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910 - 919

910

The smallest squares containing k 910's :
719104 = 8482,
91091086969 = 3018132,
91075491049104 = 95433482.

9102 = (4 + 5)2 + (6 + 7)2 + (8 + 9)2 + ... + (106 + 107)2.

(12 + 22 + 32 + 42)(52)(62 + 72 + ... + 9102) = 4344002.

the square root of 910 is 30. 1 6 6 2 0 6 2 5 7 9 9 6 7 12 17 2 5 ..., and
302 = 12 + 62 + 62 + 22 + 02 + 62 + 22 + 52 + 72 + 92 + 92 + 62 + 72 + 122 + 172 + 22 + 52.

26k + 338k + 910k + 1226k are squares for k = 1,2,3 (502, 15642, 513322).

Page of Squares : First Upload January 23, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

911

The smallest squares containing k 911's :
891136 = 9442,
91127911876 = 3018742,
911911714911376 = 301978762.

9112 = 829921, 8 - 2 * 9 + 921 = 82 + 9 * 92 + 1 = 911.

9112 + 9122 + 9132 + ... + 88312 = 4789092.

3-by-3 magic squares consisting of different squares with constant 9112:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(11, 306, 858, 534, 693, 254, 738, 506, 171),(21, 282, 866, 642, 619, 186, 646, 606, 213),
(30, 214, 885, 565, 690, 186, 714, 555, 110),(38, 171, 894, 549, 718, 114, 726, 534, 133),
(38, 186, 891, 459, 774, 142, 786, 443, 126),(54, 389, 822, 597, 606, 326, 686, 558, 219),
(66, 299, 858, 507, 726, 214, 754, 462, 219),(66, 501, 758, 606, 542, 411, 677, 534, 294),
(69, 378, 826, 434, 714, 363, 798, 421, 126),(70, 411, 810, 450, 690, 389, 789, 430, 150),
(102, 501, 754, 534, 646, 357, 731, 402, 366),(114, 245, 870, 430, 786, 165, 795, 390, 214),
(114, 326, 843, 378, 789, 254, 821, 318, 234),(117, 366, 826, 466, 738, 261, 774, 389, 282),
(126, 506, 747, 558, 549, 466, 709, 522, 234),(138, 434, 789, 619, 534, 402, 654, 597, 214),
(187, 534, 714, 606, 459, 502, 654, 578, 261) 

9112 = 829921, 8 + 2 + 9 + 9 + 21 = 72,
9112 = 829921, 8 + 29 + 9 + 2 + 1 = 72,
9112 = 829921, 82 + 9 + 9 + 21 = 112,
9112 = 829921, 829 + 9 + 2 + 1 = 292,
9112 = 829921, 8 + 29921 = 1732.

Page of Squares : First Upload January 23, 2006 ; Last Revised October 16, 2009
by Yoshio Mimura, Kobe, Japan

912

The smallest squares containing k 912's :
91204 = 3022,
30039129124 = 1733182,
1191228759129124 = 345141822.

A cubic polynomial :
(X + 9122)(X + 12112)(X + 18362) = X3 + 23812X2 + 29944922X + 20277371522.

1 / 912 = 0.00109649122807017543859649122807017...,
and the sum of the squares of its digits is 912.

the square root of 912 is 30.199337741082998788946...,
and 302 = 12 + 92 + 92 + 32 + 32 + 72 + 72 + 42 + 12 + 02 + 82 + 22 + 92 + 92 + 82 + 72 + 82 + 82 + 92 + 42 + 62 = 302.

9122 = 831744, 8 + 3 + 17 + 4 + 4 = 62,
9122 = 831744, 83 + 17 + 44 = 122.

Page of Squares : First Upload January 23, 2006 ; Last Revised October 5, 2006
by Yoshio Mimura, Kobe, Japan

913

The smallest squares containing k 913's :
309136 = 5562,
913913361 = 302312,
442179139139136 = 210280562.

9132 + 9142 + 9152 + ... + 320162 = 33074642.

the square root of 913 is 30.215..., and 30 = 22 + 12 + 52.

3-by-3 magic squares consisting of different squares with constant 9132:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(9, 212, 888, 328, 828, 201, 852, 321, 68),(12, 216, 887, 257, 852, 204, 876, 247, 72),
(12, 320, 855, 580, 663, 240, 705, 540, 212),(12, 393, 824, 447, 716, 348, 796, 408, 183),
(12, 608, 681, 639, 492, 428, 652, 471, 432),(23, 432, 804, 576, 617, 348, 708, 516, 257),
(36, 372, 833, 617, 624, 252, 672, 553, 276),(63, 140, 900, 300, 855, 112, 860, 288, 105),
(63, 276, 868, 428, 777, 216, 804, 392, 183),(63, 508, 756, 636, 567, 328, 652, 504, 393),
(68, 372, 831, 489, 688, 348, 768, 471, 148),(104, 468, 777, 567, 644, 312, 708, 447, 364),
(112, 456, 783, 504, 687, 328, 753, 392, 336) 

Page of Squares : First Upload January 23, 2006 ; Last Revised October 16, 2009
by Yoshio Mimura, Kobe, Japan

914

The smallest squares containing k 914's :
491401 = 7012,
4914991449 = 701072,
2289091491491401 = 478444512.

9142 = 835396, a zigzag square.

Komachi equation: 9142 = 982 / 72 * 652 + 432 * 22 - 102.

9142 = 835396, 83 + 5 + 396 = 222.

Page of Squares : First Upload January 23, 2006 ; Last Revised July 9, 2010
by Yoshio Mimura, Kobe, Japan

915

The smallest squares containing k 915's :
915849 = 9572,
9157915809 = 956972,
891591591539961 = 298595312.

9152 = 123 + 173 + 943 = 343 + 413 + 903.

9152 + 9162 + 9172 + ... + 245322 = 22184052.

9152 = 837225 appears in the decimal expression of π:
  π = 3.14159•••837225••• (from the 88730th digit).

3-by-3 magic squares consisting of different squares with constant 9152:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 165, 900, 549, 720, 132, 732, 540, 99),(10, 95, 910, 385, 826, 82, 830, 382, 49),
(10, 95, 910, 602, 686, 65, 689, 598, 70),(10, 190, 895, 470, 769, 158, 785, 458, 106),
(10, 190, 895, 622, 655, 146, 671, 610, 122),(10, 346, 847, 385, 770, 310, 830, 353, 154),
(10, 385, 830, 470, 710, 335, 785, 430, 190),(10, 433, 806, 470, 694, 367, 785, 410, 230),
(14, 175, 898, 298, 850, 161, 865, 290, 70),(14, 577, 710, 623, 514, 430, 670, 490, 385),
(17, 94, 910, 494, 767, 70, 770, 490, 65),(27, 300, 864, 564, 675, 252, 720, 540, 165),
(36, 477, 780, 645, 540, 360, 648, 564, 315),(38, 209, 890, 545, 710, 190, 734, 538, 95),
(45, 360, 840, 456, 720, 333, 792, 435, 144),(50, 223, 886, 335, 830, 190, 850, 314, 127),
(50, 242, 881, 575, 694, 158, 710, 545, 190),(50, 335, 850, 575, 650, 290, 710, 550, 175),
(50, 479, 778, 575, 622, 346, 710, 470, 335),(62, 466, 785, 641, 538, 370, 650, 575, 290),
(65, 322, 854, 490, 710, 305, 770, 479, 122),(70, 290, 865, 410, 785, 230, 815, 370, 190),
(70, 410, 815, 518, 655, 374, 751, 490, 182),(84, 360, 837, 612, 645, 216, 675, 540, 300),
(113, 566, 710, 634, 463, 470, 650, 550, 335),(118, 274, 865, 430, 785, 190, 799, 382, 230),
(122, 305, 854, 550, 710, 175, 721, 490, 278),(122, 454, 785, 610, 545, 410, 671, 578, 230),
(161, 410, 802, 602, 655, 214, 670, 490, 385),(175, 506, 742, 550, 658, 319, 710, 385, 430),
(190, 335, 830, 370, 802, 239, 815, 286, 302),(190, 430, 785, 463, 734, 290, 766, 337, 370)

Page of Squares : First Upload January 23, 2006 ; Last Revised October 16, 2009
by Yoshio Mimura, Kobe, Japan

916

The smallest squares containing k 916's :
2916 = 542,
6916916224 = 831682,
23829169169169 = 48815132.

The squares which begin with 916 and end in 916 are
916702842916 = 9574462,   916909662916 = 9575542,   9160029114916 = 30265542,
9162402086916 = 30269462,   9163055918916 = 30270542,...

9162 = 839056, a square with different digits.

9162 = 839056 , 8 - 3 + 905 + 6 = 916.

Page of Squares : First Upload January 23, 2006 ; Last Revised October 5, 2006
by Yoshio Mimura, Kobe, Japan

917

The smallest squares containing k 917's :
917764 = 9582,
5917917184 = 769282,
3982917917991769 = 631103632.

(675 / 917)2 = 0.541837269... (Komachic).

A cubic polynomial :
(X + 2882)(X + 6042)(X + 6272) = X3 + 9172X2 + 4541882X + 1090679042.

9172 = 283 + 663 + 813.

3-by-3 magic squares consisting of different squares with constant 9172:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(8, 339, 852, 501, 712, 288, 768, 468, 179),(8, 564, 723, 597, 552, 424, 696, 467, 372),
(12, 307, 864, 424, 768, 267, 813, 396, 152),(12, 396, 827, 507, 692, 324, 764, 453, 228),
(19, 228, 888, 552, 712, 171, 732, 531, 152),(45, 408, 820, 460, 720, 333, 792, 395, 240),
(64, 333, 852, 528, 684, 307, 747, 512, 144),(69, 332, 852, 548, 699, 228, 732, 492, 251),
(96, 228, 883, 267, 856, 192, 872, 237, 156),(98, 441, 798, 567, 658, 294, 714, 462, 343),
(108, 435, 800, 600, 640, 267, 685, 492, 360),(116, 408, 813, 552, 683, 264, 723, 456, 332),
(123, 316, 852, 492, 732, 251, 764, 453, 228),(123, 316, 852, 516, 732, 197, 748, 453, 276),
(132, 397, 816, 627, 636, 208, 656, 528, 363),(192, 397, 804, 496, 732, 243, 747, 384, 368),
(237, 548, 696, 604, 627, 288, 648, 384, 523) 

9172 = 840889, 82 + 42 + 02 + 82 + 82 + 92 = 172.

Page of Squares : First Upload January 23, 2006 ; Last Revised October 16, 2009
by Yoshio Mimura, Kobe, Japan

918

The smallest squares containing k 918's :
91809 = 3032,
4918918225 = 701352,
730949189189184 = 270360722.

9182 = 253 + 743 + 753.

9182 = 842724 is exchangeable, 272484 = 5222.

9182 = (12 + 2)(222 + 2)(242 + 2) = (12 + 2)(42 + 2)(52 + 2)(242 + 2)
= (22 + 2)(52 + 2)(72 + 2)(102 + 2).

9182 = 842724, 842 + 72 + 4 = 918.

(1 + 2 + ... + 8)(9 + 10 + ... + 25)(26 + 27 + 28) = 9182,
(1)(2 + 3 + ... + 25)(26 + 27 + ... + 76) = 9182.

(13 + 23 + ... + 4373)(4383 + 4393 + ... + 7823)(7833 + 7843 + ... + 9183) = 80760647600942402.

1 / 918 = 0.001089..., and 1089 = 332.

Komachi equation: 9182 = 92 * 82 * 7652 / 42 / 32 * 22 / 102.

The 4-by-4 magic squares consisting of different squares with constant 918:

22 42132272
72252122102
172 92222 82
242142112 52
      
22 72 92282
82232152102
112142242 52
272122 62 32

9182 = 842724, 8 + 42 + 7 + 24 = 92,
9182 = 842724, 8 + 427 + 2 + 4 = 212.

Page of Squares : First Upload January 23, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

919

The smallest squares containing k 919's :
919681 = 9592,
9198919921 = 959112,
791991976091904 = 281423522.

9192 is the 4th square which is the sum of 10 seventh powers.

9192 = 273 + 553 + 873.

(376 / 919)2 = 0.167395842... (Komachic),
(625 / 919)2 = 0.462518397... (Komachic).

1 / 919 = 0.00108813928182807399347116430903155...,
and the sum of the squares of its digits is 919.

the square root of 919 is 30.31501278244823699516853932224349406...,
and 302 = 32 + 12 + 52 + 02 + 12 + 22 + 72 + 82 + 22 + 42 + 42 + 82 + 22 + 32 + 62 + 92 + 92 + 52 + 12 + 62 + 82 + 52 + 32 + 92 + 32 + 22 + 22 + 22 + 42 + 32 + 42 + 92 + 42 + 02 + 62.

3-by-3 magic squares consisting of different squares with constant 9192:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 154, 906, 386, 822, 141, 834, 381, 62),(10, 531, 750, 630, 550, 381, 669, 510, 370),
(18, 134, 909, 339, 846, 118, 854, 333, 66),(18, 546, 739, 594, 557, 426, 701, 486, 342),
(35, 150, 906, 606, 685, 90, 690, 594, 125),(46, 147, 906, 498, 766, 99, 771, 486, 118),
(46, 318, 861, 426, 771, 262, 813, 386, 186),(78, 494, 771, 531, 606, 442, 746, 483, 234),
(98, 189, 894, 306, 854, 147, 861, 282, 154),(99, 314, 858, 638, 594, 291, 654, 627, 154),
(109, 234, 882, 342, 834, 179, 846, 307, 186),(134, 477, 774, 531, 594, 458, 738, 514, 189),
(141, 426, 802, 494, 717, 294, 762, 386, 339),(174, 381, 818, 477, 746, 246, 766, 378, 339),
(179, 462, 774, 594, 654, 253, 678, 451, 426),(186, 573, 694, 602, 606, 339, 669, 386, 498)

9192 = 844561, 8 + 4 + 45 + 6 + 1 = 82,
9192 = 844561, 8 + 44 + 5 + 6 + 1 = 82,
9192 = 844561, 84 + 4 + 5 + 6 + 1 = 102.

9192 = 844561 appears in the decimal expression of e:
  e = 2.71828•••844561••• (from the 66270th digit).

Page of Squares : First Upload January 23, 2006 ; Last Revised October 16, 2009
by Yoshio Mimura, Kobe, Japan