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560 - 569

560

The smallest squares containing k 560's :
25600 = 1602,
5601025600 = 748402,
556056015105600 = 235808402.

5602 is the 2nd square which is the sum of 10 seventh powers.

5602± 3 are primes.

Cubic polynomial :
(X + 5602)(X + 18632)(X + 42842) = X3 + 47052X2 + 83989082X + 44694115202.

Komachi equation: 5602 = 122 / 32 + 42 + 5672 - 892.

5602 + 5612 + 5622 + 5632 + ... + 24812 = 709592.

(1 + 2)(3)(4 + 5)(6 + 7 + 8)(9)(10 + 11) = 5672.

(13 + 23 + ... + 3843)(3853 + 3863 + ... + 5603) = 102453120002,
(13 + 23 + ... + 843)(853 + 863 + ... + 1043)(1053 + 1063 + ... + 5603) = 23152685220002.

5602 = 313600 appears in the decimal expression of e:
  e = 2.71828•••313600••• (from the 56394th digit)

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

561

The smallest squares containing k 561's :
6561 = 812,
1561356196 = 395142,
173056156116561 = 131550812.

The squares which begin with 561 and end in 561 are
5612856561 = 749192,   56130612561 = 2369192,   561122344561 = 7490812,
561254190561 = 7491692,   561496947561 = 7493312,...

5612± 2 are primes.

204k + 294k + 561k + 966k are squares for k = 1,2,3 (452, 11732, 333452).

Komachi equation: 5612 = 12 - 22 + 342 + 5672 - 892.

3-by-3 magic squares consisting of different squares with constant 5612:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(4, 137, 544, 292, 464, 119, 479, 284, 68),(4, 208, 521, 353, 404, 164, 436, 329, 128),
(16, 73, 556, 119, 544, 68, 548, 116, 31),(16, 367, 424, 392, 296, 271, 401, 304, 248),
(17, 136, 544, 376, 401, 112, 416, 368, 79),(23, 164, 536, 296, 452, 151, 476, 289, 68),
(31, 236, 508, 388, 376, 151, 404, 343, 184),(40, 164, 535, 364, 415, 100, 425, 340, 136),
(52, 284, 481, 311, 416, 212, 464, 247, 196),(68, 289, 476, 361, 388, 184, 424, 284, 233),
(76, 287, 476, 329, 364, 272, 448, 316, 119),(95, 236, 500, 340, 425, 136, 436, 280, 215),
(102, 261, 486, 306, 438, 171, 459, 234, 222),(116, 268, 479, 304, 439, 172, 457, 224, 236)

5612 = 314721, 3 + 1 + 4 + 7 + 21 = 62,
5612 = 314721, 3 + 1 + 4 + 72 + 1 = 92,
5612 = 314721, 31 + 47 + 2 + 1 = 92,
5612 = 314721, 314 + 7 + 2 + 1 = 182,
5612 = 314721, 3 + 1 + 4 + 721 = 272.

5122 + 5132 + 5142 + 5152 + ... + 5612 = 37952.

Page of Squares : First Upload May 16, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

562

The smallest squares containing k 562's :
5625 = 752,
155625625 = 124752,
88125625625625 = 93875252.

29k + 109k + 301k + 461k are squares for k = 1,2,3 (302, 5622, 112502).

5622 = 315844, 3 + 1 + 5 + 8 + 4 + 4 = 52,
5622 = 315844, 3 + 158 + 4 + 4 = 132,
5622 = 315844, 33 + 13 + 53 + 83 + 443 = 2932.

Page of Squares : First Upload January 9, 2006 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

563

The smallest squares containing k 563's :
1425636 = 11942,
56356386025 = 2373952,
856356353563489 = 292635672.

563 is the third integer which is the sum of a square and a prime in 9 ways :
42 + 547, 82 + 499, 102 + 463, 122 + 419, 142 + 367, 162 + 307, 182 + 239, 202 + 163, 222 + 79.

5632 = 316969, 31 + 69 + 69 = 132.

3-by-3 magic squares consisting of different squares with constant 5632:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(14, 342, 447, 378, 337, 246, 417, 294, 238),(22, 177, 534, 303, 454, 138, 474, 282, 113),
(33, 302, 474, 342, 366, 257, 446, 303, 162),(49, 258, 498, 318, 399, 238, 462, 302, 111),
(54, 257, 498, 338, 414, 177, 447, 282, 194),(63, 230, 510, 270, 462, 175, 490, 225, 162),
(78, 282, 481, 369, 338, 258, 418, 351, 138) 

5402 + 5412 + 5422 + 5432 + ... + 5632 = 27022.

Page of Squares : First Upload May 16, 2005 ; Last Revised June 16, 2009
by Yoshio Mimura, Kobe, Japan

564

The smallest squares containing k 564's :
58564 = 2422,
564442564 = 237582,
256456459290564 = 160142582.

The squares which begin with 564 and end in 564 are
564442564 = 237582,   564364542564 = 7512422,   564388582564 = 7512582,
5641774558564 = 23752422,   5641850566564 = 23752582,...

5642 = 318096, a zigzag square with different digits.

5642 = 318096, 3 + 18 + 0 + 9 + 6 = 62,
5642 = 318096, 3 + 1 + 8096 = 902.

Page of Squares : First Upload May 16, 2005 ; Last Revised August 9, 2006
by Yoshio Mimura, Kobe, Japan

565

The smallest squares containing k 565's :
556516 = 7462,
9256556521 = 962112,
3565456544565136 = 597114442.

5652 = 319225, 31 * 9 * 2 + 2 + 5 = 565.

Komachi fractions : 243 / 8619075 = (3 / 565)2, 432 / 8619075 = (4 / 565)2.

Komachi equations:
5652 = 12 * 22 * 32 / 42 * 52 * 6782 / 92 = 12 / 22 / 32 * 452 * 6782 / 92.

3-by-3 magic squares consisting of different squares with constant 5652:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 75, 560, 339, 448, 60, 452, 336, 45),(20, 165, 540, 360, 420, 115, 435, 340, 120),
(20, 192, 531, 360, 405, 160, 435, 344, 108),(20, 333, 456, 360, 344, 267, 435, 300, 200),
(21, 72, 560, 200, 525, 60, 528, 196, 45),(45, 160, 540, 384, 405, 88, 412, 360, 141),
(60, 155, 540, 300, 468, 101, 475, 276, 132),(120, 340, 435, 363, 384, 200, 416, 237, 300)

5652 = 319225, 3 + 19 + 2 + 25 = 72,
5652 = 319225, 3 + 19 + 22 + 5 = 72,
5652 = 319225, 31 + 9 + 2 + 2 + 5 = 72,
5652 = 319225, 3 + 1 + 92 + 25 = 112.

Page of Squares : First Upload May 16, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

566

The smallest squares containing k 566's :
56644 = 2382,
5660756644 = 752382,
426566566566144 = 206534882.

5662 = 320356, 3 + 2 + 0 + 3 + 56 = 82,
5662 = 320356, 3 + 20 + 35 + 6 = 82,
5662 = 320356, 3 + 2 + 0 + 356 = 192,
5662 = 320356, 320 + 35 + 6 = 192,
5662 = 320356, 320 + 356 = 262,
5662 = 320356, 32035 + 6 = 1792.

5662 = 320356 appears in the decimal expression of e:
  e = 2.71828•••320356••• (from the 24124th digit)

Page of Squares : First Upload January 9, 2006 ; Last Revised August 9, 2006
by Yoshio Mimura, Kobe, Japan

567

The smallest squares containing k 567's :
567009 = 7532,
5676567649 = 753432,
2456718567567616 = 495652962.

Cubic polynomials :
(X + 3682)(X + 5672)(X + 42842) = X3 + 43372X2 + 29032922X + 8938823042,
(X + 5672)(X + 12002)(X + 15402) = X3 + 20332X2 + 21541802X + 10478160002.

5672 = 321489, a square with different digits.

5672 = 64 + 124 + 184 + 214 = 94 + 184 + 184 + 184.

Komachi fraction : 576 / 2893401 = (8 / 567)2.

Komachi equations:
5672 = - 122 * 32 * 42 + 52 / 62 * 782 * 92 = 92 * 82 * 72 * 62 * 52 / 42 * 32 / 22 / 102
 = 92 / 82 * 72 * 62 * 52 * 42 * 32 * 22 / 102 = 92 / 82 * 72 * 62 / 52 * 42 * 32 / 22 * 102
 = 92 * 872 - 62 * 542 / 32 / 22 * 102.

(1)(2 + 3 + ... + 19)(20 + 21 + ... + 61) = 5672,
(1)(2 + 3 + ... + 7)(8 + 9 + ... + 154) = 5672.

5672 = 321489, 3 + 2 + 14 + 8 + 9 = 62,
5672 = 321489, 3 + 21 + 48 + 9 = 92,
5672 = 321489, 32 + 1489 = 392.

(13 + 23 + ... + 3033)(3043 + 3053 + ... + 5043)(5053 + 5063 + ... + 5673) = 5390886751810562.

3-by-3 magic squares consisting of different squares with constant 5672:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 106, 557, 323, 458, 86, 466, 317, 62),(2, 179, 538, 382, 398, 131, 419, 362, 122),
(7, 196, 532, 308, 448, 161, 476, 287, 112),(10, 317, 470, 355, 370, 242, 442, 290, 205),
(12, 129, 552, 384, 408, 87, 417, 372, 96),(13, 74, 562, 166, 538, 67, 542, 163, 34),
(13, 278, 494, 334, 403, 218, 458, 286, 173),(18, 306, 477, 387, 342, 234, 414, 333, 198),
(19, 178, 538, 298, 461, 142, 482, 278, 109),(24, 348, 447, 393, 312, 264, 408, 321, 228),
(29, 262, 502, 382, 362, 211, 418, 349, 158),(46, 142, 547, 173, 526, 122, 538, 157, 86),
(57, 276, 492, 348, 372, 249, 444, 327, 132),(58, 230, 515, 365, 410, 142, 430, 317, 190),
(62, 158, 541, 242, 499, 118, 509, 218, 122),(62, 262, 499, 338, 419, 178, 451, 278, 202),
(67, 190, 530, 310, 458, 125, 470, 275, 158),(74, 163, 538, 397, 386, 122, 398, 382, 131),
(74, 253, 502, 397, 382, 134, 398, 334, 227),(118, 227, 506, 254, 482, 157, 493, 194, 202),
(122, 317, 454, 386, 382, 163, 397, 274, 298) 

5672 = 321489 appears in the decimal expression of e:
  e = 2.71828•••321489••• (from the 121868th digit).

Page of Squares : First Upload May 16, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

568

The smallest squares containing k 568's :
156816 = 3962,
5684556816 = 753962,
456856810375681 = 213742092.

5682 = 163 + 163 + 683 = 243 + 363 + 643.

Komachi equations:
5682 = 122 * 32 - 42 + 5672 - 82 - 92 = - 12 * 22 + 342 + 5672 + 82 - 92.

5682 = 322624, 34 + 24 + 24 + 64 + 24 + 44 = 412.

5682 = 322624, 32 + 2 + 62 + 4 = 102,
5682 = 322624, 32 + 2 + 6 + 24 = 82,
5682 = 322624, 32 + 26 + 2 + 4 = 82.

5682 + 5692 + 5702 + 5712 + ... + 16342 = 373452,
5682 + 5692 + 5702 + 5712 + ... + 1778082 = 432880622.

Page of Squares : First Upload May 16, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

569

The smallest squares containing k 569's :
7569 = 872,
1356964569 = 368372,
5696756956944 = 23867882.

The squares which begin with 569 and end in 569 are
56923756569 = 2385872,   56960027569 = 2386632,   569024309569 = 7543372,
569138974569 = 7544132,   569401540569 = 7545872,...

(486 / 569)2 = 0.729538146... (Komachic).

Komachi equation: 5692 = 94 - 84 - 74 + 64 + 54 + 44 * 34 * 24 - 104.

3-by-3 magic squares consisting of different squares with constant 5692:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(15, 120, 556, 340, 444, 105, 456, 335, 60),(24, 232, 519, 263, 456, 216, 504, 249, 88),
(36, 209, 528, 297, 444, 196, 484, 288, 81),(48, 111, 556, 204, 524, 87, 529, 192, 84),
(81, 336, 452, 384, 367, 204, 412, 276, 279),(108, 241, 504, 279, 468, 164, 484, 216, 207),
(111, 304, 468, 372, 396, 169, 416, 273, 276)

5692 = 323761, 3 + 2 + 37+6 + 1 = 72,
5692 = 323761, 32 + 3 + 7+6 + 1 = 72,
5692 = 323761, 323 + 76 + 1 = 202.

Page of Squares : First Upload January 9, 2006 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan