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420 - 429

420

The smallest squares containing k 420's :
42025 = 2052,
20420124201 = 1428992,
62364204204201 = 78971012.

4202 = (22 - 1)(62 - 1)(412 - 1) = (62 - 1)(712 - 1) = (62 - 1)(82 - 1)(92 - 1).

Komachi equations:
4202 = 122 - 32 * 42 + 52 / 62 * 72 * 82 * 92 = 122 * 32 * 42 * 52 * 62 * 72 / 82 / 92
 = 122 / 32 - 42 + 52 / 62 * 72 * 82 * 92 = 122 / 32 / 42 * 52 / 62 * 72 * 82 * 92
 = - 122 + 32 * 42 + 52 / 62 * 72 * 82 * 92 = - 122 / 32 + 42 + 52 / 62 * 72 * 82 * 92,
4202 = 93 + 83 * 73 - 63 - 543 / 33 / 23 + 103 = 93 * 83 * 73 * 63 / 543 - 33 * 23 + 103
 = - 93 + 83 * 73 - 63 + 543 / 33 / 23 + 103.

A quartic polynomial:
(X + 420)(X + 525)(X + 1344)(X + 1680) = X4 + 632X3 + 23102X2 + 529202X + 7056002.
The next is:
(X + 27560)(X + 44096)(X + 155025)(X + 248040) = X4 + 6892X3 + 2618202X2 + 569665202X + 68359824002.

28665k + 36435k + 54705k + 56595k are squares for k = 1,2,3 (4202, 913502, 204183002).

The integral triangle of sides 245, 1443, 1448 (or 800, 1241, 2009)(or 113, 3137, 3150) has square area 4202.

(1)(2 + 3)(4 + 5 + 6)(7)(8)(9 + 10 + 11 + 12) = 4202,
(1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9)(10 + 11)(12) = 4202,
(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13) = 4202,
(1)(2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13)(14) = 4202,
(1)(2 + 3)(4)(5)(6 + 7 + 8)(9 + 10 + 11 + 12 + 13 + 14 + 15) = 4202,
(1 + 2 + 3 + 4)(5 + 6 + ... + 16)(17 + 18 + ... + 23) = 4202,
(1 + 2 + ... + 48)(49 + 50 + 51) = 4202,
(1 + 2 + ... + 7)(8 + 9 + ... + 112) = 4202.

4202 = 176400 appears in the decimal expression of e:
  e = 2.71828•••176400••• (from the 6355th digit),
  (176400 is the fifth 6-digit square in the expression of e.)

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

421

The smallest squares containing k 421's :
142129 = 3772,
4214216889 = 649172,
1653421421421225 = 406622852.

4213 - 4202 + 4192 - 4182 + .. + 13 = 61192.

4212 = 30 + 31 + 32 + 34 + 311.

Komachi Square Sum : 4212 = 82 + 762 + 1242 + 3952.

(347 / 421)2 = 0.679351842... (Komachic).

3-by-3 magic squares consisting of different squares with constant 4212:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(12, 189, 376, 291, 268, 144, 304, 264, 123), (21, 236, 348, 268, 276, 171, 324, 213, 164),
(24, 88, 411, 164, 381, 72, 387, 156, 56), (24, 101, 408, 276, 312, 61, 317, 264, 84),
(93, 236, 336, 276, 291, 128, 304, 192, 219) 

4212 = 177241, 17 + 7 + 24 + 1 = 72,
4212 = 177241, 1 + 7 + 72 + 41 = 112,
4212 = 177241, 1 + 77 + 2 + 41 = 112,
4212 = 177241, 1 + 7 + 7 + 241 = 162,
4212 = 177241, 12 + 72 + 72 + 242 + 12 = 262,
4212 = 177241, 173 + 73 + 23 + 43 + 13 = 732.

Page of Squares : First Upload February 14, 2005 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

422

The smallest squares containing k 422's :
4225 = 652,
164224225 = 128152,
422042294894224 = 205436682.

67k + 209k + 233k + 275k are squares for k = 1,2,3 (282, 4222, 65482).

4222 = 178084, 1 + 7 + 8 + 0 + 84 = 102,
4222 = 178084, 1 + 7 + 80 + 8 + 4 = 102.

4222 = 178084 appears in the decimal expression of π:
  π = 3.14159•••178084••• (from the 15510th digit),
  (178084 is the ninth 6-digit square in the expression of π.).

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

423

The smallest squares containing k 423's :
423801 = 6512,
4231242304 = 650482,
423423423335025 = 205772552.

4232 = 178929, 1 + 7 + 8 + 9 + 2 + 9 = 62.

Komachi Square Sum : 4232 = 22 + 742 + 1952 + 3682.

423, 424 and 425 are three consecutive integers having square factors (the 6th case).

6204k + 7050k + 52593k + 113082k are squares for k = 1,2,3 (4232, 1250672, 399011672).

3-by-3 magic squares consisting of different squares with constant 4232:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 146, 397, 173, 362, 134, 386, 163, 58),(2, 205, 370, 230, 310, 173, 355, 202, 110),
(12, 111, 408, 177, 372, 96, 384, 168, 57),(12, 156, 393, 201, 348, 132, 372, 183, 84),
(13, 118, 406, 274, 307, 98, 322, 266, 67),(19, 142, 398, 218, 338, 131, 362, 211, 58),
(22, 202, 371, 238, 301, 178, 349, 218, 98),(34, 142, 397, 278, 307, 86, 317, 254, 118),
(35, 98, 410, 202, 365, 70, 370, 190, 77),(58, 253, 334, 274, 278, 163, 317, 194, 202)

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

424

The smallest squares containing k 424's :
17424 = 1322,
424442404 = 206022,
42404424849424 = 65118682.

The squares which begin with 424 and end in 424 are
4242177424 = 651322,   42490401424 = 2061322,   424280271424 = 6513682,
424624263424 = 6516322,   424931889424 = 6518682,...

4242 = 43 + 163 + 563.

(261 / 424)2 = 0.378921546... (Komachic).

4242 = 179776, a square with odd digits except the last digit 6.

Komachi equation: 4242 = 13 + 23 + 33 * 43 * 563 / 73 - 893.

4242 = 179776, 1 + 7 + 9 + 7 + 76 = 102,
4242 = 179776, 1 + 7 + 9 + 77 + 6 = 102,
4242 = 179776, 1 + 79 + 7 + 7 + 6 = 102.

(13 + 23 + ... + 2553)(2563 + 2573 + ... + 4243) = 27411072002,
(13 + 23 + ... + 2793)(2803 + 2813 + ... + 4083)(4093 + 4103 + ... + 4243) = 979331118796802.

Page of Squares : First Upload February 14, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

425

The smallest squares containing k 425's :
234256 = 4842,
4425442576 = 665242,
1194254258144256 = 345579842.

4252 = 180625, a zigzag square with different digits.

4252 = 180625, 18 + 0 + 6 + 25 = 72.

4252 = 54 + 104 + 104 + 204.

Komachi Square Sums : 4252 = 82 + 262 + 1572 + 3942 = 22 + 42 + 52 + 1982 + 3762.

3-by-3 magic squares consisting of different squares with constant 4252:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 65, 420, 255, 336, 52, 340, 252, 39),(0, 119, 408, 200, 360, 105, 375, 192, 56),
(0, 180, 385, 255, 308, 144, 340, 231, 108),(0, 200, 375, 255, 300, 160, 340, 225, 120),
(20, 153, 396, 225, 340, 120, 360, 204, 97),(60, 232, 351, 295, 276, 132, 300, 225, 200),
(60, 272, 321, 295, 204, 228, 300, 255, 160) 

Page of Squares : First Upload February 14, 2005 ; Last Revised April 13, 2009
by Yoshio Mimura, Kobe, Japan

426

The smallest squares containing k 426's :
426409 = 6532,
42604262464 = 2064082,
42654261426529 = 65310232.

4262 = 303 + 373 + 473.

4262 = 181476 : 182 + 12 + 42 + 72 + 62 = 426.

The 4-by-4 magic square consisting of different squares with constant 426:

 02 12132162
 42152112 82
 72142102 92
192 22 62 52

4262 = 181476, 1 + 8 + 14 + 7 + 6 = 62,
4262 = 181476, 18 + 1 + 4 + 7 + 6 = 62.

Page of Squares : First Upload February 14, 2005 ; Last Revised August 31, 2009
by Yoshio Mimura, Kobe, Japan

427

The smallest squares containing k 427's :
427716 = 6542,
427414276 = 206742,
427427755554276 = 206743262.

4272 = 182329, a zigzag square.

Cubic Polynomial : (X + 362)(X + 4272)(X + 6722) = X3 + 7972X2 + 2883722X + 103299842.

Komachi equation: 4272 = - 93 + 83 * 73 - 63 + 53 - 43 * 33 + 213.

3-by-3 magic squares consisting of different squares with constant 4272:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(18, 166, 393, 274, 297, 138, 327, 258, 94),(41, 138, 402, 282, 311, 78, 318, 258, 121),
(54, 158, 393, 183, 366, 122, 382, 153, 114),(58, 249, 342, 279, 282, 158, 318, 202, 201),
(87, 202, 366, 246, 327, 122, 338, 186, 183) 

4272 = 182329, 1 + 8 + 2 + 3 + 2 + 9 = 52,
4272 = 182329, 182 + 3 + 2 + 9 = 142.

Page of Squares : First Upload February 14, 2005 ; Last Revised June 4, 2010
by Yoshio Mimura, Kobe, Japan

428

The smallest squares containing k 428's :
42849 = 2072,
5724284281 = 756592,
342842885544289 = 185160172.

The square root of 428 is 20.68816086557...,
and 202 = 62 + 82 + 82 + 12 + 62 + 02 + 82 + 62 + 52 + 52 + 72.

4282 = 183184, 1 * 8 * 3 * 18 - 4 = 18 * 3 * 1 * 8 - 4 = 428.

4282 = 113 + 293 + 543.

4282 = 183184.

3792 + 3802 + 3812 + 3822 + ... + 4282 = 28552,
1402 + 1412 + 1422 + 1432 + ... + 4282 = 50322.

4282 = 183184, 1 + 8 + 3 + 1 + 8 + 4 = 52,
4282 = 183184, 1 + 8 + 3 + 184 = 142,
4282 = 183184, 183 + 1 + 8 + 4 = 142,
4282 = 183184, 1 + 83 + 1 + 84 = 132.

4282 = 183184 appears in the decimal expression of e:
  e = 2.71828•••183184••• (from the 66932nd digit).

Page of Squares : First Upload February 14, 2005 ; Last Revised July 18, 2006
by Yoshio Mimura, Kobe, Japan

429

The smallest squares containing k 429's :
429025 = 6552,
4291429081 = 655092,
1494299942942976 = 386561762.

4292± 2 are primes.

4292 = 184041, 184 + 0 + 41 = 152.

A, B, C, A + B, B + C, and C + A are squares for A = 4292, B = 8802, C = 23402.

57k + 102k + 222k + 348k are squares for k = 1,2,3 (272, 4292, 73712).
20878k + 23738k + 42328k + 97097k are squares for k = 1,2,3 (4292, 1105392, 318390932).
25025k + 26884k + 62348k + 69784k are squares for k = 1,2,3 (4292, 1005292, 248455352).

Komachi equation: 4292 = 93 + 83 * 73 - 63 - 53 + 43 - 33 + 23 * 103.

3-by-3 magic squares consisting of different squares with constant 4292:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(4, 92, 419, 227, 356, 76, 364, 221, 52),(4, 155, 400, 280, 304, 115, 325, 260, 104),
(13, 64, 424, 104, 412, 59, 416, 101, 28),(13, 104, 416, 256, 332, 91, 344, 251, 52),
(18, 111, 414, 246, 342, 81, 351, 234, 78),(52, 221, 364, 244, 316, 157, 349, 188, 164),
(56, 133, 404, 208, 364, 91, 371, 184, 112),(59, 164, 392, 232, 344, 109, 356, 197, 136),
(64, 211, 368, 283, 256, 196, 316, 272, 101),(64, 244, 347, 269, 248, 224, 328, 251, 116)

4292 + 4302 + 4312 + 4322 + ... + 1214122 = 244249942.

4292 = 184041 appears in the decimal expression of π:
  π = 3.14159•••184041••• (from the 73341st digit).

Page of Squares : First Upload February 14, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan