660
The smallest squares containing k 660's :
66049 = 2572,
866066041 = 294292,
166006603766025 = 128843552.
(1 + 2 + ... + 10)(11 + 12 + ... + 21)(22 + 23) = 6602.
6602 = 435600, 4 * 3 * 5 + 600 = 660.
42k + 129k + 660k + 1194k are squares for k = 1,2,3 (452, 13712, 446312).
The integral triangle of sides 1000, 2057, 2993 has square area 6602.
Page of Squares : First Upload July 25, 2005 ; Last Revised September 30, 2011by Yoshio Mimura, Kobe, Japan
661
The smallest squares containing k 661's :
1661521 = 12892,
6661661161 = 816192,
46618661661796 = 68277862.
(235 / 661)2 = 0.126395847... (Komachic).
6612 = 436921, a square with different digits.
6612 = 436921 is exchangeable, 214369 = 4632.
(42 - 7)(52 - 7)(62 - 7)(102 - 7) = 6612 - 7.
1 / 661 = 0.00151285930408472012102874432677...,
the sum of the squares of the digits is 661.
217k + 218k + 272k + 518k are squares for k = 1,2,3 (352, 6612, 134052).
3-by-3 magic squares consisting of different squares with constant 6612:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(24, 112, 651 147, 636, 104 644, 141, 48), | (24, 301, 588, 427, 456, 216, 504, 372, 211), |
(36, 237, 616, 392, 504, 171, 531, 356, 168), | (51, 212, 624, 464, 456, 117, 468, 429, 184), |
(69, 324, 572, 436, 453, 204, 492, 356, 261), | (77, 204, 624, 336, 552, 139, 564, 301, 168), |
(141, 292, 576, 348, 531, 184, 544, 264, 267) |
6612 = 436921, 4 + 3 + 6 + 9 + 2 + 1 = 52,
6612 = 436921, 436 + 92 + 1 = 232,
6612 = 436921, 4 + 36 + 921 = 312.
6612 = 436921 appears in the decimal expression of e:
e = 2.71828•••436921••• (from the 81584th digit).
by Yoshio Mimura, Kobe, Japan
662
The smallest squares containing k 662's :
186624 = 4322,
46626628624 = 2159322,
356625662166225 = 188845352.
6622 = 438244, 4 + 3 + 8 + 2 + 4 + 4 = 52.
6622 = 438244, 43 + 82 + 44 = 132,
6622 = 438244, 438 + 2 + 44 = 222.
6622 = 438244 is the third mosaic square, 484 = 222 and 324 = 182.
Page of Squares : First Upload July 25, 2005 ; Last Revised August 21, 2006by Yoshio Mimura, Kobe, Japan
663
The smallest squares containing k 663's :
2663424 = 16322,
36631663236 = 1913942,
966366316663489 = 310864332.
6632 = 439569, 4 / 3 * 9 * 56 - 9 = 663.
67626k + 78897k + 100776k + 192270k are squares for k = 1,2,3 (6632, 2406692, 945073352).
6632 = 439569, 4 + 3 + 9 + 5 + 6 + 9 = 62.
6632 = 2212 + 4422 + 4422 : 2442 + 2442 + 1222 = 3662.
3-by-3 magic squares consisting of different squares with constant 6632:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 255, 612, 312, 540, 225, 585, 288, 12), | (13, 130, 650, 370, 538, 115, 550, 365, 62), |
(13, 218, 626, 286, 563, 202, 598, 274, 83), | (22, 374, 547, 403, 442, 286, 526, 323, 242), |
(26, 77, 658, 182, 634, 67, 637, 178, 46), | (26, 182, 637, 413, 494, 158, 518, 403, 94), |
(34, 158, 643, 323, 566, 122, 578, 307, 106), | (34, 227, 622, 323, 538, 214, 578, 314, 83), |
(34, 358, 557, 442, 403, 286, 493, 386, 218), | (38, 290, 595, 445, 430, 238, 490, 413, 170), |
(67, 358, 554, 398, 466, 253, 526, 307, 262), | (98, 317, 574, 346, 518, 227, 557, 266, 242), |
(108, 351, 552, 417, 468, 216, 504, 312, 297), | (122, 259, 598, 301, 562, 182, 578, 238, 221) |
6632 = 439569, 4 + 3 + 9 + 56 + 9 = 92,
6632 = 439569, 4 + 3956 + 9 = 632.
by Yoshio Mimura, Kobe, Japan
664
The smallest squares containing k 664's :
11664 = 1082,
1664966416 = 408042,
6646640859664 = 25781082.
The squares which begin with 664 and end in 664 are
664048971664 = 8148922, 664401051664 = 8151082, 664864113664 = 8153922,
6640372379664 = 25768922, 6641485643664 = 25771082,...
Komachi Fraction : 450 / 7936128 = (5 / 664)2.
(13 + 23 + ... + 2153)(2163 + 2173 + ... + 4553)(4563 + 4573 + ... + 6643) = 4575467191680002.
The square root of 664 is 25.7 6 8 19 7 4 5 3 4...,
where 252 = 72 + 62 + 82 + 192 + 72 + 42 + 52 + 32 + 42.
by Yoshio Mimura, Kobe, Japan
665
The smallest squares containing k 665's :
46656 = 2162,
42966656656 = 2072842,
121665665006656 = 110302162.
6652 = 442225, a square consisting of only 3 kinds of digits.
Kaprekar : 6652 = 442225 : 42 + 42 + 22 + 22 + 252
665k + 1421k + 2401k + 5117k are squares for k = 1,2,3 (982, 58662, 3885702).
230k + 610k + 665k + 2720k are squares for k = 1,2,3 (652, 28752, 1437252).
3-by-3 magic squares consisting of different squares with constant 6652:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(8, 81, 660, 444, 492, 55, 495, 440, 60), | (9, 260, 612, 388, 495, 216, 540, 360, 145), |
(12, 316, 585, 360, 495, 260, 559, 312, 180), | (15, 80, 660, 180, 636, 73, 640, 177, 36), |
(15, 80, 660, 352, 561, 60, 564, 348, 55), | (15, 180, 640, 460, 465, 120, 480, 440, 135), |
(15, 240, 620, 332, 540, 201, 576, 305, 132), | (15, 240, 620, 460, 444, 183, 480, 433, 156), |
(15, 404, 528, 460, 375, 300, 480, 372, 271), | (24, 345, 568, 432, 440, 249, 505, 360, 240), |
(35, 210, 630, 378, 525, 154, 546, 350, 147), | (36, 145, 648, 375, 540, 100, 548, 360, 111), |
(80, 240, 615, 408, 505, 144, 519, 360, 208), | (100, 207, 624, 300, 576, 143, 585, 260, 180), |
(100, 288, 591, 375, 516, 188, 540, 305, 240), | (120, 431, 492, 460, 300, 375, 465, 408, 244) |
6652 + 6662 + 6672 + ... + 10162 = 158842,
6652 + 6662 + 6672 + ... + 4082582 = 1506059832.
(13 + 23 + ... + 2143)(2153 + 2163 + ... + 2583)(2593 + 2603 + ... + 6653) = 1220199709713602.
Page of Squares : First Upload July 25, 2005 ; Last Revised March 17, 2011by Yoshio Mimura, Kobe, Japan
666
The smallest squares containing k 666's :
1666681 = 12912,
27666666889 = 1663332,
16666476665466681 = 1290987092.
666 = 22 + 32 + 52 + 72 + 112 + 132 + 172 (the sum of the squares of consecutive 7 primes).
6662 = 3! + 3! + 4! + 8! + 8! + 9!
6662± 5 are primes.
6662 + 6672 + 6682 + ... + 6842 = 6852 + 6862 + 6872 + ... + 7022.
21k + 45k + 201k + 633k are squares for k = 1,2,3 (302, 6662, 161822).
The 4-by-4 magic squares consisting of different squares with constant 666:
|
|
|
6662 = 443556, 44 + 3556 = 602.
6662 = 443556 appears in the decimal expression of e:
e = 2.71828•••443556••• (from the 105277th digit).
by Yoshio Mimura, Kobe, Japan
667
The smallest squares containing k 667's :
276676 = 5262,
16670166769 = 1291132,
95316676676676 = 97630262.
6672 = 444889, a square with 3 kinds of digits and non-descending sequence of digits.
667k + 2507k + 2553k + 2737k are squares for k = 1,2,3 (922, 45542, 2306442).
113k + 124k + 262k + 590k are squares for k = 1,2,3 (332, 6672, 150572).
3-by-3 magic squares consisting of different squares with constant 6672:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 162, 647, 438, 487, 126, 503, 426, 102), | (10, 210, 633, 258, 585, 190, 615, 242, 90), |
(42, 206, 633, 423, 498, 134, 514, 393, 162), | (57, 118, 654, 186, 633, 98, 638, 174, 87), |
(57, 234, 622, 298, 567, 186, 594, 262, 153), | (71, 318, 582, 438, 406, 297, 498, 423, 134), |
(71, 318, 582, 438, 462, 199, 498, 361, 258), | (87, 314, 582, 406, 438, 297, 522, 393, 134), |
(87, 342, 566, 406, 423, 318, 522, 386, 153), | (118, 402, 519, 438, 441, 242, 489, 298, 342), |
(174, 377, 522, 402, 486, 217, 503, 258, 354)$v |
6672 = 103 + 173 + 763 = 103 + 383 + 733.
Page of Squares : First Upload July 25, 2005 ; Last Revised March 17, 2011by Yoshio Mimura, Kobe, Japan
668
The smallest squares containing k 668's :
146689 = 3832,
17866866889 = 1336672,
186686668668889 = 136633332.
6682 = 446224, a square with 3 kind of even digits.
6682± 3 are primes.
6684 = 199115858176 : 12 + 92 + 92 + 12 + 12 + 52 + 82 + 52 + 82 + 172 + 62.
the first integer which is the sum of a square and a prime in 12 ways :
32 + 659, 52 + 643, 72 + 619, 92 + 587, 112 + 547, 132 + 499, 152 + 443, 172 + 379, 192 + 307, 212 + 227, 232 + 139, 252 + 43.
by Yoshio Mimura, Kobe, Japan
669
The smallest squares containing k 669's :
669124 = 8182,
6693366969 = 818132,
8766966966918889 = 936320832.
6692 + 6702 + 6712 + ... + 26122 = 764462.
6692 = 2232 + 4462 + 4462 : 6442 + 6442 + 3222 = 9662,
6692 = 2682 + 4042 + 4612 : 1642 + 4042 + 8622 = 9662.
6690k + 103026k + 142497k + 195348k are squares for k = 1,2,3 (6692, 2629172, 1069670792).
Komachi equations:
6692 = 13 + 23 + 33 + 43 + 53 + 63 * 73 + 83 * 93 = 93 * 83 + 73 * 63 + 53 + 43 + 33 + 23 + 13.
3-by-3 magic squares consisting of different squares with constant 6692:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 198, 639, 369, 534, 162, 558, 351, 114), | (8, 131, 656, 181, 632, 124, 644, 176, 43), |
(16, 149, 652, 184, 628, 139, 643, 176, 56), | (16, 272, 611, 404, 491, 208, 533, 364, 176), |
(28, 179, 644, 371, 532, 164, 556, 364, 77), | (28, 316, 589, 461, 436, 212, 484, 397, 236), |
(29, 128, 656, 448, 491, 76, 496, 436, 107), | (32, 259, 616, 469, 448, 164, 476, 424, 203), |
(40, 125, 656, 356, 560, 85, 565, 344, 100), | (40, 331, 580, 419, 440, 280, 520, 380, 181), |
(56, 316, 587, 349, 488, 296, 568, 331, 124), | (64, 253, 616, 341, 544, 188, 572, 296, 181), |
(92, 436, 499, 461, 404, 268, 476, 307, 356), | (104, 259, 608, 404, 512, 149, 523, 344, 236), |
(124, 397, 524, 428, 356, 371, 499, 404, 188), | (139, 392, 524, 436, 344, 373, 488, 419, 184) |
6692 = 447561, 4 + 4 + 7 + 5 + 61 = 92,
6692 = 447561, 4 + 4 + 75 + 61 = 122,
6692 = 447561, 4 + 4 + 7 + 561 = 242,
6692 = 447561, 4 + 4756 + 1 = 692,
6692 = 447561, 4 + 4 + 7561 = 872.
by Yoshio Mimura, Kobe, Japan