Squares:640s

640

The smallest squares containing k 640's :
6400 = 80²,
64064016 = 8004²,
56406408576400 = 7510420².

Page of Squares : First Upload January 30, 2006 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

641

The smallest squares containing k 641's :
14641 = 121²,
356416641 = 18879²,
17641016414641 = 4200121².

The squares which begin with 641 and end in 641 are
6419374641 = 80121², 64196863641 = 253371², 641006795641 = 800629²,
641394358641 = 800871², 641407172641 = 800879²,...

641² = 4⁴ + 10⁴ + 10⁴ + 25⁴.

3-by-3 magic squares consisting of different squares with constant 641²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 79, 636, 376, 516, 57, 519, 372, 56),	(12, 281, 576, 369, 468, 236, 524, 336, 153),
(15, 380, 516, 420, 384, 295, 484, 345, 240),	(16, 180, 615, 345, 516, 160, 540, 335, 84),
(36, 264, 583, 308, 519, 216, 561, 268, 156),	(48, 231, 596, 324, 524, 177, 551, 288, 156),
(56, 321, 552, 447, 376, 264, 456, 408, 191),	(92, 369, 516, 441, 412, 216, 456, 324, 313),
(97, 264, 576, 384, 488, 159, 504, 321, 232)

641² = 410881, 4 + 108 + 8 + 1 = 11².

Page of Squares : First Upload January 30, 2006 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan

642

The smallest squares containing k 642's :
664225 = 815²,
6426427225 = 80165²,
1896422642764225 = 43547935².

642² = 412164, a zigzag square.

642^k + 66126^k + 136746^k + 208650^k are squares for k = 1,2,3 (642², 258084², 109223460²).

642² = 412164, with 4 = 2², 121 = 11² and 64 = 8².

642² = 412164, 4 + 1 + 21 + 6 + 4 = 6²,
642² = 412164, 4 + 12 + 1 + 64 = 9²,
642² = 412164, 4 + 1 + 216 + 4 = 15²,
642² = 412164, 4 + 12 + 16 + 4 = 6²,
642² = 412164, 412 + 164 = 24².

Page of Squares : First Upload July 11, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

643

The smallest squares containing k 643's :
86436 = 294²,
64346439556 = 253666²,
485664364386436 = 22037794².

643² = 24³ + 50³ + 65³.

3-by-3 magic squares consisting of different squares with constant 643²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(14, 177, 618, 273, 558, 166, 582, 266, 63), $(18, 255, 590, 410, 450, 207, 495, 382, 150),
(42, 337, 546, 366, 462, 257, 527, 294, 222),	(58, 207, 606, 417, 474, 122, 486, 382, 177),
(66, 338, 543, 382, 417, 306, 513, 354, 158),	(81, 158, 618, 222, 591, 122, 598, 198, 129),

643² = 413449, 4 + 1 + 3 + 4 + 4 + 9 = 5².

Page of Squares : First Upload July 11, 2005 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan

644

The smallest squares containing k 644's :
56644 = 238²,
264452644 = 16262²,
75644644996449 = 8697393².

The squares which begin with 644 and end in 644 are
6441988644 = 80262², 644388296644 = 802738², 644426828644 = 802762²,
6440114156644 = 2537738², 6440235968644 = 2537762²,...

(417 / 644)² = 0.419276358... (Komachic).

612² + 613² + 614² + 615² + ... + 644² = 3608².

(1³ + 2³ + ... + 79³)(80³ + 81³ + ... + 344³)(345³ + 346³ + ... + 644³) = 37268487000000².

644² = 414736, 4 + 1 + 4 + 7 + 3 + 6 = 5²,
644² = 414736, 4³ + 1³ + 47³ + 36³ = 388²,
644² = 414736, 4 + 1 + 473 + 6 = 22².

Page of Squares : First Upload July 11, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

645

The smallest squares containing k 645's :
64516 = 254²,
36450064561 = 190919²,
1645764571264576 = 40568024².

1 + 2 + 3 + ... + 645 = 1² + 2² + 3² + ... + 85² = 208335.

645² = 416025, a square with different digits.

645² + 646² + 647² + 648² + ... + 2261² = 61369².

(1² + 2² + ... + 14²)(15² + 16² + ... + 357²)(358² + 359² + ... + 645²) = 1072570800².

645² = 2³ + 30³ + 73³ = 17³ + 58³ + 60³.

3-by-3 magic squares consisting of different squares with constant 645²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(5, 80, 640, 320, 556, 67, 560, 317, 44),	(5, 80, 640, 448, 460, 61, 464,445,52),
(5, 256, 592, 320, 515, 220, 560, 292, 131),	(11, 340, 548, 452, 395, 236, 460, 380, 245),
(30, 135, 630, 306, 558, 105, 567, 294, 90),	(30, 270, 585, 423, 450, 186, 486, 375, 198),
(32, 424, 485, 451, 332, 320, 460, 355, 280),	(35, 124, 632, 380, 515, 80, 520, 368, 101),
(35, 188, 616, 280, 560, 155, 580, 259, 112),	(35, 280, 580, 380, 460, 245, 520, 355, 140),
(35, 296, 572, 380, 472, 221, 520, 325, 200),	(40, 187, 616, 220, 584, 163, 605, 200, 100),
(40, 220, 605, 352, 515, 164, 539, 320, 152),	(52, 200, 611, 355, 520, 140, 536, 325, 152),
(53, 220, 604, 404, 460, 203, 500, 395, 100),	(54, 297, 570, 375, 450, 270, 522, 354, 135),
(80, 380, 515, 445, 340, 320, 460, 395, 220),	(88, 340, 541, 416, 445, 212, 485, 320, 280),
(100, 364, 523, 395, 380, 340, 500, 373, 164),	(140, 317, 544, 355, 500, 200, 520, 256, 283),
(157, 376, 500, 424, 443, 200, 460, 280, 355)

645² = 416025, 4 + 1 + 6 + 0 + 25 = 6²,
645² = 416025, 416 + 0 + 25 = 21².

Page of Squares : First Upload July 11, 2005 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan

646

The smallest squares containing k 646's :
166464 = 408²,
6465446464 = 80408²,
2746469646646441 = 52406771².

The square root of 646 is 25. 4 1 6 5 3 0 0 5 4 2 7 7 6 6 8 8 4 4 9 1 9 ...,
and 25² = 4² + 1² + 6² + 5² + 3² + 0² + 0² + 5² + 4² + 2² + 7² + 7² + 6² + 6² + 8² + 8² + 4² + 4² + 9² + 1² + 9².

66538^k + 70414^k + 105298^k + 175066^k are squares for k = 1,2,3 (646², 226100², 84715148²).

646² = 417316, 4 + 1 + 7 + 31 + 6 = 7²,
646² = 417316, 41 + 73 + 1 + 6 = 11².

Page of Squares : First Upload July 11, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

647

The smallest squares containing k 647's :
2647129 = 1627²,
64792647936 = 254544²,
647647376472889 = 25448917².

647² = 418609, a square with diffent digits.

647² = 243² + 422² + 426² : 624² + 224² + 342² = 746².

Komachi Fraction : 647² = 7534962 / 18.

The square root of 647 is 25.43..., and 25 = 4² + 3².

3-by-3 magic squares consisting of different squares with constant 647²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(6, 237, 602, 413, 462, 186, 498, 386, 147),	(10, 147, 630,378, 510, 125, 525, 370, 78),
(19, 78, 642, 282, 579, 62, 582, 278, 51),	(30, 397, 510, 435, 390, 278, 478, 330, 285),
(42, 154, 627, 253, 582, 126, 594, 237, 98),	(62, 294, 573, 426, 413, 258, 483, 402, 154),
(93, 258, 586, 426, 467, 138, 478, 366, 237),	(98, 333, 546, 426, 378, 307, 477, 406, 162),
(114, 278, 573, 307, 534, 198, 558, 237, 226),	(114, 397, 498, 422, 426, 243, 477, 282, 334)

647² = 418609, 4 + 1 + 86 + 0 + 9 = 10²,
647² = 418609, 41 + 8 + 6 + 0 + 9 = 8².

Page of Squares : First Upload July 11, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

648

The smallest squares containing k 648's :
36481 = 191²,
4648648761 = 68181²,
546480996486489 = 23376933².

(1 + 2 + 3)(4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13) = 648²,
(1)(2)(3)(4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13) = 648²,
(1³ + 2³)(3³ + 4³ + 5³)(6³) = 648².

648² = (24 + 25 + 26 + 27 + 28 + 29 + 30 + 31)² + (32 + 33 + 34 + 35 + 36 + 37 + 38 + 39)² + (40 + 41 + 42 + 43 + 44 + 45 + 46 + 47)² + (48 + 49 + 50 + 51 + 52 + 53 + 54 + 55)².

648² = 36³ + 72³.

648² = (1² + 8)(4² + 8)(44² + 8).

Komachi equations:
648² = - 12² * 3² / 4² * 5² * 6² + 78² * 9² = - 12² * 3² * 45² / 6² + 78² * 9²
= 9² * 8² * 7² / 6² * 54² * 3² / 21² = 9² * 8² / 7² / 6² * 54² / 3² * 21²
= 98² / 7² * 6² * 54² * 3² / 21²,
648² = 9³ * 8³ + 7³ * 6³ * 54³ / 3³ / 21³.

648² = 419904, 4 + 19 + 9 + 0 + 4 = 6²,
648² = 419904, 41 + 9 + 90 + 4 = 12²,
648² = 419904, 41 + 99 + 0 + 4 = 12².

Page of Squares : First Upload July 11, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

649

The smallest squares containing k 649's :
8649 = 93²,
164942649 = 12843²,
256496496498064 = 16015508².

The squares which begin with 649 and end in 649 are
6495231649 = 80593², 64944954649 = 254843², 64977578649 = 254907²,
649083201649 = 805657², 649382940649 = 805843²,...

649² = 204² + 424² + 447² : 744² + 424² + 402² = 946².

649² + 650² + 651² + 652² + ... + 11232² = 687246².

3-by-3 magic squares consisting of different squares with constant 649²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(8, 129, 636, 321, 552, 116, 564, 316, 57),	(12, 224, 609, 441, 444, 172, 476, 417, 144),
(24, 127, 636, 449, 456, 108, 468, 444, 71),	(36, 388, 519, 417, 384, 316, 496, 351, 228),
(48, 279, 584, 424, 456, 183, 489, 368, 216),	(63, 276, 584, 404, 441, 252, 504, 388, 129),
(84, 199, 612, 287, 564, 144, 576, 252, 161),	(116, 249, 588, 336, 532, 159, 543, 276, 224),
(116, 372, 519, 447, 424, 204, 456, 321, 332)

649² = 421201, 42 + 1 + 20 + 1 = 8².

649² = 421201 appears in the decimal expression of e:
e = 2.71828•••421201••• (from the 129492nd digit).

Page of Squares : First Upload July 11, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan