, 412, 131),
(63, 258, 666, 414, 558, 177, 582, 369, 198),	(91, 428, 568, 472, 392, 371, 532, 421, 232),
(92, 320, 635, 485, 440, 292, 520, 467, 160),	(124, 292, 643, 323, 604, 212, 628, 253, 236)

717² = 514089, 5 + 14 + 0 + 8 + 9 = 6²,
717² = 514089, 51 + 4 + 0 + 89 = 12²,
717² = 514089, 51³ + 4³ + 0³ + 8³ + 9³ = 366².

Page of Squares : First Upload August 29, 2005 ; Last Revised August 4, 2009
by Yoshio Mimura, Kobe, Japan

718

The smallest squares containing k 718's :
71824 = 268²,
30718871824 = 175268²,
371827185718041 = 19282821².

501² + 502² + 503² + ... + 718² = 9047²,
358² + 359² + 360² + ... + 718² = 10412².

1 / 718 = 0.0013927576601671309192200557103064...,
and the sum of the squares of the digits is 718.

718² = 515524, 5 + 15 + 5 + 24 = 7².

Page of Squares : First Upload August 29, 2005 ; Last Revised August 31, 2006
by Yoshio Mimura, Kobe, Japan

719

The smallest squares containing k 719's :
471969 = 687²,
71967719824 = 268268²,
327197196719649 = 18088593².

Komachi equation: 719² = 98² * 7² + 6² + 5² * 43² + 2² + 10².

The third prime for which the Legendre Symbol (a/719) = 1 for a = 1,2,...,10.

3-by-3 magic squares consisting of different squares with constant 719²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(21, 82, 714, 298, 651, 66, 654, 294, 53),	(21, 154, 702, 494, 507, 126, 522, 486, 91),
(27, 186, 694, 386, 582, 171, 606, 379, 78),	(42, 474, 539, 501, 406, 318, 514, 357, 354),
(46, 162, 699, 267, 654, 134, 666, 251, 102),	(46, 186, 693, 462, 539, 114, 549, 438, 154),
(66, 342, 629, 386, 549, 258, 603, 314, 234),	(69, 262, 666, 354, 594, 197, 622, 309, 186),
(69, 354, 622, 438, 539, 186, 566, 318, 309),	(69, 438, 566, 498, 379, 354, 514, 426, 267),
(123, 386, 594, 426, 522, 251, 566, 309, 318),	(126, 261, 658, 354, 602, 171, 613, 294, 234),
(134, 363, 606, 498, 486, 181, 501, 386, 342),	(165, 350, 606, 406, 555, 210, 570, 294, 325)

Page of Squares : First Upload August 29, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan