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520 - 529

520

The smallest squares containing k 520's :
205209 = 4532,
34520525209 = 1857972,
420520520520409 = 205065972.

5202 is the 8th square which is the sum of 4 sixth powers : 26 + 46 + 46 + 86.

5202 = (12 + 4)(32 + 4)(62 + 4)(102 + 4) = (22 + 4)(62 + 4)(292 + 4) = (32 + 4)(102 + 4)(142 + 4).

Komachi equation: 5202 = 982 / 72 * 652 * 42 * 32 / 212.

Page of Squares : First Upload April 25, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

521

The smallest squares containing k 521's :
1521 = 392,
2521144521 = 502112,
63521521661521 = 79700392.

5212 = 120 + 122 + 123 + 124 + 125.

The squares which begin with 521 and end in 521 are
5214428521 = 722112,   52115867521 = 2282892,   52194428521 = 2284612,
521227685521 = 7219612,   521340317521 = 7220392,...

3-by-3 magic squares consisting of different squares with constant 5212:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 72, 516, 324, 404, 57, 408, 321, 44),(1, 132, 504, 252, 441, 116, 456, 244, 63),
(12, 71, 516, 159, 492, 64, 496, 156, 33),(28, 276, 441, 351, 336, 188, 384, 287, 204),
(36, 289, 432, 327, 324, 244, 404, 288, 159),(96, 225, 460, 300, 404, 135, 415, 240, 204)

(12)(22 + 32 + ... + 92)(102 + 112 + ... + 5212) = 1158722.

5212 = 271441, 2 + 7 + 14 + 41 = 82.

Page of Squares : First Upload April 25, 2005 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

522

The smallest squares containing k 522's :
55225 = 2352,
6522985225 = 807652,
31725225225225 = 56325152.

5222 = (12 + 5)(132 + 5)(162 + 5) = (12 + 5)(22 + 5)(712 + 5) = (72 + 5)(712 + 5)
= (32 + 9)(1232 + 9).

Komachi equations:
5222 = 92 + 872 * 62 - 542 / 32 / 22 */ 12 = 92 * 872 * 62 / 542 * 32 * 22 */ 12
= - 92 + 872 * 62 + 542 / 32 / 22 */ 12 = 92 * 872 / 62 * 52 / 42 * 322 / 102.

5222 = 272484 is an exchangeable square, that is, 842724 = 9182.

5222 = 272484, 27 + 2 + 48 + 4 = 92,
5222 = 272484, 272 + 48 + 4 = 182.

5222 = 272484 appears in the decimal expressions of π and e:
  π = 3.14159•••272484••• (from the 19116th digit),
  e = 2.71828•••272484••• (from the 16314th digit).

Page of Squares : First Upload April 25, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

523

The smallest squares containing k 523's :
155236 = 3942,
15235952356 = 1234342,
52339523975236 = 72346062.

5232 = 273529, a zigzag square.

5232 = 403 + 413 + 523.

62k + 73k + 266k + 440k are squares for k = 1,2,3 (292, 5232, 102292).

3-by-3 magic squares consisting of different squares with constant 5232:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(42, 249, 458, 278, 378, 231, 441, 262, 102),(46, 153, 498, 342, 386, 87, 393, 318, 134),
(57, 246, 458, 298, 393, 174, 426, 242, 183),(62, 183, 486, 297, 414, 118, 426, 262, 153),
(73, 234, 462, 354, 318, 217, 378, 343, 114) 

5232 = 273529, 27 + 3 + 5 + 29 = 82,
5232 = 273529, 273 + 5 + 2 + 9 = 172.

5232 = 273529 appears in the decimal expression of e:
  e = 2.71828•••273529••• (from the 5860th digit),
  (273529 is the fourth 6-digit square in the expression of e.)

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

524

The smallest squares containing k 524's :
47524 = 2182,
2052452416 = 453042,
352452487543524 = 187737182.

The squares which begin with 524 and end in 524 are
524491711524 = 7242182,   524584415524 = 7242822,   5240519051524 = 22892182,
5240812075524 = 22892822,   5242808519524 = 22897182,...

92 + 232 + ... + (14x + 9)2 + ... + 2192 = 5242.

5242 = 274576, 27 + 4 + 5 + 7 + 6 = 72.

Page of Squares : First Upload April 25, 2005 ; Last Revised August 3, 2006
by Yoshio Mimura, Kobe, Japan

525

The smallest squares containing k 525's :
525625 = 7252,
5255525025 = 724952,
3525081525525609 = 593723972.

5252 = 275625, a zigzag square.

5252 = 103 + 653 = 83 + 493 + 543, the first square which is the sum of 2 cubes and the sum of 3 cubes.

Komachi equations:
5252 = 92 * 82 * 72 / 62 * 52 * 42 / 322 * 102 = 92 / 82 * 72 / 62 * 52 / 42 * 322 * 102
 = 982 / 72 * 62 * 52 * 42 / 322 * 102.

A quartic polynomial: (See 420)
(X + 420)(X + 525)(X + 1344)(X + 1680) = X4 + 632X3 + 23102X2 + 529202X + 7056002.

(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 37) = 5252.

3-by-3 magic squares consisting of different squares with constant 5252:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(5,116,512,160,488,109,500,155,40),(5,160,500,340,380,125,400,325,100),
(5,172,496,340,379,128,400,320,115),(5,304,428,340,328,229,400,275,200),
(15,198,486,270,414,177,450,255,90),(16,115,512,325,400,100,412,320,59),
(18,174,495,225,450,150,474,207,90),(20,100,515,131,500,92,508,125,44),
(20,100,515,268,445,76,451,260,68),(20,200,485,229,440,172,472,205,104),
(20,200,485,352,365,136,389,320,148),(32,251,460,365,320,200,376,332,155),
(37,200,484,284,400,187,440,275,80),(40,155,500,220,460,125,475,200,100),
(40,220,475,307,376,200,424,293,100),(61,148,500,260,445,100,452,236,125),
(67,244,460,356,317,220,380,340,125),(81,258,450,342,369,150,390,270,225),
(101,232,460,268,424,155,440,205,200),(104,320,403,347,260,296,380,325,160)

5252 = 275625, 2 + 7 + 5 + 62 + 5 = 92,
5252 = 275625, 2 + 75 + 62 + 5 = 122,
5252 = 275625, 2 + 7 + 562 + 5 = 242,
5252 = 275625, 275 + 625 = 302,
5252 = 275625, 2 + 7562 + 5 = 872.

Page of Squares : First Upload April 25, 2005 ; Last Revised January 11, 2011
by Yoshio Mimura, Kobe, Japan

526

The smallest squares containing k 526's :
85264 = 2922,
8152645264 = 902922,
526305264526336 = 229413442.

5262 = 276676, a square consisting of just 3 kinds of digits.

5262 = 276676, 27 + 66 + 76 = 132.

Page of Squares : First Upload April 25, 2005 ; Last Revised August 3, 2006
by Yoshio Mimura, Kobe, Japan

527

The smallest squares containing k 527's :
527076 = 7262,
25275276324 = 1589822,
278527527452736 = 166891442.

5272 = 277729, 2 * 7 + 7 * 72 + 9 = 527.

5272 = 277729, a square consisting of just 3 kinds of digits.

37417k + 43214k + 54808k + 142290k are squares for k = 1,2,3 (5272, 1628432, 563789872).

Komachi equation: 5272 = 14 + 234 - 44 - 54 - 64 - 74 - 84 + 94.

3-by-3 magic squares consisting of different squares with constant 5272:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6,102,517,363,374,78,382,357,66),(6,213,482,293,402,174,438,266,123),
(18,267,454,294,382,213,437,246,162),(22,69,522,162,498,59,501,158,42),
(22,171,498,258,438,139,459,238,102),(27,130,510,330,402,85,410,315,102),
(42,266,453,306,357,238,427,282,126),(50,123,510,165,490,102,498,150,85),
(102,293,426,357,354,158,374,258,267),(123,302,414,346,363,162,378,234,283)

(13 + 23 + ... + 4953)(4963 + 4972 + ... + 5273) = 80373427202.

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

528

The smallest squares containing k 528's :
25281 = 1592,
20528585284 = 1432782,
452805288925284 = 212792222.

5282 = 278784, 27 * 8 + 78 * 4 = 528.

5282 = (42 + 8)(62 + 8)(162 + 8).

5282 + 5292 + 5302 + ... + 5442 = 5452 + 5462 + 5472 + ... + 5602.

528529 = 7272.

Cubic Polynomials :
(X + 1962)(X + 5282)(X + 6932) = X3 + 8932X2 + 4037882X + 717171842,
(X + 2092)(X + 5282)(X + 6842) = X3 + 8892X2 + 4037882X + 754807682.
(X + 5282)(X + 17282)(X + 627712) = X3 + 627972X2 + 1134225122X + 572712560642.

5282 = 183 + 463 + 563.

A, B, C, A + B, B + C, C + A are squares for A = 5282, B = 57962, and C = 63252.

Komachi equations:
5282 = 92 * 82 * 72 / 62 * 52 - 42 + 322 * 102 = 982 / 72 * 62 * 52 - 42 + 322 * 102.

5282 = 278784, 2 + 7 + 8 + 7 + 8 + 4 = 62.

5282 = 278784 appears in the decimal expression of e:
  e = 2.71828•••278784••• (from the 142482nd digit).

Page of Squares : First Upload April 25, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

529

The square of 23.

The smallest squares containing k 529's :
529 = 232,
1829529529 = 427732,
52921652924529 = 72747272.

The squares which begin with 529 and end in 529 are
5295909529 = 727732,   52910580529 = 2300232,   529222785529 = 7274772,
529289715529 = 7275232,   529586586529 = 7277272,...

5292 = 279841, a square with different digits.

528529 = 7272.

3-by-3 magic squares consisting of different squares with constant 5292:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(9, 72, 524, 204, 484, 63, 488, 201, 36),(9, 268, 456, 372, 321, 196, 376, 324, 183),
(20, 240, 471, 321, 380, 180, 420, 279, 160),(36, 191, 492, 336, 372, 169, 407, 324, 96),
(48, 196, 489, 236, 447, 156, 471, 204, 128),(48, 344, 399, 369, 264, 272, 376, 303, 216),
(84, 321, 412, 344, 348, 201, 393, 236, 264) 

5292 = 279841, 2 + 798 + 41 = 292,
5292 = 279841, 27 + 9 + 8 + 4 + 1 = 72,
5292 = 279841, 27 + 9 + 84 + 1 = 112.

Page of Squares : First Upload April 25, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan