430
The smallest squares containing k 430's :
204304 = 4522,
12430243081 = 1114912,
358430430843081 = 189322592.
4302 = 762 + 772 + 782 + ... + 992.
430k + 740k + 1470k + 1585k 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, 2011by 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.
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.
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:
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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.
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, 2013by 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) |
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, 2010by 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) |
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.
730k + 41902k + 71978k + 77234k 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.
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) |
by Yoshio Mimura, Kobe, Japan