---

720

The smallest squares containing k 720's :
487204 = 698²,
67207204 = 8198²,
720511807207201 = 26842351².

(367 / 720)² = 0.259816743... (Komachic).

720² = (4² - 1)(11² - 1)(17² - 1).

41² + 720 = 49², 41² - 720 = 31².

Komachi equations:
720² = 9² * 8² / 7² * 6² / 54² * 3² * 210²,
720² = - 9³ + 8³ + 76³ + 5³ + 43³ + 2³ + 1³.

720² = 25³ + 26³ + 27³ + 28³ + 29³ + ... + 39³.

(1 + 2 + 3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11)(12) = 720²,
(1 + 2 + ... + 15)(16)(17 + 18 + ... + 28) = 720².

720² = 32³ + 36³ + 76³.

Page of Squares : First Upload September 5, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

---

721

The smallest squares containing k 721's :
3721 = 61²,
1172172169 = 34237²,
5721372179721 = 2391939².

The squares which begin with 721 and end in 721 are
7214633721 = 84939²,   72125010721 = 268561²,   72193778721 = 268689²,
721121957721 = 849189²,   721329174721 = 849311²,...

Komachi equations: 721² = 9³ + 8³ + 76³ + 5³ + 43³ - 2³ */ 1³.

721² = 519841, and 51984 = 228², 1 = 1².

721² = 519841, 5 + 1 + 9 + 8 + 41 = 8²,
721² = 519841, 5 + 1 + 9 + 84 + 1 = 10².

9911² = 433² + 434² + 435² + ... + 721².

The square root of 721 is 26. 8 5 14 4 3 1 6 4 1 9 5 10 4 3 9 ...,
and 26² = 8² + 5² + 14² + 4² + 3² + 1² + 6² + 4² + 1² + 9² + 5² + 10² + 4² + 3² + 9².

3-by-3 magic squares consisting of different squares with constant 721²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
( 9, 136, 708, 188, 684, 129, 696, 183, 44),	(9, 316, 648, 444, 513, 244, 568, 396, 201),
(12, 361, 624, 449, 492, 276, 564, 384, 233),	(24, 201, 692, 393, 584, 156, 604, 372, 129),
(24, 348, 631, 503, 444, 264, 516, 449, 228),	(36, 297, 656, 332, 576, 279, 639, 316, 108),
(44, 447, 564, 489, 396, 352, 528, 404, 279),	(64, 177, 696, 408, 584, 111, 591, 384, 152),
(64, 228, 681, 276, 639, 188, 663, 244, 144),	(80, 279, 660, 471, 480, 260, 540, 460, 129),
(81, 288, 656, 352, 591, 216, 624, 296, 207),	(87, 384, 604, 496, 471, 228, 516, 388, 321),
(120, 375, 604, 404, 540, 255, 585, 296, 300)

Page of Squares : First Upload September 5, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan

---

722

The smallest squares containing k 722's :
7225 = 85²,
4367227225 = 66085²,
72272257225 = 268835².

A+B, A+C, A+D, B+C, B+D, and C+D are squares if A = 722, B = 3122, C = 3767, D = 8114.

722² + 723² + 724² + ... + 1082² = 17252².

The square root of 722 is 26.87005768508880592723...,
and 26² = 8² + 7² + 0² + 0² + 5² + 7² + 6² + 8² + 5² + 0² + 8² + 8² + 8² + 0² + 5² + 9² + 2² + 7² + 2² + 3².

722² = 19⁴ + 19⁴ + 19⁴ + 19⁴.

82^k + 182^k + 722^k + 1318^k are squares for k = 1,2,3 (48², 1516², 51696²).

722² = 521284 appears in the decimal expression of π:
  π = 3.14159•••521284••• (from the 79311st digit).

Page of Squares : First Upload September 5, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

---

723

The smallest squares containing k 723's :
72361 = 269²,
723287236 = 26894²,
272367723723489 = 16503567².

(694 / 723)² = 0.921387564... (Komachic).

3-by-3 magic squares consisting of different squares with constant 723²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(17, 202, 694, 406, 577, 158, 598, 386, 127),	(22, 218, 689, 458, 529, 182, 559, 442, 122),
(27, 144, 708, 384, 603, 108, 612, 372, 99),	(34, 262, 673, 367, 574, 242, 622, 353, 106),
(38, 434, 577, 463, 458, 314, 554, 353, 302),	(46, 287, 662, 337, 578, 274, 638, 326, 97),
(48, 396, 603, 468, 477, 276, 549, 372, 288),	(49, 182, 698, 502, 511, 98, 518, 478, 161),
(50, 127, 710, 302, 650, 95, 655, 290, 98),	(62, 143, 706, 193, 686, 122, 694, 178, 97),
(62, 193, 694, 479, 518, 158, 538, 466, 127),	(62, 262, 671, 479, 518, 158, 538, 431, 218),
(70, 298, 655, 430, 545, 202, 577, 370, 230),	(86, 287, 658, 497, 502, 154, 518, 434, 257),
(118, 209, 682, 242, 662, 161, 671, 202, 178),	(122, 346, 623, 433, 538, 214, 566, 337, 298),
(127, 346, 622, 466, 518, 193, 538, 367, 314),	(178, 463, 526, 494, 302, 433, 497, 466, 242)

Page of Squares : First Upload February 20, 2006 ; Last Revised August 17, 2009
by Yoshio Mimura, Kobe, Japan

---

724

The smallest squares containing k 724's :
6724 = 82²,
724578724 = 26918²,
267243437246724 = 16347582².

The squares which begin with 724 and end in 724 are
724578724 = 26918²,   72405122724 = 269082²,   724061442724 = 850918²,
724340570724 = 851082²,   724912610724 = 851418²,...

724² = 524176 is a zigzag square with different digits.

724, 725 and 726 are three consecutive integers having square factors (the 10th case).

Komachi equation: 724² = - 9³ + 8³ - 7³ + 65³ + 4³ + 3³ * 21³.

724² = 524176, 5 + 2 + 4 + 1 + 7 + 6 = 5²,
724² = 524176, 52 + 41 + 76 = 13².

724² = 21² + 22² + 23² + ... + 116².

724² + 725² + 726² + ... + 819² = 7564²,
724² + 725² + 726² + ... + 1012² = 14824².

(1³ + 2³ + ... + 80³)(81³ + 82³ + ... + 724³) = 850273200²,
(1³ + 2³ + ... + 715³)(716³ + 717³ + ... + 724³) = 14836021200².

Page of Squares : First Upload September 5, 2005 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

---

725

The smallest squares containing k 725's :
725904 = 852²,
72572588449 = 269393²,
72572577254116 = 8518954².

725² = 525625, a square consisting of 3 kinds of digits.

725² = 525625, 5 + 2 + 5 + 6 + 2 + 5 = 5².

725² = (98 + 99 + 100 + 101 + 102)² + (103 + 104 + 105 + 106 + 107)².

Komachi equation: 725² = 98⁵ / 7⁵ - 6⁵ - 5⁵ - 4⁵ - 3⁵ - 2⁵ + 1⁵.

(1³ + 2³ + ... + 115³)(116³ + 117³ + ... + 724³)(725³) = 34161672300000².

The square root of 725 is 26.9258240356725201562535524577016...,
and the sum of the squares of the digits is 26².

725² = 525625, 5 * 25 * 6 - 25 = 725.

3-by-3 magic squares consisting of different squares with constant 725²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 85, 720, 435, 576, 68, 580, 432, 51),	(0, 120, 715, 435, 572, 96, 580, 429, 72),
(0, 203, 696, 500, 504, 147, 525, 480, 140),	(0, 435, 580, 500, 420, 315, 525, 400, 300),
(5, 156, 708, 300, 645, 140, 660, 292, 69),	(5, 300, 660, 348, 580, 261, 636, 315, 148),
(13, 84, 720, 240, 680, 75, 684, 237, 40),	(36, 315, 652, 348, 580, 261, 635, 300, 180),
(40, 372, 621, 405, 504, 328, 600, 365, 180),	(40, 405, 600, 492, 456, 275, 531, 392, 300),
(48, 420, 589, 464, 435, 348, 555, 400, 240),	(68, 180, 699, 324, 635, 132, 645, 300, 140),
(75, 216, 688, 400, 588, 141, 600, 365, 180),	(75, 400, 600, 464, 435, 348, 552, 420, 211),
(120, 275, 660, 363, 600, 184, 616, 300, 237),	(176, 432, 555, 468, 499, 240, 525, 300, 400)

725² = 525625 appears in the decimal expression of e:
  e = 2.71828•••525625••• (from the 136312nd digit).

Page of Squares : First Upload September 5, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan

---

726

The smallest squares containing k 726's :
627264 = 792²,
1537267264 = 39208²,
1187264726197264 = 34456708².

726² = 527076, a zigzag square.

726²± 5 are primes.

726² = 28² + 334² + 644² : 446² + 433² + 82² = 627².

Komachi Cubic Sum : 762² = 7³ + 14³ + 38³ + 52³ + 69³

(1² + 2² + 3² + 4²) + (1² + 2² + 3² + ... + 116²) = 726².

The 4-by-4 magic square consisting of different squares with constant 726:

Page of Squares : First Upload September 5, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

---

727

The smallest squares containing k 727's :
727609 = 853²,
37278727929 = 193077²,
272772772715556 = 16515834².

727² = 528529, the successive integers 528 and 529.

727² = 528529, 5 + 2 + 8 + 5 + 29 = 7²,
727² = 528529, 5 + 28 + 5 + 2 + 9 = 7²,
727² = 528529, 5 + 2 + 85 + 29 = 11²,
727² = 528529, 52 + 8 + 52 + 9 = 11².

727² + 728² + 729² + ... + 4205² = 157052².

727² = 528529, 5 + 2 + 8 * 5 * 2 * 9 = 727.

3-by-3 magic squares consisting of different squares with constant 727²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 466, 558, 506, 402, 333, 522, 387, 326),	(18, 317, 654, 493, 486, 222, 534, 438, 227),
(30, 227, 690, 402, 570, 205, 605, 390, 102),	(38, 282, 669, 339, 618, 178, 642, 259, 222),
(38, 282, 669, 381, 578, 222, 618, 339, 178),	(38, 339, 642, 381, 578, 222, 618, 282, 259),
(38, 339, 642, 438, 502, 291, 579, 402, 178),	(42, 237, 686, 387, 574, 222, 614, 378, 93),
(61, 378, 618, 462, 498, 259, 558, 371, 282),	(126, 317, 642, 462, 534, 173, 547, 378, 294)

727² = 528529 appears in the decimal expression of e:
  e = 2.71828•••528529••• (from the 138806th digit).

Page of Squares : First Upload September 5, 2005 ; Last Revised August 17, 2009
by Yoshio Mimura, Kobe, Japan

---

728

The smallest squares containing k 728's :
77284 = 278²,
67284728449 = 259393²,
2367281972897284 = 48654722².

The sum of the squares of the divisors of 728 is a square, 850².

728²± 3 are primes.

728² = (1² + 3)(2² + 3)(7² + 3)(19² + 3) = (5² + 3)(7² + 3)(19² + 3).

728² = 63³ + 1⁵ + 6⁷.

Komachi Fraction : 728² = 14309568 / 27.

Komachi equations:
728² = 1² * 2² * 3² / 4² * 56² * 78² / 9² = 98² / 7² * 65² / 4² * 32² / 10².

728² = 529984 appears in the decimal expression of π:
  π = 3.14159•••529984••• (from the 52994th digit).

Page of Squares : First Upload September 5, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

---

729

The square of 27.

The smallest squares containing k 729's :
729 = 27²,
577296729 = 24027²,
102789729729729 = 10138527².

The squares which begin with 729 and end in 729 are
72914580729 = 270027²,   729269884729 = 853973²,   729362116729 = 854027²,
729696933729 = 854223²,   729789192729 = 854277²,...

729² = (72 + 9)³.

729² = 9⁶ = 27⁴ = 81³ = 9³ + 54³ + 72³ = 25³ + 48³ + 74³.

Kaprekar : 729⁷ = 109418989131512359209, and 10² + 9² + 4² + 1² + 8² + 9² + 8² + 9² + 1² + 3² + 1² + 5² + 1² + 2² + 3² + 5² + 9² + 2² + 0² + 9² = 729.

Komachi equations:
729² = 12³ * 3³ / 4³ * 56³ / 7³ / 8³ * 9³ = 12³ * 3³ / 4³ / 56³ * 7³ * 8³ * 9³,
729² = 12⁶ - 3⁶ * 4⁶ + 56⁶ - 7⁶ * 8⁶ + 9⁶ = 12⁶ - 3⁶ * 4⁶ + 56⁶ / 7⁶ - 8⁶ + 9⁶
 = 12⁶ - 3⁶ * 4⁶ + 56⁶ / 7⁶ / 8⁶ * 9⁶ = 12⁶ - 3⁶ * 4⁶ - 56⁶ + 7⁶ * 8⁶ + 9⁶
 = 12⁶ - 3⁶ * 4⁶ - 56⁶ / 7⁶ + 8⁶ + 9⁶ = 12⁶ - 3⁶ * 4⁶ * 56⁶ / 7⁶ / 8⁶ + 9⁶
 = 12⁶ - 3⁶ * 4⁶ / 56⁶ * 7⁶ * 8⁶ + 9⁶ = 12⁶ / 3⁶ - 4⁶ + 56⁶ - 7⁶ * 8⁶ + 9⁶
 = 12⁶ / 3⁶ - 4⁶ + 56⁶ / 7⁶ - 8⁶ + 9⁶ = 12⁶ / 3⁶ - 4⁶ + 56⁶ / 7⁶ / 8⁶ * 9⁶
 = 12⁶ / 3⁶ - 4⁶ - 56⁶ + 7⁶ * 8⁶ + 9⁶ = 12⁶ / 3⁶ - 4⁶ - 56⁶ / 7⁶ + 8⁶ + 9⁶
 = 12⁶ / 3⁶ - 4⁶ * 56⁶ / 7⁶ / 8⁶ + 9⁶ = 12⁶ / 3⁶ - 4⁶ / 56⁶ * 7⁶ * 8⁶ + 9⁶
 = 12⁶ / 3⁶ / 4⁶ - 56⁶ / 7⁶ / 8⁶ + 9⁶ = 12⁶ / 3⁶ / 4⁶ * 56⁶ - 7⁶ * 8⁶ + 9⁶
 = 12⁶ / 3⁶ / 4⁶ * 56⁶ / 7⁶ - 8⁶ + 9⁶ = 12⁶ / 3⁶ / 4⁶ * 56⁶ / 7⁶ / 8⁶ * 9⁶
 = 12⁶ / 3⁶ / 4⁶ / 56⁶ * 7⁶ * 8⁶ * 9⁶ = - 12⁶ + 3⁶ * 4⁶ - 56⁶ + 7⁶ * 8⁶ + 9⁶
 = - 12⁶ + 3⁶ * 4⁶ - 56⁶ / 7⁶ + 8⁶ + 9⁶ = - 12⁶ + 3⁶ * 4⁶ + 56⁶ - 7⁶ * 8⁶ + 9⁶
 = - 12⁶ + 3⁶ * 4⁶ + 56⁶ / 7⁶ - 8⁶ + 9⁶ = - 12⁶ + 3⁶ * 4⁶ + 56⁶ / 7⁶ / 8⁶ * 9⁶
 = - 12⁶ + 3⁶ * 4⁶ * 56⁶ / 7⁶ / 8⁶ + 9⁶ = - 12⁶ + 3⁶ * 4⁶ / 56⁶ * 7⁶ * 8⁶ + 9⁶
 = - 12⁶ / 3⁶ + 4⁶ - 56⁶ + 7⁶ * 8⁶ + 9⁶ = - 12⁶ / 3⁶ + 4⁶ - 56⁶ / 7⁶ + 8⁶ + 9⁶
 = - 12⁶ / 3⁶ + 4⁶ + 56⁶ - 7⁶ * 8⁶ + 9⁶ = - 12⁶ / 3⁶ + 4⁶ + 56⁶ / 7⁶ - 8⁶ + 9⁶
 = - 12⁶ / 3⁶ + 4⁶ + 56⁶ / 7⁶ / 8⁶ * 9⁶ = - 12⁶ / 3⁶ + 4⁶ * 56⁶ / 7⁶ / 8⁶ + 9⁶
 = - 12⁶ / 3⁶ + 4⁶ / 56⁶ * 7⁶ * 8⁶ + 9⁶ = - 12⁶ / 3⁶ / 4⁶ + 56⁶ / 7⁶ / 8⁶ + 9⁶
 = - 12⁶ / 3⁶ / 4⁶ * 56⁶ + 7⁶ * 8⁶ + 9⁶ = - 12⁶ / 3⁶ / 4⁶ * 56⁶ / 7⁶ + 8⁶ + 9⁶.

729² = 531441, 5 + 31 + 4 + 41 = 9²,
729² = 531441, 5 + 31 + 44 + 1 = 9²,
729² = 531441, 5 + 314 + 4 + 1 = 18²,
729² = 531441, 531 + 4 + 41 = 24²,
729² = 531441, 531 + 44 + 1 = 24².

729² is the first square which is the sum of 3 eleventh powers : 3¹¹ + 3¹¹ + 3¹¹,
729² is the third square which is the sum of 9 tenth powers.

553² + 554² + 555² + ... + 729² = 8555².

(1³ + 2³ + ... + 728³)(729³) = 5223002148².

3-by-3 magic squares consisting of different squares with constant 729²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 137, 716, 241, 676, 128, 688, 236, 49),	(4, 220, 695, 320, 625, 196, 655, 304, 100),
(4, 332, 649, 424, 529, 268, 593, 376, 196),	(6, 147, 714, 483, 534, 114, 546, 474, 93),
(9, 252, 684, 396, 576, 207, 612, 369, 144),	(17, 136, 716, 436, 572, 119, 584, 431, 68),
(32, 79, 724, 116, 716, 73, 719, 112, 44),	(32, 236, 689, 271, 644, 208, 676, 247, 116),
(44, 376, 623, 508, 431, 296, 521, 452, 236),	(49, 224, 692, 496, 497, 196, 532, 484, 119),
(51, 318, 654, 402, 534, 291, 606, 381, 138),	(56, 313, 656, 359, 584, 248, 632, 304, 199),
(66, 282, 669, 339, 606, 222, 642, 291, 186),	(66, 381, 618, 438, 474, 339, 579, 402, 186),
(79, 172, 704, 308, 649, 124, 656, 284, 143),	(79, 340, 640, 460, 521, 220, 560, 380, 271),
(89, 224, 688, 416, 583, 136, 592, 376, 199),	(103, 376, 616, 424, 472, 359, 584, 409, 152),
(116, 457, 556, 488, 364, 401, 529, 436, 248),	(172, 401, 584, 439, 532, 236, 556, 296, 367),
(196, 439, 548, 464, 508, 241, 527, 284, 416)

Page of Squares : First Upload September 5, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Kobe, Japan

---