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460 - 469

460

The smallest squares containing k 460's :
124609 = 3532,
2346046096 = 484362,
64604605214601 = 80376992.

(13 + 23 + ... + 493)(503 + 513 + 523)(533 + 543 + ... + 4603) = 819617022002.

Page of Squares : First Upload March 14, 2005 ; Last Revised July 24, 2006
by Yoshio Mimura, Kobe, Japan

461

The smallest squares containing k 461's :
461041 = 6792,
4617746116 = 679542,
3461461324614225 = 588341852.

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

3-by-3 magic squares consisting of different squares with constant 4612:

3124444
236381108
39622861
     
16123444
312324101
33930472
     
72196411
264357124
371216168
     
84236387
259348156
372189196

4612 = 212521, a square with just 3 kinds of digits.

4612 = 212521, a square pegged by 2.

4612 = 212521, 2 + 1 + 25 + 21 = 72,
4612 = 212521, 2 + 1 + 252 + 1 = 162,
4612 = 212521, 21 + 2 + 5 + 21 = 72,
4612 = 212521, 21 + 25 + 2 + 1 = 72.

Page of Squares : First Upload March 14, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

462

The smallest squares containing k 462's :
4624 = 682,
36446246281 = 1909092,
246204625246281 = 156909092.

4622 = 213444, a square whose digits are 1,2,3 and 4.

4622 = 213444, 21 - 3 + 444 = 462.

4622 = (42 + 6)(62 + 6)(152 + 6).

462k + 1034k + 2200k + 2233k are squares for k = 1,2,3 (772, 33332, 1516132).

The integral triangle of sides 585, 746, 847 (or 539, 890, 1233)(or 56, 9273, 9305) has square area 4622.

Komachi Square Sum : 4622 = 32 + 92 + 282 + 612 + 4572.

4622 + 4632 + 4642 + 4652 + ... + 14222 = 304422,
4622 + 4632 + 4642 + 4652 + ... + 25102 = 723982.

(1 + 2)(3)(4)(5 + 6)(7)(8 + 9 + 10 + 11 + 12 + 13 + 14) = 4622,
(1)(2 + 3 + ... + 12)(13 + 14 + ... + 75) = 4622.

4622 = 213444, 2 + 1 + 34 + 44 = 92,
4622 = 213444, 2 + 134 + 4 + 4 = 122,
4622 = 213444, 21 + 3 + 4 + 4 + 4 = 62,
4622 = 213444, 213 + 4 + 4 + 4 = 152.

4622 = 213444 appears in the decimal expression of π:
  π = 3.14159•••213444••• (from the 5672nd digit),
  (213444 is the second 6-digit square in the expression of π.).

Page of Squares : First Upload March 14, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

463

The smallest squares containing k 463's :
463761 = 6812,
46346339524 = 2152822,
463946355463396 = 215394142.

The square root of 463 is 21.51743479135001...,
where 212 = 52 + 12 + 72 + 42 + 32 + 42 + 72 + 92 + 132 + 52 + 02 + 02 + 12.

4632 = 214369, a square with differebt digits.

Komachi equation: 4632 = 13 + 23 + 343 + 563 - 73 + 83 - 93.

3-by-3 magic squares consisting of different squares with constant 4632:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(18, 94, 453, 123, 438, 86, 446, 117, 42),(18, 149, 438, 229, 378, 138, 402, 222, 59),
(18, 283, 366, 306, 282, 203, 347, 234, 198),(30, 213, 410, 315, 310, 138, 338, 270, 165),
(86, 213, 402, 282, 346, 123, 357, 222, 194) 

4632 = 214369, a exchangeable square (436921 = 6612).

4632 + 4642 + 4652 + 4662 + ... + 89122 = 4857452.

4632 = 214369, 2 + 1 + 4 + 3 + 6 + 9 = 52.

4632 = 214369 appears in the decimal expression of π
  π = 3.14159•••214369••• (from the 71862nd digit).

Page of Squares : First Upload March 14, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

464

The smallest squares containing k 464's :
8464 = 922,
446434641 = 211292,
113704644644644 = 106632382.

The squares which begin with 464 and end in 464 are
46400606464 = 2154082,   46479910464 = 2155922,   464316862464 = 6814082,
464567654464 = 6815922,   464998520464 = 6819082,...

4642 = 215296, a zigzag square.

4642 = 215296, 2 + 1 * 52 * 9 - 6 = 2 * 1 + 52 * 9 - 6 = 464.

4642 = 215296, 2 + 1 + 5 + 2 + 9 + 6 = 52,
4642 = 215296, 2 + 152 + 9 + 6 = 132,
4642 = 215296, 21 + 52 + 96 = 132.

Page of Squares : First Upload March 14, 2005 ; Last Revised July 24, 2006
by Yoshio Mimura, Kobe, Japan

465

The smallest squares containing k 465's :
465124 = 6822,
1465664656 = 382842,
465770465504656 = 215817162.

4652 is the third square which is the sum of 10 sixth powers.

4652 + 4662 + 4672 + ... + 4802 = 4812 + 4822 + 4832 + ... + 4952.

3-by-3 magic squares consisting of different squares with constant 4652:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 68, 460, 268, 376, 55, 380, 265, 40),(6, 117, 450, 315, 330, 90, 342, 306, 75),
(16, 188, 425, 215, 380, 160, 412, 191, 100),(20, 65, 460, 152, 436, 55, 439, 148, 40),
(20, 224, 407, 260, 343, 176, 385, 220, 140),(20, 260, 385, 328, 265, 196, 329, 280, 172),
(23, 160, 436, 236, 380, 127, 400, 215, 100),(27, 114, 450, 210, 405, 90, 414, 198, 75),
(40, 145, 440, 320, 328, 79, 335, 296, 128),(40, 265, 380, 320, 260, 215, 335, 280, 160),
(44, 233, 400, 260, 320, 215, 383, 244, 100),(49, 232, 400, 280, 335, 160, 368, 224, 175),
(55, 140, 440, 244, 385, 92, 392, 220, 119),(65, 260, 380, 292, 320, 169, 356, 215, 208)

4652 = 216225, 622521 = 7892.

4652 = 216225, 2 + 1 + 6 + 2 + 25 = 62,
4652 = 216225, 2 + 1 + 6 + 22 + 5 = 62,
4652 = 216225, 21 + 6 + 2 + 2 + 5 = 62,
4652 = 216225, 216 + 2 + 2 + 5 = 152,
4652 = 216225, 2163 + 23 + 253 = 31772,
4652 = 216225, 216 + 225 = 212.

Page of Squares : First Upload March 14, 2005 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

466

The smallest squares containing k 466's :
46656 = 2162,
46678466704 = 2160522,
1466701646646681 = 382975412.

15292 = 4562 + 4572 + 4582 + 4592 + ... + 4662,
56762 = 1702 + 1712 + 1722 + 1732 + ... + 4662,
56812 = 1682 + 1692 + 1702 + 1712 + ... + 4662.

4662 = 217156, 21 + 7 + 15 + 6 = 72.

Page of Squares : First Upload March 14, 2005 ; Last Revised July 24, 2006
by Yoshio Mimura, Kobe, Japan

467

The smallest squares containing k 467's :
224676 = 4742,
467467641 = 216212,
1467467467621444 = 383075382.

1 / 467 = 0.0021413, 212 + 42 + 12 + 32 = 467.

1 / 467 = 0.0021413276231263383297644, the sum of the squares of its digits is 467.

4672 = 573 + 85 + 27.

3-by-3 magic squares consisting of different squares with constant 4672:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 162, 438, 318, 321, 118, 342, 298, 111),(18, 134, 447, 174, 417, 118, 433, 162, 66),
(18, 161, 438, 242, 378, 129, 399, 222, 98),(42, 206, 417, 287, 318, 186, 366, 273, 98),
(63, 222, 406, 298, 294, 207, 354, 287, 102) 

4672 = 218089, 2 + 1 + 8 + 0 + 89 = 102,
4672 = 218089, 2 + 1 + 80 + 8 + 9 = 102.

Page of Squares : First Upload March 14, 2005 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

468

The smallest squares containing k 468's :
84681 = 2912,
4686634681 = 684592,
510746846884681 = 225997092.

4682 = (13 + 1)(23 + 1)(233 + 1).

Cubic polynomial:
(X + 4682)(X + 11042)(X + 18172) = X3 + 21772X2 + 22391882X + 9387930242.

Komachi Fraction : 4682 = 5913648 / 27.

(1)(2 + 3 + ... + 14)(15 + 16 + ... + 66) = 4682.

4682 = 219024, 2 + 1 + 9 + 0 + 24 = 62,
4682 = 219024, 21 + 9 + 0 + 2 + 4 = 62,
4682 = 219024, 219 + 0 + 2 + 4 = 152.

4682 = 219024 appears in the decimal expression of π:
  π = 3.14159•••219024••• (from the 68005th digit).

Page of Squares : First Upload March 14, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

469

The smallest squares containing k 469's :
346921 = 5892,
4690469169 = 684872,
8469694469469249 = 920309432.

Cubic polynomial : (X + 642)(X + 1712)(X + 4322) = X3 + 4692X2 + 796322X + 47278082.

3-by-3 magic squares consisting of different squares with constant 4692:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(8, 9, 459, 324, 333, 64, 339, 316, 72),(12, 109, 456, 269, 372, 96, 384, 264, 53),
(12, 251, 396, 276, 324, 197, 379, 228, 156),(24, 188, 429, 296, 339, 132, 363, 264, 136),
(24, 237, 404, 309, 296, 192, 352, 276, 141),(45, 100, 456, 156, 435, 80, 440, 144, 75),
(64, 171, 432, 243, 384, 116, 396, 208, 141),(69, 228, 404, 324, 316, 123, 332, 261, 204)

4692 = 219961, 21 + 9 + 9 + 61 = 102,
4692 = 219961, 219 + 9 + 61 = 172.

Page of Squares : First Upload March 14, 2005 ; Last Revised April 27, 2009
by Yoshio Mimura, Kobe, Japan