Squares:660s

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.

42^k + 129^k + 660^k + 1194^k 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, 2011
by 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.

217^k + 218^k + 272^k + 518^k 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).

Page of Squares : First Upload July 25, 2005 ; Last Revised March 17, 2011
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, 2006
by 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.

67626^k + 78897^k + 100776^k + 192270^k 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.

Page of Squares : First Upload February 5, 2006 ; Last Revised August 17, 2013
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.

Page of Squares : First Upload July 25, 2005 ; Last Revised August 24, 2006
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

665^k + 1421^k + 2401^k + 5117^k are squares for k = 1,2,3 (98², 5866², 388570²).
230^k + 610^k + 665^k + 2720^k 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, 2011
by 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.

21^k + 45^k + 201^k + 633^k are squares for k = 1,2,3 (30², 666², 16182²).

The 4-by-4 magic squares consisting of different squares with constant 666:

0²	3²	9²	24²
4²	19²	15²	8²
11²	14²	18²	5²
23²	10²	6²	1²

0²	3²	9²	24²
11²	22²	6²	5²
16²	13²	15²	4²
17²	2²	18²	7²

0²	7²	16²	19²
9²	18²	15²	6²
12²	17²	8²	13²
21²	2²	11²	10²

666² = 443556, 44 + 3556 = 60².

666² = 443556 appears in the decimal expression of e:
e = 2.71828•••443556••• (from the 105277th digit).

Page of Squares : First Upload July 25, 2005 ; Last Revised January 16, 2014
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.

667^k + 2507^k + 2553^k + 2737^k are squares for k = 1,2,3 (92², 4554², 230644²).
113^k + 124^k + 262^k + 590^k 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, 2011
by 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.

Page of Squares : First Upload July 25, 2005 ; Last Revised January 16, 2014
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.

6690^k + 103026^k + 142497^k + 195348^k 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.

Page of Squares : First Upload July 25, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan