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960 - 969

960

The smallest squares containing k 960's :
9604 = 982,
96059601 = 98012,
96096064179600 = 98028602.

Komachi equation: 9602 = 92 * 82 * 72 * 62 * 52 * 42 / 32 / 212.

342 + 960 = 462, 342 - 960 = 142.

960k + 2080k + 2745k + 3240k are squares for k = 1,2,3 (952, 48252, 2541252).

(13 + 23 + ... + 5603)(5613 + 5623 + ... + 9593)(9603) = 20216422195200002,
(13 + 23 + ... + 4893)(4903 + 4913 + ... + 8153)(8163 + 8173 + ... + 9603) = 118807902520650002.

9602 = (32 - 1)(112 - 1)(312 - 1) = (32 - 1)(72 - 1)(492 - 1).

9602 = 183 + 443 + 943 = 163 + 323 + 963.

9602 = 921600 appears in the decimal expression of π:
  π = 3.14159•••921600••• (from the 88777th digit).

Page of Squares : First Upload February 27, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

961

the square of 31.

The smallest squares containing k 961's :
961 = 312,
9610076961 = 980312,
21961229610961 = 46862812.

The squares which begin with 961 and end in 961 are
9610076961 = 980312,   96119220961 = 3100312,   961319459961 = 9804692,
961441041961 = 9805312,   961809756961 = 9807192,...

961 is the sum of the consecutive primes : 3 + 5 + 7 + ... + 89 = 961.

403k + 961k + 5363k + 8649k are squares for k = 1,2,3 (1242, 102302, 8956522).

3-by-3 magic squares consisting of different squares with constant 9612:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(24, 263, 924, 468, 804, 241, 839, 456, 108)(25, 420, 864, 636, 655, 300, 720, 564, 295),
(36, 272, 921, 596, 729, 192, 753, 564, 196)(36, 519, 808, 556, 648, 441, 783, 484, 276),
(39, 288, 916, 592, 729, 204, 756, 556, 207)(52, 591, 756, 624, 596, 423, 729, 468, 416),
(56, 324, 903, 441, 812, 264, 852, 399, 196)(56, 417, 864, 528, 736, 321, 801, 456, 272),
(57, 456, 844, 564, 668, 399, 776, 519, 228)(88, 249, 924, 519, 792, 164, 804, 484, 207),
(129, 504, 808, 552, 704, 351, 776, 417, 384)(132, 276, 911, 384, 857, 204, 871, 336, 228),
(153, 484, 816, 636, 561, 452, 704, 612, 231)(168, 479, 816, 641, 552, 456, 696, 624, 223),
(196, 591, 732, 633, 636, 344, 696, 412, 519) 

9612 = 923521, 9 + 2 + 35 + 2 + 1 = 72,
9612 = 923521, 92 + 3 + 5 + 21 = 112,
9612 = 923521, 923 + 521 = 382.

Page of Squares : First Upload February 27, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

962

The smallest squares containing k 962's :
196249 = 4432,
20962696225 = 1447852,
229629622996225 = 151535352.

9622 is the tenth square which is the sum of 6 eighth powers : 38 + 38 + 48 + 48 + 58 + 58.

Komachi equation: 9622 = - 92 * 82 * 72 - 62 + 5432 * 22 + 102.

9622 = 925444, and 9 = 32, 25 = 52, 4 = 22.

9622 + 9632 + 9642 + ... + 116902 = 7295722,
9622 + 9632 + 9642 + ... + 133692 = 8923422.

9622 = (52 + 1)(62 + 1)(312 + 1).

(13 + 23 + ... + 2483)(2493 + 2503 + ... + 3323)(3333 + 3343 + ... + 9623) = 6510705492085202.

9622 = 925444, 9 + 2 + 5 + 4 + 44 = 82,
9622 = 925444, 9 + 2 + 5 + 44 + 4 = 82,
9622 = 925444, 925 + 444 = 372.

Page of Squares : First Upload February 27, 2006 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

963

The smallest squares containing k 963's :
649636 = 8062,
13996309636 = 1183062,
69630963963289 = 83445172.

172672 = 32 + 42 + 52 + ... + 9632.

9632 = 3212 + 6422 + 6422 : 2462 + 2462 + 1232 = 3692.

Komachi equation: 9632 = 983 + 73 - 63 - 53 - 43 * 33 * 23 - 13.

(12)(22)(32 + 42 + ... + 9632) = 345342.

(13 + 23 + ... + 1963)(1973 + 1983 + ... + 4273)(4283 + 4293 + ... + 9633) = 7847082243960962.

3-by-3 magic squares consisting of different squares with constant 9632:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 98, 958, 562, 778, 79, 782, 559, 58)(1, 218, 938, 278, 898, 209, 922, 271, 62),
(12, 309, 912, 516, 768, 267, 813, 492, 156)(12, 552, 789, 636, 597, 408, 723, 516, 372),
(14, 97, 958, 337, 898, 86, 902, 334, 47)(14, 458, 847, 658, 623, 326, 703, 574, 322),
(21, 168, 948, 372, 876, 147, 888, 363, 84)(22, 194, 943, 623, 722, 134, 734, 607, 142),
(34, 482, 833, 593, 646, 398, 758, 527, 274)(47, 274, 922, 506, 778, 257, 818, 497, 106),
(48, 267, 924, 651, 672, 228, 708, 636, 147)(54, 558, 783, 657, 594, 378, 702, 513, 414),
(62, 145, 950, 350, 890, 113, 895, 338, 110)(62, 257, 926, 673, 674, 142, 686, 638, 223),
(62, 454, 847, 506, 737, 358, 817, 422, 286)(62, 575, 770, 625, 562, 470, 730, 530, 337),
(65, 362, 890, 470, 790, 287, 838, 415, 230)(79, 422, 862, 502, 719, 398, 818, 482, 161),
(97, 518, 806, 586, 673, 362, 758, 454, 383)(110, 415, 862, 610, 638, 385, 737, 590, 190),
(113, 254, 922, 482, 817, 166, 826, 442, 223)(113, 502, 814, 658, 554, 433, 694, 607, 278),
(132, 429, 852, 597, 708, 264, 744, 492, 363)(142, 239, 922, 278, 902, 191, 911, 238, 202),
(142, 362, 881, 562, 751, 218, 769, 482, 322)(154, 458, 833, 623, 686, 262, 718, 497, 406),
(194, 383, 862, 433, 818, 266, 838, 334, 337)(202, 586, 737, 623, 482, 554, 706, 593, 278),
(238, 449, 818, 482, 782, 289, 799, 338, 418) 

9632 = 927369, 9 + 2 + 7 + 3 + 6 + 9 = 62,
9632 = 927369, 9 + 27 + 36 + 9 = 92,
9632 = 927369, 92 + 7 + 36 + 9 = 122,
9632 = 927369, 927 + 369 = 362.

Page of Squares : First Upload February 27, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

964

The smallest squares containing k 964's :
24964 = 1582,
764964964 = 276582,
52964305964964 = 72776582.

The squares which begin with 964 and end in 964 are
964013712964 = 9818422,   964634336964 = 9821582,   964995804964 = 9823422,
9640043844964 = 31048422,   9642006204964 = 31051582,...

9642 = 929296, a zigzag square with 3 kinds of digits.

9642 = 929296, a square pegged by 9.

9642 = 929296 : 929 + 29 + 6 = 964.

9642 + 9652 + 9662 + ... + 28122 = 843662.

9642 = 313 + 413 + 943.

9642 = 929296, 929 + 296 = 352.

9642 = 929296 appears in the decimal expression of π:
  π = 3.14159•••929296••• (from the 14494th digit),
  (929296 is the eighth 6-digit square in the expression of π.)

Page of Squares : First Upload February 27, 2006 ; Last Revised October 16, 2006
by Yoshio Mimura, Kobe, Japan

965

The smallest squares containing k 965's :
1965604 = 14022,
19659965796 = 1402142,
965409650965764 = 310710422.

9652 is the 10th square which is the sum of 8 sixth powers : 36, 46, 66, 66, 66, 86, 86, 86.

9652 + 9662 + 9672 + ... + 2176362 = 586188042,
9652 + 9662 + 9672 + ... + 4189012 = 1565334812.

3-by-3 magic squares consisting of different squares with constant 9652:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 475, 840, 579, 672, 380, 772, 504, 285)(11, 180, 948, 348, 885, 164, 900, 340, 75),
(12, 291, 920, 515, 780, 240, 816, 488, 165)(16, 315, 912, 588, 720, 259, 765, 560, 180),
(60, 227, 936, 285, 900, 200, 920, 264, 123)(60, 285, 920, 480, 808, 219, 835, 444, 192),
(60, 324, 907, 380, 843, 276, 885, 340, 180)(60, 380, 885, 565, 732, 276, 780, 501, 268),
(60, 480, 835, 565, 660, 420, 780, 515, 240)(75, 340, 900, 660, 675, 200, 700, 600, 285),
(75, 516, 812, 660, 565, 420, 700, 588, 309)(84, 387, 880, 637, 684, 240, 720, 560, 315),
(88, 360, 891, 675, 660, 200, 684, 605, 312)(108, 381, 880, 605, 660, 360, 744, 592, 165),
(133, 456, 840, 600, 700, 285, 744, 483, 380)(189, 348, 880, 420, 835, 240, 848, 336, 315),
(192, 556, 765, 600, 675, 340, 731, 408, 480)(200, 549, 768, 660, 480, 515, 675, 632, 276)

9652 = 931225, 9 + 3 + 12 + 25 = 72,
9652 = 931225, 9 + 31 + 2 + 2 + 5 = 72,
9652 = 931225, 9 + 3122 + 5 = 562,
9652 = 931225, 93 + 1 + 2 + 25 = 112,
9652 = 931225, 93 + 1 + 22 + 5 = 112,
9652 = 931225, 931 + 225 = 342.

Page of Squares : First Upload February 27, 2006 ; Last Revised November 5, 2009
by Yoshio Mimura, Kobe, Japan

966

The smallest squares containing k 966's :
369664 = 6082,
11966609664 = 1093922,
1396685966996644 = 373722622.

9662 + 9672 + 9682 + ... + 30302 = 30312 + 30322 + 30332 + ... + 37972.

9662 = 1642 + 4042 + 8622 : 2682 + 4042 + 4612 = 6692,
9662 = 3222 + 6442 + 6442 : 4462 + 4462 + 2232 = 6692.

9662 = 933156, a square with odd digits except the last digit 6.

9662 = (32 + 5)(82 + 5)(312 + 5).

Komachi square sum : 9662 = 52 + 292 + 4372 + 8612.

204k + 294k + 561k + 966k are squares for k = 1,2,3 (452, 11732, 333452).

9662 + 9672 + 9682 + ... + 13412 = 224662,
9662 + 9672 + 9682 + ... + 3282382 = 1085734522,
9662 + 9672 + 9682 + ... + 7593572 = 3820399702.

(13 + 23 + ... + 413)(423 + 433 + ... + 693)(703 + 713 + ... + 9663) = 9073365472082.

The 4-by-4 magic squares consisting of different squares with constant 966:

02 12172262
22 92252162
112282 62 52
292102 42 32
     
12 42182252
62292 52 82
202 32192142
232102162 92
     
12 82152262
92102232162
202192142 32
222212 42 52
     
12 82152262
102272 42112
172 22232122
242132142 52
12 92202222
102 82192212
172252 62 42
242142132 52
     
12 92202222
102162132212
172252 62 42
242 22192 52

9662 = 933156, 93 + 33 + 33 + 13 + 563 = 4202,
9662 = 933156, 9 + 3 + 3 + 15 + 6 = 62,
9662 = 933156, 933 + 156 = 332,
9662 = 933156, 93 + 3156 = 572.

Page of Squares : First Upload February 27, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

967

The smallest squares containing k 967's :
96721 = 3112,
159672967281 = 3995912,
1196796796752001 = 345947512.

1042 + 1052 + 1062 + ... + 9672 = 173642.

9672 + 9682 + 9692 + ... + 502642 = 65062372.

9672 = 4462 + 4622 + 7232 : 3272 + 2642 + 6442 = 7692.

3-by-3 magic squares consisting of different squares with constant 9672:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 158, 954, 338, 894, 147, 906, 333, 58)(3, 446, 858, 522, 723, 374, 814, 462, 243),
(14, 138, 957, 642, 717, 94, 723, 634, 102)(18, 157, 954, 563, 774, 138, 786, 558, 77),
(18, 222, 941, 509, 798, 198, 822, 499, 102)(18, 291, 922, 669, 662, 222, 698, 642, 189),
(30, 445, 858, 630, 642, 355, 733, 570, 270)(42, 310, 915, 590, 717, 270, 765, 570, 158),
(51, 402, 878, 618, 662, 339, 742, 579, 222)(58, 525, 810, 675, 558, 410, 690, 590, 333),
(66, 402, 877, 558, 733, 294, 787, 486, 282)(126, 618, 733, 653, 486, 522, 702, 563, 354),
(138, 381, 878, 418, 822, 291, 861, 338, 282)(234, 598, 723, 627, 654, 338, 698, 387, 546)

9672 = 935089, 9 + 3 + 508 + 9 = 232,
9672 = 935089, 935 + 0 + 89 = 322.

Page of Squares : First Upload February 27, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

968

The smallest squares containing k 968's :
459684 = 6782,
2596819681 = 509592,
96899689687696 = 98437642.

9682 = 937024, a square with different digits.

(13 + 23 + ... + 6663)(6673 + 6683 + ... + 7593)(7603 + 7613 + ... + 9683) = 151134849369614882.

9682 = 483 + 683 + 803 = 224 + 224 + 224 + 224.

9682 = (22 + 7)(92 + 7)(312 + 7).

9682 = 937024, 9 + 3 + 7 + 0 + 2 + 4 = 52,
9682 = 937024, 93 + 70 + 2 + 4 = 132,
9682 = 937024, 937 + 0 + 24 = 312.

Page of Squares : First Upload February 27, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

969

The smallest squares containing k 969's :
3969 = 632,
1996927969 = 446872,
96949879693969 = 98463132.

The squares which begin with 969 and end in 969 are
96915783969 = 3113132,   96993004969 = 3114372,   969116206969 = 9844372,
969364300969 = 9845632,   969608487969 = 9846872,...

9692 = (32 + 8)(2352 + 8) = (32 + 8)(72 + 8)(312 + 8).

Komachi equation: 9692 = - 16 + 26 + 36 - 46 - 56 + 66 + 76 + 86 + 96.

969k + 5016k + 5092k + 6612k are squares for k = 1,2,3 (1332, 97852, 7404112).

9692 + 9702 + 9712 + ... + 193302 = 15515892.

(13 + 23 + ... + 5703)(5713 + 5723 + ... + 9503)(9513 + 9523 + ... + 9693) = 88914796490520002.

3-by-3 magic squares consisting of different squares with constant 9692:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(7, 316, 916, 476, 799, 272, 844, 448, 161)(7, 596, 764, 644, 568, 449, 724, 511, 392),
(16, 97, 964, 548, 796, 71, 799, 544, 68)(28, 136, 959, 484, 833, 104, 839, 476, 92),
(30, 531, 810, 594, 630, 435, 765, 510, 306)(41, 176, 952, 568, 776, 119, 784, 553, 136),
(44, 272, 929, 313, 884, 244, 916, 289, 128)(44, 545, 800, 580, 656, 415, 775, 460, 356),
(51, 162, 954, 306, 909, 138, 918, 294, 99)(51, 306, 918, 666, 678, 189, 702, 621, 246),
(64, 169, 952, 343, 896, 136, 904, 328, 119)(64, 268, 929, 436, 839, 212, 863, 404, 176),
(68, 391, 884, 449, 772, 376, 856, 436, 127)(68, 391, 884, 544, 748, 289, 799, 476, 272),
(68, 544, 799, 596, 607, 464, 761, 524, 292)(71, 308, 916, 524, 784, 223, 812, 479, 224),
(97, 196, 944, 356, 889, 148, 896, 332, 161)(99, 462, 846, 558, 666, 429, 786, 531, 198),
(104, 376, 887, 649, 632, 344, 712, 631, 184)(121, 236, 932, 292, 904, 191, 916, 257, 184),
(121, 316, 908, 604, 737, 176, 748, 544, 289)(124, 377, 884, 481, 796, 272, 832, 404, 289),
(124, 583, 764, 671, 604, 352, 688, 484, 481)(136, 476, 833, 511, 748, 344, 812, 391, 356),
(140, 344, 895, 455, 820, 244, 844, 385, 280)(160, 425, 856, 656, 680, 215, 695, 544, 400),
(191, 412, 856, 548, 761, 244, 776, 436, 383)(236, 569, 748, 656, 652, 289, 673, 436, 544)

9692 = 938961, 9 + 3 + 8 + 9 + 6 + 1 = 62,
9692 = 938961, 9 + 38 + 96 + 1 = 122.

Page of Squares : First Upload February 27, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan