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990

The smallest squares containing k 990's :
419904 = 648²,
9909904 = 3148²,
869900990999056 = 29494084².

990² + 991² + 992² + ... + 1012² = 1013² + 1014² + 1015² + ... + 1034².

The integral triangle of sides 1090, 29161, 30249 has square area 990².

Komachi square sum : 990² = 1³ + 2³ + 5³ + 39³ + 64³ + 87³.

990^k + 1020^k + 1785^k + 1830^k are squares for k = 1,2,3 (75², 2925², 117675²).

(1)(2 + 3)(4)(5 + 6)(7 + 8)(9)(10 + 11 + 12) = 990²,
(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11)(12 + 13) = 990²,
(1)(2)(3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11)(12 + 13) = 990²,
(1 + 2 + ... + 54)(55 + 56 + ... + 65) = 990²,
(1³ + ... + 262³)(263³ + 264² + ... + 274³)(275³ + 276³ + ... + 990³) = 256883348531040².

990² = 980100, 980 + 10 + 0 = 990.

The 4-by-4 magic squares consisting of different squares with constant 990:

9909904 = 3148².

Page of Squares : First Upload March 20, 2006 ; Last Revised September 30, 2011
by Yoshio Mimura, Kobe, Japan

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991

The smallest squares containing k 991's :
7991929 = 2827²,
99199171681 = 314959²,
3999199199165284 = 63239222².

991² = 982081, 982 + 0 + 8 + 1 = 991.

991² + 992² + 993² + ... + 1232² = 17325².

3-by-3 magic squares consisting of different squares with constant 991²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 198, 971, 546, 811, 162, 827, 534, 114)	(6, 334, 933, 658, 699, 246, 741, 618, 226),
(26, 138, 981, 267, 946, 126, 954, 261, 62)	(26, 603, 786, 699, 546, 442, 702, 566, 411),
(27, 294, 946, 674, 699, 198, 726, 638, 219)	(37, 426, 894, 534, 762, 341, 834, 469, 258),
(37, 426, 894, 534, 811, 198, 834, 378, 379)	(42, 134, 981, 314, 933, 114, 939, 306, 82),
(42, 501, 854, 651, 658, 354, 746, 546, 357)	(69, 422, 894, 498, 789, 334, 854, 426, 267),
(75, 334, 930, 566, 750, 315, 810, 555, 134)	(75, 530, 834, 666, 645, 350, 730, 534, 405),
(82, 426, 891, 594, 693, 386, 789, 566, 198)	(107, 246, 954, 414, 882, 181, 894, 379, 198),
(107, 594, 786, 666, 539, 498, 726, 582, 341)	(114, 523, 834, 558, 726, 379, 811, 426, 378),
(118, 294, 939, 411, 874, 222, 894, 363, 226)	(126, 354, 917, 539, 798, 234, 822, 469, 294),
(261, 622, 726, 674, 414, 597, 678, 651, 314)

991² = 982081, 9 + 8 + 2 + 0 + 81 = 10²,
991² = 982081, 9 + 82 + 0 + 8 + 1 = 10².

Page of Squares : First Upload March 20, 2006 ; Last Revised November 12, 2009
by Yoshio Mimura, Kobe, Japan

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992

The smallest squares containing k 992's :
29929 = 173²,
4199299204 = 64802²,
9923589929929 = 3150173².

992 is the second integer which is the sum of a square and a prime in 11 ways:
1² + 991, 3² + 983, 5² + 967, 9² + 911, 13² + 823, 19² + 631, 23² + 463, 25² + 367, 27² + 263, 29² + 151, 31² + 31.

992² = 984064, 9 + 8 + 40 + 64 = 11².

Page of Squares : First Upload March 20, 2006 ; Last Revised October 19, 2006
by Yoshio Mimura, Kobe, Japan

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993

The smallest squares containing k 993's :
1089936 = 1044²,
78993599364 = 281058²,
899344499399329 = 29989073².

1² + 2² + 3² + ... + 993² = 326875409, which is an integer with different digits.

Komachi equation: 993² = 987² + 654² / 3² / 2² - 1².

(2² + 1)(6² + 1)(8² + 1)(9² + 1) = 993² + 1.

993² + 994² + 995² + ... + 2528² = 71152².

The square root of 993 is 31. 5 11 9 0 25 1 3 1 7 7, and 31² = 5² + 11² + 9² + 0² + 25² + 1² + 3² + 1² + 7² + 7².

993² = 5³ + 25³ + 99³.

993² = 9⁴ + 16⁴ + 26⁴ + 26⁴.

993² = 152² + 452² + 871² : 178² + 254² + 251² = 399²,
993² = 331² + 662² + 662² : 266² + 266² + 133² = 399²,
993² = 431² + 452² + 772² : 277² + 254² + 134² = 399².

3-by-3 magic squares consisting of different squares with constant 993²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(7, 160, 980, 460, 868, 145, 880, 455, 68)	(7, 320, 940, 560, 775, 268, 820, 532, 175),
(7, 428, 896, 688, 644, 313, 716, 623, 292)	(16, 353, 928, 577, 752, 296, 808, 544, 193),
(17, 268, 956, 436, 857, 248, 892, 424, 103)	(17, 452, 884, 604, 697, 368, 788, 544, 263),
(18, 171, 978, 654, 738, 117, 747, 642, 126)	(18, 438, 891, 534, 747, 378, 837, 486, 222),
(28, 224, 967, 593, 772, 196, 796, 583, 112)	(28, 281, 952, 401, 868, 268, 908, 392, 89),
(40, 332, 935, 500, 815, 268, 857, 460, 200)	(44, 217, 968, 623, 748, 196, 772, 616, 103),
(44, 388, 913, 647, 704, 268, 752, 583, 284)	(54, 453, 882, 522, 738, 411, 843, 486, 198),
(56, 488, 863, 673, 616, 392, 728, 607, 296)	(68, 431, 892, 472, 772, 409, 871, 452, 152),
(112, 191, 968, 232, 952, 161, 959, 208, 152)	(112, 428, 889, 628, 721, 268, 761, 532, 352),
(114, 522, 837, 693, 642, 306, 702, 549, 438)	(116, 488, 857, 593, 724, 332, 788, 473, 376),
(121, 332, 928, 592, 772, 199, 788, 529, 292)	(136, 472, 863, 688, 577, 424, 703, 656, 248),
(164, 313, 928, 367, 892, 236, 908, 304, 263)	(172, 401, 892, 551, 788, 248, 808, 452, 359),
(208, 647, 724, 676, 628, 367, 697, 416, 572)

993² = 986049, 9 + 8 + 6 + 0 + 4 + 9 = 6²,
993² = 986049, 9 + 86 + 0 + 49 = 12².

Page of Squares : First Upload March 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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994

The smallest squares containing k 994's :
399424 = 632²,
2899499409 = 53847²,
9949409946199441 = 99746729².

994² = 988036, 988 + 0 * 3 + 6 = 994.

994^k + 1470^k + 2338^k + 4802^k are squares for k = 1,2,3 (98², 5628², 357308²).

994² = 988036, 988 + 0 + 36 = 32².

Page of Squares : First Upload March 20, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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995

The smallest squares containing k 995's :
749956 = 866²,
31199569956 = 176634²,
2699569959952996 = 51957386².

995² = 990025, and 9 = 3², 900 = 30², 25 = 5².

1 / 995 = 0.00 1 0 0 5 0 2 5 1 2 5 6 2 8 1 4 0 7 0 3 5 1 7 5 8 7 9 3 9 6 9 8 4 9 2 4 6 2 3 ...,
and the sum of the squares of its digits is 995.

3-by-3 magic squares consisting of different squares with constant 995²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(14, 198, 975, 702, 689, 150, 705, 690, 130)	(15, 238, 966, 630, 750, 175, 770, 609, 162),
(18, 590, 801, 625, 630, 450, 774, 495, 382)	(30, 95, 990, 310, 942, 81, 945, 306, 58),
(30, 310, 945, 670, 705, 210, 735, 630, 230)	(30, 319, 942, 670, 690, 255, 735, 642, 194),
(30, 518, 849, 570, 705, 410, 815, 474, 318)	(30, 570, 815, 670, 591, 438, 735, 562, 366),
(33, 294, 950, 410, 870, 255, 906, 383, 150)	(42, 194, 975, 481, 858, 150, 870, 465, 130),
(54, 175, 978, 465, 870, 130, 878, 450, 129)	(66, 390, 913, 513, 770, 366, 850, 495, 150),
(95, 570, 810, 618, 670, 399, 774, 465, 418)	(122, 255, 954, 450, 870, 175, 879, 410, 222),
(130, 417, 894, 690, 606, 383, 705, 670, 210)	(130, 465, 870, 690, 670, 255, 705, 570, 410),
(150, 495, 850, 607, 630, 474, 774, 590, 207)	(175, 426, 882, 630, 735, 230, 750, 518, 399),
(177, 486, 850, 670, 690, 255, 714, 527, 450)	(210, 618, 751, 670, 465, 570, 705, 626, 318),
(234, 625, 738, 662, 450, 591, 705, 630, 310)	(257, 570, 774, 630, 705, 310, 726, 410, 543)

995² = 990025, 9 + 9 + 0 + 0 + 2 + 5 = 5².

Page of Squares : First Upload March 20, 2006 ; Last Revised November 12, 2009
by Yoshio Mimura, Kobe, Japan

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996

The smallest squares containing k 996's :
12996 = 114²,
478996996 = 21886²,
42534996996996 = 6521886².

996²± 5 are primes.

The squares which begin with 996 and end in 996 are
99612196996 = 315614²,   996231556996 = 998114²,   996774604996 = 998386²,
9961055580996 = 3156114²,   9962772580996 = 3156386²,...

996² = 332² + 664² + 664² : 466² + 466² + 233² = 699².

Komachi cube sum : 996² = 7³ + 35³ + 41³ + 69³ + 82³.

996² = 992016, 9 + 9 + 2 + 0 + 16 = 6²,
996² = 992016, 9 + 9 + 201 + 6 = 15²,
996² = 992016, 9 + 9201 + 6 = 96².

Page of Squares : First Upload March 20, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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997

The smallest squares containing k 997's :
5997601 = 2449²,
99799759921 = 315911²,
299749975997521 = 17313289².

997² = 994009, a square with 3 kinds of digits.

997² = 156² + 492² + 853² : 358² + 294² + 651² = 799²,
997² = 432² + 453² + 776² : 677² + 354² + 234² = 799².

3-by-3 magic squares consisting of different squares with constant 997²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 99, 992, 632, 768, 69, 771, 628, 72)	(16, 453, 888, 492, 776, 387, 867, 432, 236),
(20, 555, 828, 660, 628, 405, 747, 540, 380)	(43, 372, 924, 588, 756, 277, 804, 533, 252),
(56, 252, 963, 603, 776, 168, 792, 573, 196)	(69, 288, 952, 448, 861, 228, 888, 412, 189),
(93, 516, 848, 624, 632, 453, 772, 573, 264)	(96, 443, 888, 547, 768, 324, 828, 456, 317),
(120, 453, 880, 628, 720, 285, 765, 520, 372)	(132, 317, 936, 576, 792, 187, 803, 516, 288),
(204, 448, 867, 483, 816, 308, 848, 357, 384)	(204, 597, 772, 628, 684, 363, 747, 412, 516),
(232, 588, 771, 672, 669, 308, 699, 448, 552)

997² = 94009, and 9 = 3², 400 = 20².

997² = 994009 appears in the decimal expression of e:
  e = 2.71828•••994009••• (from the 86195th digit)

Page of Squares : First Upload March 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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998

The smallest squares containing k 998's :
99856 = 316²,
23799849984 = 154272²,
7299869199889984 = 85439272².

Page of Squares : First Upload October 19, 2006 ; Last Revised October 19, 2006
by Yoshio Mimura, Kobe, Japan

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999

The smallest squares containing k 999's :
1999396 = 1414²,
6999999556 = 83666²,
239994999699984 = 15491772².

A cubic polynomial :
(X + 504²)(X + 999²)(X +1472²) = X³ + 1849²X² + 1722312²X + 741146112².

999² = 998001, 998 + 0 + 0 + 1 = 999.

Komachi square sum : 999² = 132² + 456² + 879²,
Komachi cube sum : 999² = 2³ + 5³ + 7³ + 13³ + 48³ + 96³.

(1³ + 2³ + ... + 215³)(216³ + 217³ + ... + 999³) = 11585851200²,
(1³ + 2³ + ... + 430³)(431³ + 432³ + ... + 444³)(445³ + 446³ + ... + 999³) = 1553721213371250².

999² = 998001, 998 + 0 + 0 + 1 = 998 + 0 * 0 + 1 = 999.

3-by-3 magic squares consisting of different squares with constant 999²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(14, 226, 973, 269, 938, 214, 962, 259, 74)	(14, 362, 931, 403, 854, 326, 914, 371, 158),
(15, 276, 960, 480, 840, 249, 876, 465, 120)	(18, 414, 909, 531, 774, 342, 846, 477, 234),
(19, 98, 994, 518, 851, 74, 854, 514, 67)	(22, 221, 974, 461, 862, 206, 886, 454, 83),
(29, 626, 778, 694, 547, 466, 718, 554, 419)	(34, 466, 883, 622, 701, 346, 781, 538, 314),
(39, 168, 984, 456, 879, 132, 888, 444, 111)	(48, 384, 921, 591, 732, 336, 804, 561, 192),
(50, 451, 890, 701, 650, 290, 710, 610, 349)	(62, 349, 934, 554, 766, 323, 829, 538, 146),
(67, 146, 986, 314, 941, 118, 946, 302, 109)	(74, 259, 962, 637, 754, 154, 766, 602, 221),
(74, 370, 925, 650, 685, 326, 755, 626, 190)	(74, 518, 851, 557, 686, 466, 826, 509, 238),
(77, 346, 934, 574, 781, 242, 814, 518, 259)	(94, 206, 973, 322, 931, 166, 941, 298, 154),
(94, 307, 946, 419, 874, 242, 902, 374, 211)	(98, 451, 886, 509, 742, 434, 854, 494, 157),
(98, 509, 854, 514, 826, 227, 851, 238, 466)	(109, 326, 938, 686, 707, 166, 718, 626, 301),
(111, 444, 888, 696, 672, 249, 708, 591, 384)	(115, 550, 826, 626, 605, 490, 770, 574, 275),
(158, 659, 734, 694, 454, 557, 701, 598, 386)	(202, 349, 914, 386, 886, 253, 899, 302, 314),
(204, 624, 753, 687, 456, 564, 696, 633, 336)	(206, 413, 886, 518, 814, 259, 829, 406, 382),
(206, 566, 797, 659, 682, 314, 722, 461, 514)	(214, 589, 778, 643, 514, 566, 734, 622, 269),
(227, 466, 854, 514, 802, 301, 826, 371, 422)

999² = 998001, 99 + 800 + 1 = 30²,
999² = 998001, 99 + 8001 = 90².

Page of Squares : First Upload March 20, 2006 ; Last Revised November 12, 2009
by Yoshio Mimura, Kobe, Japan

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