Squares:650s

650

The smallest squares containing k 650's :
65025 = 255²,
13650650896 = 116836²,
6506506501600561 = 80662919².

650 = (1² + 2² + 3² + ... + 1000²) / (1² + 2² + 3² + ... + 115²).

650² = 121² + 122² + 123² + ... + 144²

650² = 422500, 4 = 2², 22500 = 150²

650² = (1² + 1)(3² + 1)(8² + 1)(18² + 1) = (1² + 1)(8² + 1)(57² + 1)
= (2² + 1)(3² + 1)(5² + 1)(18² + 1) = (2² + 1)(5² + 1)(57² + 1)
= (2² + 1)(5² + 1)(7² + 1)(8² + 1) = (5² + 1)(7² + 1)(18² + 1).

Komachi Square Sum : 650² = 4² + 7² + 9² + 185² + 623².

650 = 1² + 2² + 3² + ... + 12².

(1 + 2 + 3 + 4)(5)(6 + 7)(8 + 9 + 10 + 11 + 12)(13) = 650².

(1³ + 2³ + ... + 216³)(217³ + 218³ + ... + 650³) = 4927958028².

Page of Squares : First Upload July 18, 2005 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

651

The smallest squares containing k 651's :
465124 = 682²,
6512651401 = 80701²,
6519065136516 = 2553246².

651² = 423801, a square with different digits.

Komachi Fraction : 576/3814209 = (8 / 651)².

651² = 423801, 4 + 23 + 8 + 0 + 1 = 6²,
651² = 423801, 42 + 38 + 0 + 1 = 9².

(2² + 7)(3² + 7)(6² + 7)(7² + 7) = 651² + 7.

3-by-3 magic squares consisting of different squares with constant 651²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(22, 254, 599, 346, 503, 226, 551, 326, 118),	(23, 146, 634, 454, 458, 89, 466, 439, 118),
(25, 226, 610, 274, 550, 215, 590, 265, 74),	(26, 158, 631, 206, 601, 142, 617, 194, 74),
(26, 370, 535, 410, 425, 274, 505, 326, 250),	(27, 324, 564, 396, 456, 243, 516, 333, 216),
(34, 359, 542, 446, 382, 281, 473, 386, 226),	(36, 192, 621, 348, 531, 144, 549, 324, 132),
(38, 281, 586, 314, 506, 263, 569, 298, 106),	(41, 142, 634, 326, 554, 103, 562, 311, 106),
(41, 254, 598, 422, 466, 169, 494, 377, 194),	(46, 121, 638, 286, 578, 89, 583, 274, 94),
(56, 371, 532, 448, 364, 301, 469, 392, 224),	(58, 169, 626, 386, 514, 103, 521, 362, 146),
(73, 274, 586, 334, 521, 202, 554, 278, 199),	(74, 313, 566, 359, 454, 298, 538, 346, 121),
(82, 311, 566, 391, 478, 206, 514, 314, 247),	(94, 302, 569, 454, 439, 158, 457, 374, 274),
(110, 199, 610, 230, 590, 151, 599, 190, 170),	(137, 386, 506, 446, 311, 358, 454, 422, 199),
(137, 386, 506, 446, 382, 281, 454, 359, 298),	(151, 334, 538, 386, 487, 194, 502, 274, 311)

651² = 423801 appears in the decimal expression of π:
π = 3.14159•••423801••• (from the 44347th digit).

Page of Squares : First Upload July 18, 2005 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan

652

The smallest squares containing k 652's :
265225 = 515²,
65246528356 = 255434²,
652265294065225 = 25539485².

652²± 3 are primes.

652² = 13² + 14² + 15² + ... + 108².

652² = 36³ + 47³ + 65³.

652² = 425104, 4 + 2 + 5 + 1 + 0 + 4 = 4²,
652² = 425104, 425 + 104 = 23².

Page of Squares : First Upload July 18, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

653

The smallest squares containing k 653's :
1653796 = 1286²,
65316536041 = 255571²,
4653653365371481 = 68217691².

3-by-3 magic squares consisting of different squares with constant 653²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(3, 148, 636, 284, 573, 132, 588, 276, 67),	(3, 300, 580, 420, 445, 228, 500, 372, 195),
(13, 192, 624, 384, 507, 148, 528, 364, 123),	(24, 237, 608, 283, 552, 204, 588, 256, 123),
(27, 328, 564, 372, 456, 283, 536, 333, 168),	(36, 248, 603, 347, 504, 228, 552, 333, 104),
(48, 176, 627, 396, 507, 112, 517, 372, 144),	(108, 357, 536, 384, 472, 237, 517, 276, 288),
(120, 347, 540, 435, 360, 328, 472, 420, 165),	(132, 288, 571, 328, 531, 192, 549, 248, 252)

653² = 426409, 4 + 2 + 6 + 4 + 0 + 9 = 5²,
653² = 426409, 42² + 6² + 40² + 9² = 59²,
653² = 426409, 4³ + 26³ + 40³ + 9³ = 287².

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

654

The smallest squares containing k 654's :
654481 = 809²,
16541046544 = 128612²,
10654654940646544 = 103221388².

146^k + 154^k + 654^k + 1546^k are squares for k = 1,2,3 (50², 1692², 63100²).

Komachi equation: 654² = 9² * 8² * 7² + 6² * 5² * 4² * 3² + 210².

654² = 427716, 4 + 2 + 7 + 7 + 16 = 6²,
654² = 427716, 427 + 7 + 1 + 6 = 21²,
654² = 427716, 4 + 2 + 7 + 716 = 27².

Page of Squares : First Upload January 30, 2006 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

655

The smallest squares containing k 655's :
65536 = 256²,
6555655089 = 80967²,
5655419465565529 = 75202523².

655² = 67² + 68² + 69² + ... + 116².

9196² = 304² + 305² + 306² + 307² + ... + 655².

655² = 429025, 4 + 2 + 90 + 25 = 11².

3-by-3 magic squares consisting of different squares with constant 655²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 114, 645, 411, 502, 90, 510, 405, 70),	(21, 178, 630, 322, 546, 165, 570, 315, 70),
(30, 75, 650, 110, 642, 69, 645, 106, 42),	(30, 110, 645, 254, 597, 90, 603, 246, 70),
(30, 299, 582, 330, 510, 245, 565, 282, 174),	(30, 330, 565, 450, 421, 222, 475, 378, 246),
(54, 250, 603, 453, 450, 146, 470, 405, 210),	(70, 267, 594, 405, 450, 250, 510, 394, 117),
(70, 315, 570, 405, 470, 210, 510, 330, 245),	(75, 366, 538, 450, 362, 309, 470, 405, 210),
(75, 414, 502, 450, 398, 261, 470, 315, 330),	(110, 330, 555, 411, 470, 198, 498, 315, 286)

Page of Squares : First Upload July 18, 2005 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan

656

The smallest squares containing k 656's :
6561 = 81²,
216560656 = 14716²,
465665665609 = 682397².

The squares which begin with 656 and end in 656 are
65646638656 = 256216², 65681488656 = 256284², 656449966656 = 810216²,
656560160656 = 810284², 6560175728656 = 2561284²,...

(1³ + 2³ + ... + 176³)(177³ + 178³ + ... + 591³)(592³ + 593³ + ... + 656³) = 341529673605120².

Page of Squares : First Upload July 18, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

657

The smallest squares containing k 657's :
226576 = 476²,
3657346576 = 60476²,
81216576576576 = 9012024².

657² = 12⁴ + 16⁴ + 16⁴ + 23⁴.

Kaprekar : 657²=431649, 4 + 3 + 1 + 649 = 657.

657² + 658² + 659² + ... + 752² = 6908².

657² = 431649, 4 + 3 + 1 + 649 = 657.

Komachi equations:
657² = 9² * 876² * 5² / 4² / 3² * 2² / 10² = 9² * 876² / 5² / 4² / 3² / 2² * 10².

3-by-3 magic squares consisting of different squares with constant 657²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 193, 628, 457, 452, 136, 472, 436, 137),	(16, 208, 623, 392, 497, 176, 527, 376, 112),
(20, 343, 560, 385, 460, 268, 532, 320, 215),	(21, 138, 642, 222, 606, 123, 618, 213, 66),
(23, 284, 592, 416, 452, 233, 508, 383, 164),	(28, 191, 628, 329, 548, 152, 568, 308, 119),
(28, 224, 617, 257, 572, 196, 604, 233, 112),	(32, 215, 620, 380, 500, 193, 535, 368, 100),
(32, 268, 599, 361, 508, 208, 548, 319, 172),	(44, 352, 553, 383, 436, 308, 532, 343, 176),
(73, 164, 632, 292, 577, 116, 584, 268, 137),	(73, 248, 604, 292, 556, 193, 584, 247, 172),
(75, 210, 618, 318, 555, 150, 570, 282, 165),	(88, 271, 592, 431, 472, 152, 488, 368, 241),
(114, 402, 507, 453, 318, 354, 462, 411, 222),	(138, 357, 534, 402, 474, 213, 501, 282, 318)

657² = 431649, 4 + 3 + 1 + 64 + 9 = 9²,
657² = 431649, 4 + 3 + 16 + 4 + 9 = 6².

Page of Squares : First Upload July 18, 2005 ; Last Revised June 22, 2010
by Yoshio Mimura, Kobe, Japan

658

The smallest squares containing k 658's :
665856 = 816²,
76658658129 = 276873²,
520465865865841 = 22813721².

658² + 659² + 660² + ... + 7403² = 367657².

1 / 658 = 0.00 1 5 1 9 7 5 6 8 3 8 9 0 5 7 7 5 0 7 5 ...,
the sum of the squares of the digits is 658.

The square root of 658 is 25.6515106767613201326961955114 ...,
where 25² = 6² + 5² + 1² + 5² + ... + 1² + 4².

5922^k + 95410^k + 151998^k + 179634^k are squares for k = 1,2,3 (658², 253988², 100880612²).
253^k + 295^k + 341^k + 407^k are squares for k = 1,2,3 (36², 658², 12204²).

Komachi equations:
658² = 987² * 6² / 54² * 3² * 2² */ 1² = 9² * 8² - 7² + 654² + 3² + 2² + 10²
= 987² / 6² * 5² / 4² * 32² / 10².

658² = 432964, 43 + 2 + 9 + 6 + 4 = 8²,
658² = 432964, 4 + 3 + 29 + 64 = 10².

658² = 432964 appears in the decimal expressions of π:
π = 3.14159•••432964••• (from the 18410th digit).

Page of Squares : First Upload July 18, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

659

The smallest squares containing k 659's :
659344 = 812²,
6591166596 = 81186²,
2065965936598404 = 45452898².

659² = 434281, a zigzag square.

659² + 660² + 661² + ... + 2580² = 75051².

1 / 659 = 0.001517450682, 15² + 17² + 4² + 5² + 0² + 6² + 8² + 2² = 659.

3-by-3 magic squares consisting of different squares with constant 659²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 81, 654, 369, 542, 66, 546, 366, 47),	(18, 79, 654, 129, 642, 74, 646, 126, 33),
(30, 191, 630, 290, 570, 159, 591, 270, 110),	(30, 385, 534, 465, 366, 290, 466, 390, 255),
(42, 321, 574, 414, 434, 273, 511, 378, 174),	(74, 282, 591, 366, 511, 198, 543, 306, 214),
(79, 378, 534, 438, 369, 326, 486, 394, 207),	(81, 214, 618, 394, 513, 126, 522, 354, 191),
(114, 303, 574, 438, 466, 159, 479, 354, 282),	(129, 326, 558, 402, 486, 191, 506, 303, 294)

659² = 434281, 4³ + 3³ + 4³ + 2³ + 8³ + 1³ = 26²,
659² = 434281, 4 + 34 + 2 + 8 + 1 = 7²,
659² = 434281, 4 + 34 + 2 + 81 = 11².

Page of Squares : First Upload July 18, 2005 ; Last Revised July 13, 2009
by Yoshio Mimura, Kobe, Japan