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900 - 909

900

the square of 30.

The smallest squares containing k 900's :
900 = 302,
225900900 = 150302,
1269009009009 = 11265032.

The squares which begin with 900 and end in 900 are
9000316900 = 948702,   90018000900 = 3000302,   90042004900 = 3000702,
90078016900 = 3001302,   900088612900 = 9487302,...

9002 = (42 - 1)(92 - 1)(262 - 1).

9002 + 9012 + 9022 + ... + 167412 = 12505392.

(1)(2 + 3)(4)(5)(6)(7 + 8)(9)(10) = 9002,
(1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13) = 9002,
(1)(2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13) = 9002,
(1 + 2 + 3)(4 + 5)(6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14) = 9002,
(1)(2)(3)(4 + 5)(6 + 7 + 8 + 9)(10)( 11 + 12 + 13 + 14) = 9002,
(1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9)(10)(11 + 12 + 13 + 14) = 9002,
(1)(2)(3)(4 + 5 + 6 + 7 + 8)(9)(10)(11 + 12 + 13 + 14) = 9002,
(1 + 2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14)(15) = 9002,
(1)(2)(3)(4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14)(15) = 9002,
(1 + 2)(3 + 4 + ... + 17)(18 + 19 + ... + 62) = 9002,
(1 + 2 + 3)(4 + 5 + ... + 12 )( 13 + 14 + ... + 62) = 9002.

(13 + 23 + ... + 4763)(4773 + 4783 + ... + 9003) = 441879520322.

Page of Squares : First Upload January 16, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

901

The smallest squares containing k 901's :
1590121 = 12612,
39010990144 = 1975122,
190110901901041 = 137880712.

9012 = 811801, a square with 3 kinds of digits.

9012 = (20 + 21 + 22 + ... + 36)2 + (37 + 38 + 39 + ... + 53)2.

(13 + 23 + ... + 5613)(5623 + 5633 + ... + 5823)(5833 + 5843 + ... + 9013) = 36496894207934162.

46852k + 54961k + 144160k + 565828k are squares for k = 1,2,3 (9012, 5883532, 4294427292).

3-by-3 magic squares consisting of different squares with constant 9012:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 424, 795, 476, 675, 360, 765, 420, 224),(3, 336, 836, 444, 728, 291, 784, 411, 168),
(24, 244, 867, 568, 669, 204, 699, 552, 136),(24, 477, 764, 512, 636, 381, 741, 424, 288),
(28, 264, 861, 381, 784, 228, 816, 357, 136),(36, 316, 843, 501, 708, 244, 748, 459, 204),
(48, 379, 816, 564, 624, 323, 701, 528, 204),(69, 136, 888, 216, 867, 116, 872, 204, 99),
(69, 424, 792, 568, 636, 291, 696, 477, 316),(116, 564, 693, 612, 459, 476, 651, 532, 324),
(132, 309, 836, 459, 748, 204, 764, 396, 267),(136, 357, 816, 588, 656, 189, 669, 504, 332),
(192, 451, 756, 539, 672, 264, 696, 396, 413) 

9012 = 811801, 8 + 11 + 80 + 1 = 102,
9012 = 811801, 81 + 18 + 0 + 1 = 102.

Page of Squares : First Upload January 16, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

902

The smallest squares containing k 902's :
9025 = 952,
3679029025 = 606552,
205690239029025 = 143419052.

9022 = 813604, a square with different digits.

902 is the first integer which is the sum of a square and a prime in 13 ways:
52 + 877, 72 + 853, 92 + 821, 132 + 733, 152 + 677, 172 + 613, 192 + 541, 212 + 461, 232 + 373, 252 + 277, 272 + 173, 292 + 61, 302 + 2.

9022 = 813604, 8 + 1 + 36 + 0 + 4 = 72,
9022 = 813604, 81 + 36 + 0 + 4 = 112.
9022 = 813604, 8 + 13 + 604 = 252.

Page of Squares : First Upload January 16, 2006 ; Last Revised October 5, 2006
by Yoshio Mimura, Kobe, Japan

903

The smallest squares containing k 903's :
790321 = 8892,
19039032324 = 1379822,
9034490903903721 = 950499392.

9032 = 815409, a square with different digits.

9032 = 3012 + 6022 + 6022 : 2062 + 2062 + 1032 = 3092.

9032 + 9042 + 9052 + ... + 9242 = 9252 + 9262 + 9272 + ... + 9452.

A cubic polynomial:
(X + 5762)(X + 6882)(X + 9032) = X3 + 12732X2 + 9019682X + 3578480642.

Komachi equations:
9032 = 92 * 82 * 72 / 62 * 52 * 432 / 22 / 102 = 982 / 72 * 62 * 52 * 432 / 22 / 102.

8812 + 8822 + 8832 + 8842 + ... + 9032 = 42782.

3-by-3 magic squares consisting of different squares with constant 9032:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 134, 893, 563, 698, 106, 706, 557, 82),(2, 211, 878, 269, 838, 202, 862, 262, 61),
(2, 227, 874, 634, 622, 163, 643, 614, 158),(2, 302, 851, 358, 781, 278, 829, 338, 118),
(13, 94, 898, 326, 838, 83, 842, 323, 46),(13, 478, 766, 614, 557, 358, 662, 526, 317),
(19, 382, 818, 458, 701, 338, 778, 422, 179),(22, 173, 886, 214, 862, 163, 877, 206, 62),
(22, 355, 830, 605, 622, 250, 670, 550, 253),(26, 442, 787, 547, 634, 338, 718, 467, 286),
(36, 228, 873, 297, 828, 204, 852, 279, 108),(38, 253, 866, 526, 698, 227, 733, 514, 118),
(46, 422, 797, 622, 563, 334, 653, 566, 262),(49, 392, 812, 532, 644, 343, 728, 497, 196),
(62, 578, 691, 611, 482, 458, 662, 499, 358),(72, 444, 783, 528, 657, 324, 729, 432, 312),
(74, 202, 877, 418, 787, 146, 797, 394, 158),(81, 288, 852, 468, 744, 207, 768, 423, 216),
(83, 194, 878, 422, 787, 134, 794, 398, 163),(83, 242, 866, 502, 734, 157, 746, 467, 202),
(83, 382, 814, 586, 643, 242, 682, 506, 307),(96, 423, 792, 612, 552, 369, 657, 576, 228),
(110, 410, 797, 445, 722, 310, 778, 355, 290),(110, 547, 710, 578, 590, 365, 685, 410, 422),
(112, 532, 721, 623, 476, 448, 644, 553, 308),(115, 422, 790, 530, 610, 403, 722, 515, 170),
(130, 478, 755, 610, 605, 278, 653, 470, 410),(157, 286, 842, 358, 803, 206, 814, 298, 253),
(163, 346, 818, 482, 733, 214, 746, 398, 317),(173, 514, 722, 566, 502, 493, 682, 547, 226),
(202, 478, 739, 509, 682, 302, 718, 349, 422) 

9032 = 815409, 8 + 15 + 4 + 0 + 9 = 62,
9032 = 815409, 81 + 54 + 0 + 9 = 122.

Page of Squares : First Upload January 16, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

904

The smallest squares containing k 904's :
19044 = 1382,
1949045904 = 441482,
6909049049049 = 26285072.

The squares which begin with 904 and end in 904 are
904119525904 = 9508522,   904682517904 = 9511482,   9041158949904 = 30068522,
9042939093904 = 30071482,   9044166051904 = 30073522,...

9042 is the 8th square which is the sum of 5 fifth powers : 9042 = 35 + 95 + 95 + 115 + 145.

Komachi equations:
9042 = - 14 * 24 * 34 + 44 + 54 * 64 - 74 + 84 + 94 = 94 + 84 - 74 + 64 * 54 + 44 - 34 * 24 */ 14.

9042 = 817216, 8 + 1 + 7 + 2 + 1 + 6 = 52,
9042 = 817216, 81 + 72 + 16 = 132,
9042 = 817216, 8 + 172 + 16 = 142,
9042 = 817216, 8 + 1 + 7216 = 852.

Page of Squares : First Upload January 16, 2006 ; Last Revised July 9, 2010
by Yoshio Mimura, Kobe, Japan

905

The smallest squares containing k 905's :
290521 = 5392,
69054905089 = 2627832,
637905390590521 = 252567892.

9052 = 819025, a square with different digits.

9052 = 819025 with 81 = 92 and 9025 = 952.

9052 = 819025 , 8 + 1 * 902 - 5 = 8 * 1 + 902 - 5 = 905.

9052 = 819025, 8 + 1 + 9 + 0 + 2 + 5 = 52,
9052 = 819025, 81 + 90 + 25 = 142.

9052 = 819025 appears in the decimal expression of e:
  e = 2.71828•••819025••• (from the 406th digit)
  (819025 is the first 6-digit square in the expression of e.)

3-by-3 magic squares consisting of different squares with constant 9052:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 95, 900, 543, 720, 76, 724, 540, 57),(4, 372, 825, 528, 671, 300, 735, 480, 220),
(24, 132, 895, 255, 860, 120, 868, 249, 60),(32, 225, 876, 420, 780, 185, 801, 400, 132),
(48, 265, 864, 489, 720, 248, 760, 480, 105),(57, 220, 876, 276, 840, 193, 860, 255, 120),
(60, 351, 832, 455, 732, 276, 780, 400, 225),(60, 441, 788, 545, 612, 384, 720, 500, 225),
(60, 545, 720, 633, 540, 356, 644, 480, 417),(95, 252, 864, 540, 711, 148, 720, 500, 225),
(105, 320, 840, 480, 735, 220, 760, 420, 255),(108, 319, 840, 356, 792, 255, 825, 300, 220),
(112, 375, 816, 480, 720, 265, 759, 400, 288),(112, 516, 735, 609, 588, 320, 660, 455, 420),
(120, 489, 752, 535, 648, 336, 720, 400, 375),(120, 535, 720, 585, 504, 472, 680, 528, 279),
(144, 508, 735, 545, 540, 480, 708, 519, 220),(185, 372, 804, 420, 760, 255, 780, 321, 328),
(204, 447, 760, 540, 680, 255, 697, 396, 420) 

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

906

The smallest squares containing k 906's :
90601 = 3012,
906190609 = 301032,
90690690663556 = 95231662.

2582 + 2592 + 2602 + ... + 9062 = 155762.

9062 = 3022 + 6042 + 6042 : 4062 + 4062 + 2032 = 6092.

The 4-by-4 magic squares consisting of different squares with constant 906:

22 72182232
92202192 82
142212102132
252 42112122
     
32 72 82282
102142232 92
112252122 42
262 62132 52

9062 = 820836, 8 + 208 + 3 + 6 = 152.

Page of Squares : First Upload January 16, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

907

The smallest squares containing k 907's :
599076 = 7742,
5369079076 = 732742,
179070590739076 = 133817262.

1 / 907 = 0.0011025..., and 11025 = 1052.

3-by-3 magic squares consisting of different squares with constant 9072:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 153, 894, 327, 834, 142, 846, 322, 57),(2, 246, 873, 414, 777, 218, 807, 398, 114),
(9, 282, 862, 498, 722, 231, 758, 471, 162),(30, 407, 810, 582, 630, 295, 695, 510, 282),
(42, 279, 862, 538, 702, 201, 729, 502, 198),(42, 342, 839, 601, 618, 282, 678, 569, 198),
(57, 250, 870, 630, 615, 218, 650, 618, 135),(106, 183, 882, 222, 866, 153, 873, 198, 146),
(110, 282, 855, 393, 790, 210, 810, 345, 218),(142, 471, 762, 582, 538, 441, 681, 558, 218),
(153, 474, 758, 538, 663, 306, 714, 398, 393),(169, 462, 762, 618, 498, 439, 642, 601, 222),
(183, 574, 678, 614, 582, 327, 642, 393, 506) 

9072 = 822649, 8 + 2 + 26 + 4 + 9 = 72,
9072 = 822649, 8 + 22 + 6 + 4 + 9 = 72,
9072 = 822649, 82 + 26 + 4 + 9 = 112,
9072 = 822649, 82 + 22 + 262 + 42 + 92 = 292,
9072 = 822649, 822 + 6 + 4 + 9 = 292,
9072 = 822649, 822 + 262 + 492 = 992.

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

908

The smallest squares containing k 908's :
259081 = 5092,
1908990864 = 436922,
3819089085890841 = 617987792.

9082 = 824464 a square with even digits.

Komachi fraction : 9082 = 59361408/72.

908 is the first integer which is the sum of a square and a prime in 14 ways :
12 + 907, 52 + 883, 72 + 859, 92 + 827, 112 + 787, 132 + 739, 152 + 683, 172 + 619, 192 + 547, 212 + 467, 232 + 379, 252 + 283, 272 + 179, 292 + 67.

9082 = 824464, 8 + 2 + 4 + 46 + 4 = 82,
9082 = 824464, 8 + 2 + 44 + 6 + 4 = 82,
9082 = 824464, 8 + 24 + 4 + 64 = 102,
9082 = 824464, 82 + 4 + 4 + 6 + 4 = 102.

Page of Squares : First Upload January 16, 2006 ; Last Revised October 5, 2006
by Yoshio Mimura, Kobe, Japan

909

The smallest squares containing k 909's :
190969 = 4372,
90952909056 = 3015842,
629099099095104 = 250818482.

9092 = 826281, a zigzag square.

9092 = 86281, 826 + 2 + 81 = 909.

9092 + 9102 + 9112 + ... + 28522 = 865262.

3-by-3 magic squares consisting of different squares with constant 9092:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(5, 460, 784, 560, 616, 365, 716, 485, 280),(19, 152, 896, 544, 716, 133, 728, 539, 76),
(19, 256, 872, 416, 773, 236, 808, 404, 101),(19, 256, 872, 416, 787, 184, 808, 376, 179),
(28, 91, 904, 296, 856, 77, 859, 292, 56),(30, 534, 735, 591, 570, 390, 690, 465, 366),
(42, 159, 894, 366, 822, 129, 831, 354, 102),(56, 512, 749, 604, 581, 352, 677, 476, 376),
(61, 164, 892, 332, 836, 131, 844, 317, 116),(66, 222, 879, 318, 831, 186, 849, 294, 138),
(66, 255, 870, 570, 690, 159, 705, 534, 210),(67, 556, 716, 604, 508, 451, 676, 509, 332),
(68, 536, 731, 571, 544, 452, 704, 493, 296),(76, 259, 868, 637, 604, 236, 644, 628, 131),
(77, 376, 824, 616, 628, 229, 664, 539, 308),(88, 284, 859, 584, 677, 164, 691, 536, 248),
(101, 404, 808, 556, 613, 376, 712, 536, 179),(109, 368, 824, 584, 604, 347, 688, 571, 164),
(116, 229, 872, 283, 844, 184, 856, 248, 179),(116, 340, 835, 515, 716, 220, 740, 445, 284),
(116, 485, 760, 635, 500, 416, 640, 584, 275),(124, 443, 784, 581, 644, 272, 688, 464, 371),
(129, 438, 786, 618, 534, 399, 654, 591, 222),(184, 416, 787, 508, 709, 256, 731, 388, 376),
(186, 417, 786, 474, 726, 273, 753, 354, 366) 

9092 = 826281, 8 + 2 + 62 + 8 + 1 = 92,
9092 = 826281, 82 + 62 + 81 = 152.

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