Squares:730s

730

The smallest squares containing k 730's :
173056 = 416²,
1373073025 = 37055²,
1730730730610704 = 41602052².

730² is the 10th square which is the sum of 4 sixth powers : 1⁶ + 3⁶ + 3⁶ + 9⁶.

Kaprekar : 730⁶ = 151334226289000000,
and 1² + 5² + 1² + 3² + 3² + 4² + 22² + 6² + 2² + 8² + 9² + 0² + 0² + 0² + 0² + 0² + 0².

(1³ + 2³ + ... + 221³)(222³ + 223³ + ... + 578³)(579³ + 580³ + ... + 730³) = 843856036143120².

1 / 730 = 0.001369..., and 1369 = 37².

730^k + 3970^k + 4190^k + 8010^k are squares for k = 1,2,3 (130², 9900², 806500²).
730^k + 41902^k + 71978^k + 77234^k are squares for k = 1,2,3 (438², 113588², 30119508²).

The square root of 730 is 27. 0 1 8 5 12 1 7 2 21..., and 27² = 0² + 1² + 8² + 5² + 12² + 1² + 7² + 2² + 21²,
the square root of 730 is 27. 0 18 5 1 2 1 7 2 2 1 2 5 9 2 0 6 1 7 4 6 8 ..., and 27² = 0² + 18² + 5² + 1² + 2² + 1² + 7² + 2² + 2² + 1² + 2² + 5² + 9² + 2² + 0² + 6² + 1² + 7² + 4² + 6² + 8².

Page of Squares : First Upload September 12, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

731

The smallest squares containing k 731's :
417316 = 646²,
7312473169 = 85513²,
273173197314304 = 16527952².

731 = (1² + 2² + 3² + ... + 85²) / (1² + 2² + 3² + ... + 9²).

731² = 534361, a zigzag square.

731² = 3⁰ + 3¹ + 3⁶ + 3⁷ + 3¹².

3-by-3 magic squares consisting of different squares with constant 731²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(7, 426, 594, 486, 441, 322, 546, 398, 279),	(30, 394, 615, 425, 510, 306, 594, 345, 250),
(34, 153, 714, 441, 574, 102, 582, 426, 119),	(34, 201, 702, 306, 642, 169, 663, 286, 114),
(34, 306, 663, 471, 498, 254, 558, 439, 174),	(42, 249, 686, 414, 574, 183, 601, 378, 174),
(54, 393, 614, 439, 474, 342, 582, 394, 201),	(57, 146, 714, 246, 678, 119, 686, 231, 102),
(78, 254, 681, 474, 537, 146, 551, 426, 222),	(114, 426, 583, 502, 471, 246, 519, 362, 366),
(153, 306, 646, 366, 601, 198, 614, 282, 279)

731² = 534361, 5 + 3 + 4 + 36 + 1 = 7²,
731² = 534361, 5 + 34 + 3 + 6 + 1 = 7²,
731² = 534361, 53 + 4 + 3 + 61 = 11²,
731² = 534361, 5 + 34 + 361 = 20².

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

Page of Squares : First Upload September 12, 2005 ; Last Revised August 29, 2011
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

732

The smallest squares containing k 732's :
677329 = 823²,
732947329 = 27073²,
70327327327321 = 8386139².

732² = 535824, 53 + 5 + 82 + 4 = 12².

732² is the 7th square which is the sum of 10 sixth powers.

Page of Squares : First Upload September 12, 2005 ; Last Revised September 4, 2006
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

733

The smallest squares containing k 733's :
373321 = 611²,
173327338276 = 416326²,
2273307733733089 = 47679217².

733² = 537289, a square with different digits.

733² = 537289, 5 + 3 + 72 + 89 = 13².

Komachi Fraction : 729 / 4835601 = (9 / 733)².

Kaprekar : 733⁷ = 113691454465110461077, and
11² + 3² + 6² + 9² + 1² + 4² + 5² + 4² + 4² + 6² + 5² + 1² + 10² + 4² + 6² + 10² + 7² + 7² =733.

3-by-3 magic squares consisting of different squares with constant 733²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(21, 252, 688, 508, 501, 168, 528, 472, 189),	(24, 192, 707, 437, 564, 168, 588, 427, 96),
(32, 363, 636, 456, 508, 267, 573, 384, 248),	(60, 267, 680, 405, 580, 192, 608, 360, 195),
(67, 336, 648, 504, 492, 203, 528, 427, 276),	(99, 328, 648, 472, 468, 309 552, 459, 148),
(108, 333, 644, 364, 588, 243, 627, 284, 252)

Page of Squares : First Upload September 12, 2005 ; Last Revised August 17, 2009
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

734

The smallest squares containing k 734's :
73441 = 271²,
3867347344 = 62188²,
2734734393017344 = 52294688².

734² = 538756, 53 + 87 + 56 = 14².

1 / 734 = 0.0013623, 13²+6²+23² = 734.

734² = 538756 appears in the decimal expression of π:
π = 3.14159•••538756••• (from the 24873rd digit).

Page of Squares : First Upload September 12, 2005 ; Last Revised September 4, 2006
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

735

The smallest squares containing k 735's :
273529 = 523²,
7357350625 = 85775²,
7357735073572081 = 85777241².

735 = (1² + 2² + 3² + ... + 49²) / (1² + 2² + 3² + 4² + 5²).

(326 / 735)² = 0.196725438... (Komachic).

Komachi equations:
735² = 9² * 8² * 7² / 6² * 5² / 4² / 3² * 21² = 98² * 7² * 6² * 5² / 4² * 3² / 21²
= 98² / 7² * 6² * 5² / 4² / 3² * 21².

3-by-3 magic squares consisting of different squares with constant 735²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(5, 190, 710, 242, 670, 181, 694, 235, 58),	(5, 190, 710, 410, 590, 155, 610, 395, 110),
(5, 274, 682, 410, 565, 230, 610, 382, 149),	(5, 410, 610, 454, 478, 325, 578, 379, 250),
(7, 224, 700, 476, 532, 175, 560, 455, 140),	(10, 85, 730, 293, 670, 74, 674, 290, 43),
(10, 85, 730, 470, 562, 59, 565, 466, 62),	(10, 170, 715, 370, 619, 142, 635, 358, 94),
(10, 170, 715, 506, 517, 130, 533, 494, 110),	(10, 286, 677, 470, 523, 214, 565, 430, 190),
(10, 370, 635, 470, 485, 290, 565, 410, 230),	(11, 230, 698, 302, 635, 214, 670, 290, 85),
(26, 118, 725, 155, 710, 110, 718, 149, 50),	(34, 187, 710, 437, 566, 170, 590, 430, 85),
(36, 345, 648, 423, 540, 264, 600, 360, 225),	(38, 166, 715, 341, 638, 130, 650, 325, 110),
(50, 347, 646, 485, 470, 290, 550, 446, 197),	( 50, 485, 550, 514, 370, 373, 523, 410, 314),
(56, 217, 700, 308, 644, 175, 665, 280, 140),	(60, 324, 657, 495, 468, 276, 540, 465, 180),
(72, 225, 696, 396, 600, 153, 615, 360, 180),	(85, 290, 670, 430, 565, 190, 590, 370, 235),
(85, 362, 634, 430, 491, 338, 590, 410, 155),	(106, 250, 683, 283, 650, 194, 670, 235, 190),
(106, 458, 565, 485, 470, 290, 542, 331, 370),	(107, 326, 650, 370, 590, 235, 626, 293, 250),
(110, 302, 661, 395, 586, 202, 610, 325, 250),	(110, 395, 610, 475, 506, 242, 550, 358, 331),
(122, 421, 590, 454, 422, 395, 565, 430, 190),	(135, 384, 612, 480, 513, 216, 540, 360, 345),
(155, 410, 590, 502, 370, 389, 514, 485, 202),	(166, 370, 613, 410, 565, 230, 587, 290, 334),
(190, 395, 590, 422, 554, 235, 571, 278, 370)

735² = 540225, 5 + 4 + 0 + 2 + 25 = 6²,
735² = 540225, 5 + 4 + 0 + 22 + 5 = 6²,
735² = 540225, 54 + 0 + 2 + 25 = 9²,
735² = 540225, 54 + 0 + 22 + 5 = 9²,
735² = 540225, 540² + 225² = 585².

Page of Squares : First Upload September 12, 2005 ; Last Revised June 29, 2010
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

736

The smallest squares containing k 736's :
20736 = 144²,
736796736 = 27144²,
298717367367364 = 17283442².

The squares which begin with 736 and end in 736 are
736796736 = 27144², 73634078736 = 271356², 736411124736 = 858144²,
736775022736 = 858356², 7361150364736 = 2713144²,...

736² = 541696, 5 + 4 + 16 + 96 = 11²,
736² = 541696, 5 + 41 + 69 + 6 = 11².

Page of Squares : First Upload September 12, 2005 ; Last Revised September 4, 2006
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

737

The smallest squares containing k 737's :
737881 = 859²,
2173797376 = 46624²,
573773707373476 = 23953574².

737² = 543169, a square with different digits.

Komachi equations:
737² = 9 - 8 - 7 * 6 + 543210 = 9 - 8 * 7 + 6 + 543210.

10318^k + 88440^k + 159929^k + 284482^k are squares for k = 1,2,3 (737², 338283², 166752883²).

3-by-3 magic squares consisting of different squares with constant 737²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(17, 84, 732, 372, 633, 64, 636, 368, 57),	(17, 204, 708, 444, 568, 153, 588, 423, 136),
(28, 81, 732, 192, 708, 71, 711, 188, 48),	(28, 273, 684, 504, 492, 217, 537, 476, 168),
(36, 332, 657, 487, 504, 228, 552, 423, 244),	(48, 199 ,708, 249, 672, 172, 692, 228, 111),
(48, 216, 703, 447, 568, 144, 584, 417, 168),	(48, 456, 577, 496, 447, 312, 543, 368, 336),
(57, 288, 676, 388, 564, 273, 624, 377, 108),	(108, 217, 696, 316, 648, 153, 657, 276, 188),
(108, 244, 687, 336, 633, 172, 647, 288, 204),	(108, 431, 588, 512, 468, 249, 519, 372, 368),
(120, 388, 615, 487, 420, 360, 540, 465, 188)

737² = 543169, 5 + 43 + 1 + 6 + 9 = 8²,
737² = 543169, 54 + 31 + 6 + 9 = 10²,
737² = 543169, 5³ + 4³ + 3³ + 16³ + 9³ = 71².

Page of Squares : First Upload September 12, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

738

The smallest squares containing k 738's :
173889 = 417²,
67380738084 = 259578²,
1380738738673849 = 37158293².

738 = (1² + 2² + 3² + ... + 40²) / (1² + 2² + 3² + 4²).

738² = 544644, a square with just 3 kinds of digits.

738² = 544644, 5 + 4 + 4 + 64 + 4 = 9²,
738² = 544644, 54 + 46 + 44 = 12²,
738² = 544644, 54⁴ + 4⁴ + 64⁴ + 4⁴ = 5028².

Kaprekar : 738⁷ = 119232467787562584192,
and 11² + 9² + 2² + 3² + 2² + 4² + 6² + 7² + 7² + 8² + 7² + 5² +6² + 2² + 5² + 8² + 4² + 1² + 9² + 2² = 738.

738² = 3⁴ + 9⁴ + 9⁴ + 27⁴.

Page of Squares : First Upload September 12, 2005 ; Last Revised November 25, 2008
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University

739

The smallest squares containing k 739's :
7396 = 86²,
92739739024 = 304532²,
157397398739344 = 12545812².

739² = 546121, a zigzag square.

3-by-3 magic squares consisting of different squares with constant 739²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(1, 138, 726, 294, 666, 127, 678, 289, 54),	(1, 366, 642, 498, 474, 271, 546, 433, 246),
(6, 303, 674, 399, 566, 258, 622, 366, 159),	(15, 386, 630, 450, 495, 314, 586, 390, 225),
(33, 134, 726, 474, 561, 82, 566, 462, 111),	(54, 334, 657, 369, 558, 314, 638, 351, 126),
(54, 447, 586, 478, 426, 369, 561, 406, 258),	(62, 366, 639, 414, 513, 334, 609, 386, 162),
(63, 246, 694, 314, 639, 198, 666, 278, 159),	(111, 282, 674, 414, 586, 177, 602, 351, 246),
(114, 271, 678, 426, 582, 161, 593, 366, 246)

739² = 546121, 5 + 46 + 12 + 1 = 8²,
739² = 546121, 54 + 6 + 1 + 2 + 1 = 8².

Page of Squares : First Upload September 12, 2005 ; Last Revised August 17, 2009
by Yoshio Mimura, Mathematics, Kobe pharmaceutical University