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).
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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.
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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.)
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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:
|
|
|
|
|
|
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.
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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.
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
by Yoshio Mimura, Kobe, Japan