590
The smallest squares containing k 590's :
59049 = 2432,
105908590096 = 3254362,
990590959059025 = 314736552.
5902 + 5912 + 5922 + 5932 + ... + 8072 = 103552,
5902 + 5912 + 5922 + 5932 + ... + 8862 = 128042,
5902 + 5912 + 5922 + 5932 + ... + 23282 = 643432.
113k + 124k + 262k + 590k are squares for k = 1,2,3 (332, 6672, 150572).
254k + 362k + 394k + 590k are squares for k = 1,2,3 (402, 8362,181762).
3245k + 7965k + 101185k + 235705k are squares for k = 1,2,3 (5902, 2566502,1188761502).
by Yoshio Mimura, Kobe, Japan
591
The smallest squares containing k 591's :
591361 = 7692,
459159184 = 214282,
1591105914591561 = 398886692.
1 / 591 = 0.00169..., 169 = 132.
5912 = 349281, a square with different digits.
(13 + 23 + ... + 2553)(2563 + 2573 + ... + 5913) = 56096409602.
3-by-3 magic squares consisting of different squares with constant 5912:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(9, 132, 576, 288, 504, 111, 516, 279, 72), | (14, 242, 539, 286, 469, 218, 517, 266, 106), |
(22, 106, 581, 139, 566, 98, 574, 133, 46), | (26, 181, 562, 274, 502, 149, 523, 254, 106), |
(26, 254, 533, 341, 442, 194, 482, 299, 166), | (34, 203, 554, 358, 434, 181, 469, 346, 98), |
(37, 166, 566, 386, 422, 149, 446, 379, 82), | (43, 134, 574, 266, 518, 101, 526, 251, 98), |
(46, 238, 539, 406, 379, 202, 427, 386, 134), | (50, 166, 565, 334, 475, 110, 485, 310, 134), |
(72, 324, 489, 369, 408, 216, 456, 279, 252), | (86, 187, 554, 229, 526, 142, 538, 194, 149), |
(86, 293, 506, 331, 446, 202, 482, 254, 229), | (98, 314, 491, 389, 406, 182, 434, 293, 274), |
(106, 331, 478, 373, 334, 314, 446, 358, 149), | (149, 274, 502, 358, 434, 181, 446, 293, 254) |
5912 = 349281, 3 + 49 + 28 + 1 = 92,
5912 = 349281, 34 + 9 + 281 = 182,
5912 = 349281, 3 + 492 + 81 = 242,
5912 = 349281, 3 + 49281 = 2222.
by Yoshio Mimura, Kobe, Japan
592
The smallest squares containing k 592's :
5929 = 772,
59259204 = 76982,
255925925359225 = 159976852.
5922 + 5932 + 5942 + 5952 + ... + 25352 = 732422.
5922 = 350464, 35 + 0 + 4 + 6 + 4 = 72,
5922 = 350464, 3 + 50 + 4 + 64 = 112,
5922 = 350464, 350 + 46 + 4 = 202.
by Yoshio Mimura, Kobe, Japan
593
The smallest squares containing k 593's :
295936 = 5442,
2593559329 = 509272,
1593432593059329 = 399178232.
5932 = 351649, a zigzag square with different digits.
3-by-3 magic squares consisting of different squares with constant 5932:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(8, 183, 564, 393, 424, 132, 444, 372, 127), | (24, 73, 588, 172, 564, 63, 567, 168, 44), |
(36, 312, 503, 388, 393, 216, 447, 316, 228), | (48, 332, 489, 361, 372, 288, 468, 321, 172), |
(60, 255, 532, 343, 420, 240, 480, 332, 105), | (63, 348, 476, 388, 336, 297, 444, 343, 192), |
(73, 204, 552, 372, 447, 116, 456, 332, 183) |
5932 = 351649, 3 + 5 + 1 + 6 + 49 = 82,
5932 = 351649, 35 + 16 + 4 + 9 = 82,
5932 = 351649, 35 + 16 + 49 = 102,
5932 = 351649, 33 + 53 + 163 + 43 + 93 = 712.
by Yoshio Mimura, Kobe, Japan
594
The smallest squares containing k 594's :
594441 = 7712,
9594594304 = 979522,
13594594335947881 = 1165958592.
5942 = 352836, a zigzag square.
5942± 5 are primes.
5942 = (12 + 2)(22 + 2)(1402 + 2) = (12 + 2)(22 + 2)(32 + 2)(52 + 2)(82 + 2)
= (22 + 2)(32 + 2)(52 + 2)(142 + 2) = (32 + 2)(42 + 2)(52 + 2)(82 + 2)
= (32 + 2)(82 + 2)(222 + 2) = (42 + 2)(1402 + 2) = (52 + 2)(82 + 2)(142 + 2).
(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + ... + 73) = 5942,
(1 + 2)(3 + 4 + 5 + 6)(7 + 8 + ... + 114) = 5942.
Komachi equations:
5942 = - 15 + 25 + 35 * 45 + 55 - 65 + 75 + 85 + 95 = 95 + 85 + 75 - 65 + 55 + 45 * 35 + 25 - 15.
The 4-by-4 magic square consisting of different squares with constant 594:
|
5942 = 352836, 35 + 2 + 8 + 36 = 92,
5942 = 352836, 3 + 52 + 83 + 6 = 122,
5942 = 352836, 35 + 283 + 6 = 182,
5942 = 352836, 352 + 83 + 6 = 212.
by Yoshio Mimura, Kobe, Japan
595
The smallest squares containing k 595's :
59536 = 2442,
9559559529 = 977732,
89595595595025 = 94654952.
595 = (12 + 22 + 32 + ... + 3992) / (12 + 22 + 32 + ... + 472).
5952 = 423 + 15 + 67.
5952 + 5962 + 5972 + ... + 6122 = 6132 + 6142 + 6152 + ... + 6292.
Komachi square sums : 5952 = 142 + 322 + 972 + 5862 = 82 + 912 + 3762 + 4522.
3-by-3 magic squares consisting of different squares with constant 5952:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 280, 525, 357, 420, 224, 476, 315, 168), | (10, 198, 561, 330, 465, 170, 495, 314, 102), |
(15, 170, 570, 262, 510, 159, 534, 255, 62), | (15, 170, 570, 270, 510, 145, 530, 255, 90), |
(15, 206, 558, 270, 495, 190, 530, 258, 81), | (15, 270, 530, 354, 422, 225, 478, 321, 150), |
(18, 390, 449, 415, 330, 270, 426, 305, 282), | (30, 150, 575, 305, 498, 114, 510, 289, 102), |
(30, 225, 550, 370, 438, 159, 465, 334, 162), | (30, 305, 510, 370, 390, 255, 465, 330, 170), |
(46, 303, 510, 390, 370, 255, 447, 354, 170), | (90, 226, 543, 255, 510, 170, 530, 207, 174), |
(90, 255, 530, 369, 442, 150, 458, 306, 225), | (102, 289, 510, 414, 402, 145, 415, 330, 270), |
(129, 278, 510, 330, 465, 170, 478, 246, 255) |
5952 = 354025, 3 + 54 + 0 + 2 + 5 = 82,
5952 = 354025, 35 + 4 + 0 + 25 = 82,
5952 = 354025, 35 + 40 + 25 = 102,
5952 = 354025, 354 + 0 + 2 + 5 = 192,
5952 = 354025, 353 + 43 + 03 + 253 = 2422,
5952 = 354025, 353 + 403 + 253 = 3502.
by Yoshio Mimura, Kobe, Japan
596
The smallest squares containing k 596's :
34596 = 1862,
3059638596 = 553142,
175965159614596 = 132651862.
The squares which begin with 596 and end in 596 are
59626802596 = 2441862, 59689330596 = 2443142, 596271218596 = 7721862,
596468914596 = 7723142, 5960014046596 = 24413142,...
5962 = 355216, 35 + 5 + 2 + 1 + 6 = 72,
5962 = 355216, 32 + 52 + 52 + 22 + 12 + 62 = 102,
5962 = 355216, 35 + 5 + 216 = 162.
5962 = 355216 appears in the decimal expression of e:
e = 2.71828•••355216••• (from the 101323rd digit).
by Yoshio Mimura, Kobe, Japan
597
The smallest squares containing k 597's :
597529 = 7732,
37597597801 = 1939012,
5975975979886969 = 773044372.
5972 = 356409, a square with different digits.
5972 = 173 + 403 + 663.
3-by-3 magic squares consisting of different squares with constant 5972:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(4, 77, 592, 307, 508, 64, 512, 304, 43), | (13, 76, 592, 332, 493, 56, 496, 328, 53), |
(18, 186, 567, 402, 423, 126, 441, 378, 138), | (20, 365, 472, 403, 340, 280, 440, 328, 235), |
(28, 172, 571, 389, 428, 148, 452, 379, 92), | (32, 104, 587, 196, 557, 88, 563, 188, 64), |
(32, 181, 568, 269, 512, 148, 532, 248, 109), | (53, 112, 584, 176, 563, 92, 568, 164, 83), |
(53, 200, 560, 400, 428, 115, 440, 365, 172), | (59, 308, 508, 412, 389, 188, 428, 332, 251), |
(78, 279, 522, 414, 402, 153, 423, 342, 246), | (83, 356, 472, 412, 307, 304, 424, 368, 203), |
(88, 301, 508, 328, 452, 211, 491, 248, 232), | (92, 197, 556, 284, 508, 133, 517, 244, 172) |
5972 = 356409, 3 + 5 + 64 + 0 + 9 = 92,
5972 = 356409, 3 + 564 + 0 + 9 = 242.
5972 = 356409 appears in the decimal expression of e:
e = 2.71828•••356409••• (from the 61250th digit).
by Yoshio Mimura, Kobe, Japan
598
The smallest squares containing k 598's :
595984 = 7722,
3059859856 = 553162,
5987598559849 = 24469572.
5982 = 357604, a square with different digits.
(13 + 23 + ... + 73)(83 + 93 + ... + 1153)(1163 + 1173 + ... + 5983) = 334254044882.
5982 = 357604, 3 + 5 + 7 + 6 + 0 + 4 = 52,
5982 = 357604, 357 + 604 = 312.
5982 = 357604 appears in the decimal expression of e:
e = 2.71828•••357604••• (from the 8189th digit),
(357604 is the sixth 6-digit square in the expression of e.)
by Yoshio Mimura, Kobe, Japan
599
The smallest squares containing k 599's :
599076 = 7742,
59940259929 = 2448272,
1599259925599504 = 399907482.
599 is the 10th prime for which the Legendere symbol (n/p) = 1 for n = 1, 2,..., 6.
3-by-3 magic squares consisting of different squares with constant 5992:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 77, 594, 418, 426, 51, 429, 414, 58), | (13, 246, 546, 366, 429, 202, 474, 338, 141), |
(14, 141, 582, 174, 558, 131, 573, 166, 54), | (14, 354, 483, 381, 378, 266, 462, 301, 234), |
(30, 285, 526, 365, 426, 210, 474, 310, 195), | (42, 274, 531, 419, 366, 222, 426, 387, 166), |
(51, 222, 554, 282, 499, 174, 526, 246, 147), | (54, 211, 558, 387, 414, 194, 454, 378, 99), |
(78, 266, 531, 306, 477, 194, 509, 246, 198), | (90, 274, 525, 355, 450, 174, 474, 285, 230), |
(131, 306, 498, 342, 454, 189, 474, 243, 274) |
5992 = 358801, 3 + 5 + 8 + 8 + 0 + 1 = 52,
5992 = 358801, 35 + 8801 = 942.
by Yoshio Mimura, Kobe, Japan