870
The smallest squares containing k 870's :
87025 = 2952,
7387058704 = 859482,
1870870798702096 = 432535642.
8702 = (12 + 9)(72 + 9)(362 + 9).
(1 + 2 + 3 + 4)(5 + 6 + ... + 24)(25 + 26 + ... + 33) = 8702.
The integral triangle of sides 841, 1898, 2307 has square area 8702.
The 4-by-4 magic square consisting of different squares with constant 870:
|
Page of Squares : First Upload December 19, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan
871
The smallest squares containing k 871's :
1871424 = 13682,
1871687169 = 432632,
87146687187121 = 93352392.
8712 = 758641, a square with different digits.
Komach square sum : 8712 = 12 + 22 + 62 + 3582 + 7942.
The square root of 871 is 29. 5 1 27 0 9 1 2 ...,
and 292 = 52 + 12 + 272 + 02 + 92 + 12 + 22.
3-by-3 magic squares consisting of different squares with constant 8712:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (6, 93, 866, 429, 754, 78, 758, 426, 51), | (6, 349, 798, 594, 582, 259, 637, 546, 234), |
| (30, 435, 754, 579, 646, 78, 650, 390, 429), | (30, 579, 650, 610, 450, 429, 621, 470, 390), |
| (38, 126, 861, 294, 813, 106, 819, 286, 78), | (61, 198, 846, 558, 659, 114, 666, 534, 173), |
| (78, 286, 819, 531, 666, 182, 686, 483, 234), | (78, 429, 754, 579, 538, 366, 646, 534, 237), |
| (83, 366, 786, 474, 642, 349, 726, 461, 138), | (99, 182, 846, 258, 819, 146, 826, 234, 147), |
| (99, 362, 786, 506, 666, 243, 702, 429, 286), | (106, 474, 723, 579, 502, 414, 642, 531, 254), |
| (146, 285, 810, 390, 754, 195, 765, 330, 254), | (146, 378, 771, 477, 686, 246, 714, 381, 322), |
| (162, 414, 749, 546, 637, 234, 659, 426, 378) |
Page of Squares : First Upload December 19, 2005 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan
872
The smallest squares containing k 872's :
287296 = 5362,
1087218729 = 329732,
29898728728729 = 54679732.
8722 = 760384, a square with different digits.
Komachi square sums : 8722 = 12 + 72 + 92 + 5422 + 6832 = 12 + 22 + 42 + 52 + 932 + 8672.
8722 = 760384, 7 + 6 + 0 + 3 + 84 = 102.
8722 + 8732 + 8742 + ... + 26102 = 26112 + 26122 + 26132 + ... + 32682.
The square root of 872 is 29. 5 2 ..., and 29 = 52 + 22.
8722 = 760384 appears in the decimal expression of e:
e = 2.71828•••760384••• (from the 138743rd digit)
by Yoshio Mimura, Kobe, Japan
873
The smallest squares containing k 873's :
887364 = 9422,
56873187361 = 2384812,
8738787348736 = 29561442.
8732 = 124 + 184 + 184 + 274.
8732 + 8742 + 8752 + ... + 104982 = 6208772.
3-by-3 magic squares consisting of different squares with constant 8732:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (3, 258, 834, 354, 762, 237, 798, 339, 102), | (3, 294, 822, 426, 717, 258, 762, 402, 141), |
| (6, 318, 813, 498, 669, 258, 717, 462, 186), | (8, 223, 844, 556, 652, 167, 673, 536, 148), |
| (25, 148, 860, 580, 640, 127, 652, 575, 80), | (28, 364, 793, 416, 703, 308, 767, 368, 196), |
| (28, 383, 784, 487, 644, 332, 724, 448, 193), | (32, 329, 808, 368, 728, 311, 791, 352, 112), |
| (32, 412, 769, 449, 652, 368, 748, 409, 188), | (52, 217, 844, 256, 812, 193, 833, 236, 112), |
| (52, 295, 820, 545, 652, 200, 680, 500, 223), | (64, 412, 767, 577, 596, 272, 652, 487, 316), |
| (76, 193, 848, 452, 736, 127, 743, 428, 164), | (78, 402, 771, 573, 606, 258, 654, 483, 318), |
| (97, 232, 836, 388, 764, 167, 776, 353, 188), | (97, 316, 808, 388, 743, 244, 776, 332, 223), |
| (102, 435, 750, 525, 570, 402, 690, 498, 195), | (104, 272, 823, 472, 713, 176, 727, 424, 232), |
| (140, 548, 665, 577, 560, 340, 640, 385, 452), | (193, 496, 692, 596, 428, 473, 608, 577, 244), |
| (217, 508, 676, 548, 616, 287, 644, 353, 472), | (232, 479, 692, 508, 652, 281, 671, 328, 452) |
8732 = 762129, 7 + 6 + 2 + 12 + 9 = 62,
8732 = 762129, 7 + 62 + 1 + 2 + 9 = 92,
8732 = 762129, 7 + 6 + 2 + 129 = 122.
by Yoshio Mimura, Kobe, Japan
874
The smallest squares containing k 874's :
187489 = 4332,
38748741409 = 1968472,
187425874787449 = 136903572.
8742 is the 10th square which is the sum of 10 sixth powers.
8742 = 763876, 7 + 6 + 38 + 7 + 6 = 82,
8742 = 763876, 7 + 6 + 3 + 8 + 76 = 102,
8742 = 763876, 76 + 3 + 8 + 7 + 6 = 102,
8742 = 763876, 763 + 8 + 7 + 6 = 282,
8742 = 763876, 7 + 6387 + 6 = 802,
8742 = 763876, 762 + 382 + 762 = 1142.
by Yoshio Mimura, Kobe, Japan
875
The smallest squares containing k 875's :
538756 = 7342,
87587586304 = 2959522,
387568755538756 = 196867662.
8752 = 765625, 7 * 6 * 5 / 6 * 25 = 7 / 6 * 5 * 6 * 25 = 875.
8752 is the third square which is the sum of 3 fifth powers : 55 + 55 + 155.
8752 = 765625, 7 + 6 + 5 + 6 + 25 = 72.
(13 + 23 + ... + 3993)(4003 + 4013 + ... + 8753) = 299130300002.
3-by-3 magic squares consisting of different squares with constant 8752:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (0, 245, 840, 525, 672, 196, 700, 504, 147), | (2, 186, 855, 510, 695, 150, 711, 498, 110), |
| (9, 150, 862, 430, 750, 135, 762, 425, 66), | (9, 470, 738, 538, 585, 366, 690, 450, 295), |
| (25, 90, 870, 330, 807, 74, 810, 326, 57), | (25, 222, 846, 450, 729, 178, 750, 430, 135), |
| (25, 330, 810, 450, 690, 295, 750, 425, 150), | (25, 450, 750, 594, 542, 345, 642, 519, 290), |
| (30, 169, 858, 585, 642, 106, 650, 570, 135), | (30, 263, 834, 375, 750, 250, 790, 366, 87), |
| (30, 290, 825, 375, 750, 250, 790, 345, 150), | (30, 290, 825, 407, 726, 270, 774, 393, 110), |
| (30, 375, 790, 486, 650, 327, 727, 450, 186), | (34, 537, 690, 570, 510, 425, 663, 466, 330), |
| (54, 178, 855, 295, 810, 150, 822, 279, 110), | (74, 282, 825, 615, 570, 250, 618, 601, 150), |
| (90, 450, 745, 502, 585, 414, 711, 470, 198), | (90, 450, 745, 542, 615, 306, 681, 430, 342), |
| (102, 439, 750, 614, 498, 375, 615, 570, 250), | (135, 430, 750, 570, 615, 250, 650, 450, 375), |
| (142, 519, 690, 606, 558, 295, 615, 430, 450), | (150, 295, 810, 345, 774, 218, 790, 282, 249), |
| (150, 345, 790, 510, 682, 201, 695, 426, 318), | (150, 425, 750, 471, 678, 290, 722, 354, 345) |
Page of Squares : First Upload December 19, 2005 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan
876
The smallest squares containing k 876's :
15876 = 1262,
8765827876 = 936262,
87658765615876 = 93626262.
The squares which begin with 876 and end in 876 are
8765827876 = 936262, 87690607876 = 2961262, 876331887876 = 9361262,
876796267876 = 9363742, 8760854095876 = 29598742,...
8762 = 767376, a zigzag square with 3 kinds of digits.
Komachi equations:
8762 = 92 + 8762 - 542 / 32 / 22 */ 12 = 92 * 8762 / 542 * 32 * 22 */ 12
= - 92 + 8762 + 542 / 32 / 22 */ 12.
8762 = 767376, a square pegged by 7.
8762 = 23 + 603 + 823.
8762 = 203 + 155 + 17.
8762 = 767376, 7 + 6 + 7 + 3 + 7 + 6 = 62,
8762 = 767376, 76 + 73 + 76 = 152.
8762 = 767376 appears in the decimal expression of e:
e = 2.71828•••767376••• (from the 121440th digit).
by Yoshio Mimura, Kobe, Japan
877
The smallest squares containing k 877's :
877969 = 9372,
877877641 = 296292,
877877818774009 = 296290032.
(794 / 877)2 = 0.819675243... (Komachic).
8772 = 769129, 76 + 91 + 29 = 142.
1 / 877 = 0.0011402508551, 112 + 42 + 02 + 252 + 082 + 52 + 52+12 = 877,
1 / 877 = 0.0011402508551, 112 + 42 + 0252 + 02 + 82 + 52 + 52+12 = 877,
1 / 877 = 0.0011402508551, 112+42 + 0252 + 082 + 52 + 52 + 12 = 877.
3-by-3 magic squares consisting of different squares with constant 8772:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (4, 132, 867, 483, 724, 108, 732, 477, 76), | (24, 328, 813, 517, 636, 312, 708, 507, 104), |
| (24, 517, 708, 563, 552, 384, 672, 444, 347), | (27, 276, 832, 384, 752, 237, 788, 357, 144), |
| (27, 400, 780, 500, 648, 315, 720, 435, 248), | (32, 312, 819, 468, 699, 248, 741, 428, 192), |
| (32, 504, 717, 603, 508, 384, 636, 507, 328), | (45, 248, 840, 552, 645, 220, 680, 540, 123), |
| (48, 165, 860, 435, 752, 120, 760, 420, 123), | (48, 347, 804, 384, 732, 293, 787, 336, 192), |
| (59, 372, 792, 588, 571, 312, 648, 552, 211), | (77, 312, 816, 528, 636, 293, 696, 517, 132), |
| (99, 372, 788, 508, 669, 252, 708, 428, 291), | (123, 356, 792, 612, 603, 176, 616, 528, 333) |
Page of Squares : First Upload December 19, 2005 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan
878
The smallest squares containing k 878's :
278784 = 5282,
8785687824 = 937322,
54878878848784 = 74080282.
(701 / 878)2 = 0.637451289... (Komachic).
8782 = 770884, 7 + 70 + 8 + 84 = 132,
8782 = 770884, 7 + 70 + 88 + 4 = 132,
8782 = 770884, 77 + 0 + 8 + 84 = 132,
8782 = 770884, 77 + 0 + 88 + 4 = 132,
8782 = 770884, 772 + 02 + 882 + 42 = 1172,
8782 = 770884, 7 + 70 + 884 = 312,
8782 = 770884, 77 + 0 + 884 = 312,
8782 = 770884, 72 + 7082 + 842 = 7132.
8782 = 770884 appears in the decimal expression of π
π = 3.14159•••770884••• (from the 101624th digit).
by Yoshio Mimura, Kobe, Japan
879
The smallest squares containing k 879's :
879844 = 9382,
8793187984 = 937722,
879720879487921 = 296600892.
(518 / 879)2 = 0.347281596... (Komachic).
(614 / 879)2 = 0.487931652... (Komachic).
42502 = 8562 + 8572 + 8582 + ... + 8792.
3-by-3 magic squares consisting of different squares with constant 8792:
| A2 | B2 | C2 |
| D2 | E2 | F2 |
| G2 | H2 | K2 |
| where (A, B, C, D, E, F, G, H, K) = | |
| (2, 314, 821, 614, 587, 226, 629, 574, 218), | (9, 468, 744, 552, 576, 369, 684, 471, 288), |
| (22, 91, 874, 149, 862, 86, 866, 146, 37), | (22, 331, 814, 539, 638, 274, 694, 506, 187), |
| (26, 251, 842, 446, 722, 229, 757, 434, 106), | (26, 338, 811, 523, 646, 286, 706, 491, 182), |
| (46, 203, 854, 581, 634, 182, 658, 574, 101), | (46, 434, 763, 469, 658, 346, 742, 389, 266), |
| (50, 310, 821, 421, 730, 250, 770, 379, 190), | (58, 139, 866, 334, 806, 107, 811, 322, 106), |
| (70, 379, 790, 610, 590, 229, 629, 530, 310), | (86, 194, 853, 302, 811, 154, 821, 278, 146), |
| (86, 251, 838, 554, 658, 181, 677, 526, 194), | (86, 251, 838, 566, 658, 139, 667, 526, 226), |
| (86, 398, 779, 554, 581, 358, 677, 526, 194), | (91, 542, 686, 602, 539, 346, 634, 434, 427), |
| (96, 279, 828, 369, 768, 216, 792, 324, 201), | (107, 314, 814, 554, 658, 181, 674, 491, 278), |
| (118, 469, 734, 506, 566, 443, 709, 482, 194), | (134, 229, 838, 266, 818, 181, 827, 226, 194), |
| (134, 502, 709, 533, 614, 334, 686, 379, 398), | (146, 490, 715, 590, 475, 446, 635, 554, 250), |
| (154, 379, 778, 517, 674, 226, 694, 418, 341), | (187, 446, 734, 566, 629, 238, 646, 422, 421) |
8792 = 772641, 7 + 7 + 2 + 64 + 1 = 92,
8792 = 772641, 7 + 7 + 26 + 41 = 92,
8792 = 772641, 7 + 72 + 64 + 1 = 122,
8792 = 772641, 77 + 2 + 64 + 1 = 122,
8792 = 772641, 77 + 26 + 41 = 122.
8792 = 772641 appears in the decimal expression of e:
e = 2.71828•••772641••• (from the 74430th digit).
by Yoshio Mimura, Kobe, Japan