Squares:980s

980

The smallest squares containing k 980's :
9801 = 99²,
98029801 = 9901²,
98008039809801 = 9899901².

Komachi equations:
980² = 9² * 8² * 7² / 6² / 54² * 3² * 210² = 98² / 7² * 6² / 54² * 3² * 210²,
980² = 9³ * 8³ + 76³ + 54³ - 3³ - 21³.

980 = (1² + 2² + 3² + ... + 24²) / (1² + 2²).

980² = 23³ + 49³ + 94³ = 49³ + 63³ + 84³.

980² = 1⁵ + 4⁵ + 10⁵ + 10⁵ + 15⁵.

Page of Squares : First Upload March 13, 2006 ; Last Revised July 13, 2010
by Yoshio Mimura, Kobe, Japan

981

The smallest squares containing k 981's :
298116 = 546²,
9819819025 = 99095²,
981981998174464 = 31336592².

981 = (1² + 2² + 3² + ... + 54²) / (1² + 2² + 3² + 4² + 5²).

1² + 2² + 3² + ... + 981² = 315173391, the second sum with odd digits,
(there are 2 such sums in all. Cf. 693)

1² + 2² + 3² + 4² + ... + 981² = 315173391, which consists of odd digits (the second 9-digit, cf. 693).

981² = 962361 is exchangeable : 236196 = 486².

981² = 962361, 962 + 3 * 6 + 1 = 981.

3-by-3 magic squares consisting of different squares with constant 981²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 256, 947, 488, 821, 224, 851, 472, 124)	(4, 388, 901, 604, 709, 308, 773, 556, 236),
(5, 500, 844, 644, 635, 380, 740, 556, 325)	(16, 371, 908, 683, 656, 256, 704, 628, 269),
(19, 236, 952, 556, 787, 184, 808, 536, 149)	(19, 280, 940, 340, 880, 269, 920, 331, 80),
(19, 532, 824, 584, 656, 437, 788, 499, 304)	(32, 136, 971, 179, 956, 128, 964, 173, 56),
(32, 499, 844, 536, 716, 403, 821, 448, 296)	(39, 402, 894, 534, 759, 318, 822, 474, 249),
(44, 397, 896, 592, 704, 341, 781, 556, 208)	(44, 584, 787, 632, 619, 424, 749, 488, 404),
(52, 571, 796, 676, 556, 443, 709, 572, 364)	(56, 149, 968, 436, 872, 109, 877, 424, 116),
(56, 275, 940, 565, 760, 256, 800, 556, 115)	(56, 440, 875, 685, 644, 280, 700, 595, 344),
(59, 172, 964, 404, 884, 133, 892, 389, 124)	(59, 212, 956, 668, 709, 116, 716, 644, 187),
(72, 459, 864, 621, 648, 396, 756, 576, 243)	(105, 294, 930, 390, 870, 231, 894, 345, 210),
(109, 436, 872, 488, 784, 331, 844, 397, 304)	(114, 249, 942, 318, 906, 201, 921, 282, 186),
(114, 366, 903, 546, 777, 246, 807, 474, 294)	(136, 557, 796, 661, 536, 488, 712, 604, 301),
(148, 371, 896, 616, 736, 203, 749, 532, 344)	(179, 604, 752, 676, 467, 536, 688, 616, 331),
(201, 534, 798, 642, 681, 294, 714, 462, 489)

The 4-by-4 magic square consisting of different squares with constant 981:

0²	7²	16²	26²
9²	28²	4²	10²
18²	2²	22²	13²
24²	12²	15²	6²

981² = 962361, 9 + 6 + 2 + 3 + 61 = 9²,
981² = 962361, 9 + 62 + 3 + 6 + 1 = 9².

Page of Squares : First Upload March 13, 2006 ; Last Revised November 12, 2009
by Yoshio Mimura, Kobe, Japan

982

The smallest squares containing k 982's :
589824 = 768²,
2982798225 = 54615²,
2973982459829824 = 54534232².

982² = 964324, with 9 = 3², 64 = 8², 324 = 18².

Komachi square sum : 982² = 2² + 5² + 6² + 7² + 43² + 981².

982² = 964324, 9 + 6 + 43 + 2 + 4 = 8²,
982² = 964324, 9 + 64 + 3 + 24 = 10²,
982² = 964324, 9 + 643 + 24 = 26².

Page of Squares : First Upload March 13, 2006 ; Last Revised October 19, 2006
by Yoshio Mimura, Kobe, Japan

983

The smallest squares containing k 983's :
1098304 = 1048²,
29835998361 = 172731²,
7983298303983025 = 89349305².

1² + 2² + 3² + ... + 993² = 326875409, the third 9-digit sum with different digits (see 982).

983 is the third integer which is the sum of a square and a prime in 12 ways :
4² + 967, 6² + 947, 8² + 919, 10² + 883, 12² + 839, 14² + 787, 16² + 727, 18² + 659, 22² + 499, 26² + 307, 28² + 199, 30² + 83.

(661 / 983)² = 0.452163897... (Komachic).

983² = 99² + 462² + 862² : 268² + 264² + 99² = 389².

983² = 6⁴ + 12⁴ + 12⁴ + 31⁴ = 4⁴ + 8⁴ + 14⁴ + 31⁴ = 40³ + 67³ + 67³ + 67³.

3-by-3 magic squares consisting of different squares with constant 983²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 99, 978, 621, 758, 78, 762, 618, 61)	(2, 237, 954, 666, 702, 173, 723, 646, 162),
(3, 182, 966, 434, 867, 162, 882, 426, 83)	(3, 294, 938, 574, 762, 237, 798, 547, 174),
(30, 317, 930, 358, 870, 285, 915, 330, 142)	(34, 93, 978, 222, 954, 83, 957, 218, 54),
(34, 222, 957, 573, 786, 142, 798, 547, 174)	(34, 462, 867, 573, 714, 358, 798, 493, 294),
(42, 291, 938, 349, 882, 258, 918, 322, 141)	(42, 330, 925, 525, 790, 258, 830, 483, 210),
(93, 398, 894, 614, 723, 258, 762, 534, 317)	(110, 342, 915, 450, 835, 258, 867, 390, 250),
(126, 538, 813, 637, 666, 342, 738, 483, 434)	(142, 534, 813, 573, 618, 506, 786, 547, 222),
(182, 438, 861, 582, 749, 258, 771, 462, 398)	(218, 498, 819, 531, 762, 322, 798, 371, 438),
(243, 538, 786, 582, 726, 317, 754, 387, 498)

983² = 966289, 96 + 6 + 2 + 8 + 9 = 11².

Page of Squares : First Upload March 13, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

984

The smallest squares containing k 984's :
51984 = 228²,
4629849849 = 68043²,
69843259843984 = 8357228².

The squares which begin with 984 and end in 984 are
9846195984 = 99228², 98425257984 = 313728², 98452867984 = 313772²,
984516403984 = 992228², 984603721984 = 992272²,...

Komachi equation: 984² = 98³ + 7³ + 6³ * 5³ - 4³ - 3³ * 2³ + 1³.

984² = 968256, 968 + 2 * 5 + 6 = 984.

984² = 4³ + 30³ + 98³.

984² = 968256, 9 + 6 + 8 + 2 + 5 + 6 = 6²,
984² = 968256, 9 + 6 + 8 + 2 + 56 = 9².

Page of Squares : First Upload March 13, 2006 ; Last Revised July 13, 2010
by Yoshio Mimura, Kobe, Japan

985

The smallest squares containing k 985's :
98596 = 314²,
3059859856 = 55316²,
249859857985936 = 15806956².

985² = 696² + 697².

126080^k + 250190^k + 262010^k + 331945^k are squares for k = 1,2,3 (985², 507275², 268752325²).

985² = 970225, and 9 = 3², 70225 = 265².

3-by-3 magic squares consisting of different squares with constant 985²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 140, 975, 591, 780, 112, 788, 585, 84)	(15, 220, 960, 400, 876, 207, 900, 393, 76),
(15, 220, 960, 480, 840, 185, 860, 465, 120)	(15, 400, 900, 480, 783, 356, 860, 444, 183),
(15, 480, 860, 636, 652, 375, 752, 561, 300)	(32, 276, 945, 399, 868, 240, 900, 375, 140),
(36, 265, 948, 552, 780, 239, 815, 540, 120)	(48, 536, 825, 680, 615, 360, 711, 552, 400),
(81, 392, 900, 508, 756, 375, 840, 495, 140)	(81, 580, 792, 680, 540, 465, 708, 585, 356),
(113, 384, 900, 540, 780, 265, 816, 463, 300)	(120, 428, 879, 465, 804, 328, 860, 375, 300),
(120, 465, 860, 657, 680, 276, 724, 540, 393)	(120, 540, 815, 615, 680, 360, 760, 465, 420),
(140, 495, 840, 612, 616, 465, 759, 588, 220)	(175, 360, 900, 504, 815, 228, 828, 420, 329),
(185, 480, 840, 672, 679, 240, 696, 528, 455)	(220, 465, 840, 564, 760, 273, 777, 420, 436)

985² = 970225, 9 + 7 + 0 + 2 + 2 + 5 = 5².

985² = 970225 appears in the decimal expression of e:
e = 2.71828•••970225••• (from the 13050th digit),
(970225 is the tenth 6-digit square in the expression of e.)

Page of Squares : First Upload March 13, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

986

The smallest squares containing k 986's :
986049 = 993²,
2698698601 = 51949²,
198698681986704 = 14096052².

986² = 972196, 972 - 1 + 9 + 6 = 986.

986² = (5² + 9)(7² + 9)(22² + 9).

986² = 102² + 464² + 864² : 468² + 464² + 201² = 689².

986² = 972196, 9² + 72² + 196² = 209²,
986² = 972196, 97 + 2 + 1 + 96 = 14².

Page of Squares : First Upload March 13, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

987

The smallest squares containing k 987's :
139876 = 374²,
19879872016 = 140996²,
987798761298756 = 31429266².

987 = (1² + 2² + 3² + ... + 94²) / (1² + 2² + 3² + ... + 9²).

(391 / 987)² = 0.156934782... (Komachic).

Komachi equations:
987² = 987² + 6² - 5² - 4² + 3² - 2² */ 1² = 987² - 6² + 5² + 4² - 3² + 2² */ 1².

987² + 988² + 989² + ... + 1947² = 46283².

987² = 42³ + 58³ + 89³.

3-by-3 magic squares consisting of different squares with constant 987²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 171, 972, 477, 852, 144, 864, 468, 93)	(12, 405, 900, 675, 660, 288, 720, 612, 285),
(22, 409, 898, 521, 758, 358, 838, 482, 199)	(23, 158, 974, 622, 754, 137, 766, 617, 82),
(28, 259, 952, 413, 868, 224, 896, 392, 133)	(28, 364, 917, 469, 812, 308, 868, 427, 196),
(38, 194, 967, 254, 937, 178, 953, 242, 86)	(38, 358, 919, 631, 698, 298, 758, 599, 202),
(38, 425, 890, 610, 710, 313, 775, 538, 290)	(38, 551, 818, 601, 662, 418, 782, 482, 361),
(46, 233, 958, 407, 878, 194, 898, 386, 137)	(55, 262, 950, 662, 695, 230, 730, 650, 137),
(62, 313, 934, 583, 766, 218, 794, 538, 233)	(62, 562, 809, 599, 622, 478, 782, 521, 302),
(73, 142, 974, 226, 953, 122, 958, 214, 103)	(73, 394, 902, 694, 662, 233, 698, 617, 326),
(86, 298, 937, 662, 713, 166, 727, 614, 262)	(86, 482, 857, 538, 697, 446, 823, 506, 202),
(89, 202, 962, 442, 871, 142, 878, 418, 169)	(99, 528, 828, 612, 684, 363, 768, 477, 396),
(103, 262, 946, 566, 793, 158, 802, 526, 233)	(103, 302, 934, 358, 886, 247, 914, 313, 202),
(103, 638, 746, 682, 586, 407, 706, 473, 502)	(106, 478, 857, 578, 727, 334, 793, 466, 358),
(108, 579, 792, 636, 648, 387, 747, 468, 444)	(121, 382, 902, 662, 638, 359, 722, 649, 178),
(122, 391, 898, 502, 802, 281, 841, 422, 298)	(130, 505, 838, 670, 662, 295, 713, 530, 430),
(142, 526, 823, 634, 583, 482, 743, 598, 254)	(167, 298, 926, 374, 887, 218, 898, 314, 263),
(167, 494, 838, 562, 743, 326, 794, 422, 407)	(185, 562, 790, 638, 535, 530, 730, 610, 263)

The 4-by-4 magic square consisting of different squares with constant 987:

0²	1²	19²	25²
5²	23²	17²	12²
11²	21²	16²	13²
29²	4²	9²	7²

987² = 974169, 9 + 7 + 4 + 1 + 6 + 9 = 6²,
987² = 974169, 9 + 7 + 416 + 9 = 21².

987² = 974169 appears in the decimal expression of π:
π = 3.14159•••974169••• (from the 64719th digit).

Page of Squares : First Upload March 13, 2006 ; Last Revised July 13, 2010
by Yoshio Mimura, Kobe, Japan

988

The smallest squares containing k 988's :
19881 = 141²,
9883739889 = 99417²,
256498898829889 = 16015583².

988 = (1² + 2² + 3² + ... + 104²) / (1² + 2² + 3² + ... + 10²).

988² = (7² + 3)(137² + 3).

988² = 976144, 9³ + 7³ + 6³ + 14³ + 4³ = 64²,
988² = 976144, 9 + 7 + 61 + 44 = 11²,
988² = 976144, 97 + 6 + 14 + 4 = 11²,
988² = 976144, 97 + 6144 = 79².

Page of Squares : First Upload March 13, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

989

The smallest squares containing k 989's :
198916 = 446²,
19899898489 = 141067²,
69899899899456 = 8360616².

989² = 978121, a zigzag square.

989² = 978121, 978 + 12-1 = 989.

3-by-3 magic squares consisting of different squares with constant 989²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(8, 99, 984, 459, 872, 84, 876, 456, 53)	(24, 181, 972, 224, 948, 171, 963, 216, 64),
(24, 636, 757, 669, 568, 456, 728, 501, 444)	(36, 179, 972, 627, 756, 116, 764, 612, 141),
(36, 235, 960, 440, 864, 195, 885, 420, 136)	(36, 388, 909, 444, 819, 332, 883, 396, 204),
(48, 291, 944, 564, 784, 213, 811, 528, 204)	(60, 525, 836, 636, 620, 435, 755, 564, 300),
(64, 405, 900, 675, 640, 336, 720, 636, 235)	(84, 269, 948, 676, 708, 141, 717, 636, 244),
(99, 424, 888, 584, 693, 396, 792, 564, 181)	(116, 288, 939, 504, 829, 192, 843, 456, 244),
(116, 333, 924, 372, 876, 269, 909, 316, 228)	(136, 456, 867, 651, 612, 424, 732, 629, 216),
(168, 451, 864, 501, 792, 316, 836, 384, 363)	(171, 532, 816, 624, 696, 323, 748, 459, 456)

989² = 978121, 9 + 7 + 81 + 2 + 1 = 10²,
989² = 978121, 9 + 78 + 12 + 1 = 10².

Page of Squares : First Upload March 13, 2006 ; Last Revised November 12, 2009
by Yoshio Mimura, Kobe, Japan