Squares:430s

430

The smallest squares containing k 430's :
204304 = 452²,
12430243081 = 111491²,
358430430843081 = 18932259².

430² = 76² + 77² + 78² + ... + 99².

430^k + 740^k + 1470^k + 1585^k are squares for k = 1,2,3 (65², 2325², 87425²).

132² + 133² + 134² + ... + 430² = 5083².

Page of Squares : First Upload February 21, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

431

The smallest squares containing k 431's :
431649 = 657²,
4431431761 = 66569²,
443143127431225 = 21050965².

431² = 185761, 1 + 8 - 5 + 7 * 61 = 431.

431 is the 8th prime for which the Legendre Symbol (a/431) = 1 for a = 1, 2,...,6.

Komachi Square Sums : 431² = 5² + 82² + 194² + 376² = 5² + 94² + 176² + 382².

3-by-3 magic squares consisting of different squares with constant 431²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(21, 62, 426, 118, 411, 54, 414, 114, 37),	(21, 154, 402, 206, 357, 126, 378, 186, 91),
(27, 226, 366, 294, 258, 181, 314, 261, 138),	(27, 226, 366, 294, 309, 62, 314, 198, 219),
(30, 181, 390, 219, 330, 170, 370, 210, 69),	(54, 198, 379, 267, 314, 126, 334, 219, 162)
(54, 226, 363, 258, 309, 154, 341, 198, 174),	(62, 219, 366, 294, 246, 197, 309, 278, 114)

431² = 185761, 1 + 85 + 7 + 6 + 1 = 10²,
431² = 185761, 18 + 5 + 76 + 1 = 10²,
431² = 185761, 18 + 5 + 761 = 28².

Page of Squares : First Upload February 21, 2005 ; Last Revised April 13, 2009
by Yoshio Mimura, Kobe, Japan

432

The smallest squares containing k 432's :
43264 = 208²,
43204329 = 6573²,
45684324324324 = 6759018².

432² = 186624, 18 * 6 / 6 * 24 = 18 / 6 * 6 * 24 = 432.

432²± 5 are primes.

432² = (1² + 8)(2² + 8)(4² + 8)(8² + 8) = (4² + 8)(8² + 8)(10² + 8).

432² is the 7th square which is the sum of 4 sixth powers : 432² = 6⁶ + 6⁶ + 6⁶ + 6⁶.

Cubic Polynomials :
(X + 9²)(X + 28²)(X + 432²) = X³ + 433²X² + 12708² + 108864²,
(X + 64²)(X + 171²)(X + 432²) = X³ + 469²X² + 79632² + 4727808².

432² = 186624, 18 + 6 + 6 + 2 + 4 = 6²,
432² = 186624, 1 + 8 + 6 + 62 + 4 = 9²,
432² = 186624, 1 + 8 + 66 + 2 + 4 = 9²,
432² = 186624, 1³ + 8³ + 6³ + 62³ + 4³ = 489².

Page of Squares : First Upload February 21, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

433

The smallest squares containing k 433's :
24336 = 156²,
43321843321 = 208139²,
2433433509064336 = 49329844².

433² = 187489, 1 * 8 - 7 + 48 * 9 = 433.

433² = 2⁴ + 14⁴ + 16⁴ + 17⁴.

Cubic Polynomial : (X + 9²)(X + 28²)(X + 432²) = X³ + 433²X² + 12708² + 108864².

Komachi Square Sum : 433² = 5² + 46² + 178² + 392².

3-by-3 magic squares consisting of different squares with constant 433²:

9²	192²	388²
228²	332²	159²
368²	201²	108²

12²	264²	343²
287²	252²	204²
324²	233²	168²

33²	172²	396²
276²	297²	152²
332²	264²	87²

433² + 434² + 435² + 436² + ... + 721² = 9911²,
433² + 434² + 435² + 436² + ... + 5089² = 209565²,
433² + 434² + 435² + 436² + ... + 6216² = 282934²,
433² + 434² + 435² + 436² + ... + 23568² = 2088988².

433² = 187489, 1 + 8 + 74 + 8 + 9 = 10²,
433² = 187489, 187 + 489 = 26².

Page of Squares : First Upload February 21, 2005 ; Last Revised April 13, 2009
by Yoshio Mimura, Kobe, Japan

434

The smallest squares containing k 434's :
434281 = 659²,
43443481761 = 208431²,
434434442434281 = 20843091².

1 / 434 = 0.002304..., 2304 = 48².

434² = (1² + 3)(2² + 3)(82² + 3) = (5² + 3)(82² + 3).

(1² + 2² + ... + 304²) + (1² + 2² + ... + 377²) = (1² + 2² + ... + 434²).

Komachi equation: 434² = - 1³ + 23³ - 4³ + 5³ - 6³ + 7³ * 8³ + 9³.

Page of Squares : First Upload February 21, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

435

The smallest squares containing k 435's :
4356 = 66²,
43543586241 = 208671²,
43543554352516 = 6598754².

Komachi equations:
435² = 9 + 876 * 5 * 432 / 10,
435² = 9² * 87² / 6² * 5² * 4² / 3² / 2² * 1² = 9² * 87² / 6² * 5² * 4² / 3² / 2² / 1².

The square root of 435 is 20.85665361461421020547...,
20² = 8² + 5² + 6² + 6² + ... + 0² + 5² + 4² + 7² (the sum of the squares of their digits).

435² = 189225, 18 + 9 + 2 + 2 + 5 = 6²,
435² = 189225, 1 + 892 + 2 + 5 = 30².

3-by-3 magic squares consisting of different squares with constant 435²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 261, 348, 300, 252, 189, 315, 240, 180),	(10, 65, 430, 145, 406, 58, 410, 142, 31),
(10, 65, 430, 254, 350, 47, 353, 250, 46),	(10, 130, 415, 206, 367, 110, 383, 194, 70),
(10, 130, 415, 305, 298, 86, 310, 289, 98),	(10, 241, 362, 305, 262, 166, 310, 250, 175),
(14, 175, 398, 202, 350, 161, 385, 190, 70),	(26, 143, 410, 257, 326, 130, 350, 250, 65),
(46, 95, 422, 122, 410, 79, 415, 110, 70),	(58, 175, 394, 290, 310, 95, 319, 250, 158),
(58, 194, 385, 290, 305, 110, 319, 242, 180)

Page of Squares : First Upload February 21, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

436

The smallest squares containing k 436's :
42436 = 206²,
1174364361 = 34269²,
43682436714361 = 6609269².

The squares which begin with 436 and end in 436 are
436532418436 = 660706², 436648710436 = 660794², 4360604298436 = 2088206²,
4360971830436 = 2088294², 4362692754436 = 2088706²,...

Komachi equation: 436² = 9² * 8² + 7² - 6² + 5² * 43² * 2² - 1².

436² = 190096, 1 + 9 + 0 + 0 + 9 + 6 = 5².

Page of Squares : First Upload February 21, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

437

The smallest squares containing k 437's :
1437601 = 1199²,
437437225 = 20915²,
14370214378437241 = 119875829².

437 = (1² + 2² + 3² + ... + 322²) / (1² + 2² + 3² + ... + 42²).

437² = 190969, a zigzag square.

437² = 190969, a square pegged by 9.

Cubic Polynomial :
(X + 112²)(X + 147²)(X + 396²) = X³ + 437²X² + 75012²X + 6519744²,
(X + 192²)(X + 252²)(X + 301²) = X³ + 437²X² + 106932²X + 14563584².

437 is the second integer which is the sum of a prime and a square in 10 ways :
2² + 433, 4² + 421, 6² + 401, 8² + 373, 10² + 337, 12² + 293, 14² + 241, 16² + 181, 18² + 113, 20² + 37.

(1³ + 2³ + ... + 42³)(43³ + 44³ + ... + 69³)(70³ + 71³ + ... + 437³) = 193503609936².

437² = 190969, 1 + 90 + 9 + 69 = 13²,
437² = 190969, 1 + 90 + 96 + 9 = 14².

3-by-3 magic squares consisting of different squares with constant 437²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 148, 411, 229, 348, 132, 372, 219, 68),	(13, 156, 408, 264, 328, 117, 348, 243, 104),
(21, 192, 392, 288, 301, 132, 328, 252, 141),	(48, 203, 384, 237, 336, 148, 364, 192, 147),
(77, 168, 396, 216, 363, 112, 372, 176, 147)

Page of Squares : First Upload February 21, 2005 ; Last Revised April 13, 2009
by Yoshio Mimura, Kobe, Japan

438

The smallest squares containing k 438's :
394384 = 628²,
24384384025 = 156155²,
1438743843823729 = 37930777².

438² = 191844, 1² + 9² + 18² + 4² + 4² = 438.

(1 + 2 + 3 + ... + 72)(73) = 438².

730^k + 41902^k + 71978^k + 77234^k are squares for k = 1,2,3 (438², 113588², 30119508²).

438² = 191844, 1 + 9 + 18 + 4 + 4 = 6²,
438² = 191844, 19 + 1 + 8 + 4 + 4 = 6²,
438² = 191844, 19 + 18 + 44 = 9².

438² = 191844, 1 + 91 + 8 + 44 = 12²,

Page of Squares : First Upload February 21, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

439

The smallest squares containing k 439's :
439569 = 663²,
44397439849 = 210707²,
1904394439439424 = 43639368².

(282 / 439)² = 0.412637958... (Komachic).

439² = 192721, 19 + 2 + 7 + 21 = 7²,
439² = 192721, 19 + 27 + 2 + 1 = 7²,
439² = 192721, 1 + 92 + 7 + 21 = 11²,
439² = 192721, 1² + 9² + 2² + 7² + 21² = 24².

3-by-3 magic squares consisting of different squares with constant 439²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(3, 66, 434, 294, 322, 51, 326, 291, 42),	(3, 106, 426, 146, 402, 99, 414, 141, 38),
(6, 218, 381, 286, 291, 162, 333, 246, 146),	(18, 234, 371, 259, 294, 198, 354, 227, 126),
(29, 102, 426, 258, 349, 66, 354, 246, 83),	(29, 174, 402, 258, 326, 141, 354, 237, 106),
(29, 174, 402, 258, 339, 106, 354, 218, 141),	(42, 189, 394, 214, 354, 147, 381, 178, 126),
(106, 237, 354, 258, 326, 141, 339, 174, 218)

Page of Squares : First Upload February 21, 2005 ; Last Revised April 13, 2009
by Yoshio Mimura, Kobe, Japan