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830 - 839

830

The smallest squares containing k 830's :
283024 = 5322,
83044830625 = 2881752,
3830551830668304 = 618914522.

Komachi square sum : 8302 = 42 + 52 + 132 + 692 + 8272.

830k + 2010k + 4570k + 4690k are squares for k = 1,2,3 (1102, 69002, 4553002).
170k + 370k + 830k + 1130k are squares for k = 1,2,3 (502, 14602, 455002).

Page of Squares : First Upload November 21, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

831

The smallest squares containing k 831's :
183184 = 4282,
28318831524 = 1682822,
831026831831296 = 288275362.

3-by-3 magic squares consisting of different squares with constant 8312:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 91, 826, 469, 682, 74, 686, 466, 53),(10, 206, 805, 245, 770, 194, 794, 235, 70),
(11, 146, 818, 202, 794, 139, 806, 197, 46),(11, 322, 766, 374, 686, 283, 742, 341, 154),
(34, 178, 811, 437, 694, 134, 706, 421, 122),(36, 249, 792, 519, 612, 216, 648, 504, 129),
(46, 326, 763, 571, 542, 266, 602, 539, 194),(48, 216, 801, 396, 711, 168, 729, 372, 144),
(53, 386, 734, 454, 629, 298, 694, 382, 251),(59, 134, 818, 286, 773, 106, 778, 274, 101),
(59, 238, 794, 574, 586, 133, 598, 539, 206),(70, 550, 619, 581, 410, 430, 590, 469, 350),
(85, 230, 794, 430, 694, 155, 706, 395, 190),(98, 494, 661, 581, 514, 298, 586, 427, 406),
(101, 406, 718, 458, 574, 389, 686, 443, 154),(106, 235, 790, 410, 706, 155, 715, 370, 206),
(106, 326, 757, 491, 638, 206, 662, 421, 274),(108, 384, 729, 441, 648, 276, 696, 351, 288),
(134, 293, 766, 389, 706, 202, 722, 326, 251),(134, 454, 683, 538, 571, 274, 619, 398, 386),
(139, 454, 682, 514, 587, 286, 638, 374, 379),(146, 482, 661, 517, 466, 454, 634, 491, 218),
(194, 427, 686, 469, 634, 262, 658, 326, 389) 

8312 = 690561, 6 + 9 + 0 + 5 + 61 = 92,
8312 = 690561, 69 + 0 + 5 + 6 + 1 = 92,
8312 = 690561, 6 + 9 + 0 + 561 = 242.

Page of Squares : First Upload November 21, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan

832

The smallest squares containing k 832's :
268324 = 5182,
483208324 = 219822,
4418328328324 = 21019822.

A cubic polynomial :
(X + 5312)(X + 8322)(X + 30242) = X3 + 31812X2 + 30172322X + 13359790082.

8322 = 692224, 6 + 9 + 2 + 2 + 2 + 4 = 52.

A, B, C, A+B, B+C, and C+A are squares for A = 8322, B = 8552, C = 26402.

Page of Squares : First Upload November 21, 2005 ; Last Revised September 21, 2006
by Yoshio Mimura, Kobe, Japan

833

The smallest squares containing k 833's :
118336 = 3442,
42833683369 = 2069632,
8334383365608336 = 912928442.

8332 = 44 + 74 + 224 + 264 = 74 + 144 + 144 + 284.

10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).

3-by-3 magic squares consisting of different squares with constant 8332:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(12, 167, 816, 284, 768, 153, 783, 276, 68),(12, 356, 753, 456, 633, 292, 697, 408, 204),
(15, 292, 780, 508, 615, 240, 660, 480, 167),(21, 238, 798, 378, 714, 203, 742, 357, 126),
(32, 204, 807, 537, 612, 176, 636, 527, 108),(40, 255, 792, 417, 680, 240, 720, 408, 95),
(42, 427, 714, 518, 546, 357, 651, 462, 238),(48, 423, 716, 473, 576, 372, 684, 428, 207),
(57, 264, 788, 408, 697, 204, 724, 372, 177),(68, 153, 816, 207, 796, 132, 804, 192, 103),
(68, 249, 792, 561, 572, 228, 612, 552, 121),(68, 519, 648, 561, 508, 348, 612, 408, 391),
(72, 391, 732, 428, 612, 369, 711, 408, 148),(76, 192, 807, 348, 743, 144, 753, 324, 148),
(103, 384, 732, 528, 537, 356, 636, 508, 177),(144, 447, 688, 552, 464, 417, 607, 528, 216),
(153, 444, 688, 544, 468, 423, 612, 527, 204) 

8332 = 693889, 69 + 3 + 8 + 89 = 132,
8332 = 693889, 69 + 3 + 88 + 9 = 132,
8332 = 693889, 6 + 93 + 8 + 89 = 142,
8332 = 693889, 6 + 93 + 88 + 9 = 142,
8332 = 693889, 69 + 38 + 89 = 142,
8332 = 693889, 6 + 938 + 8 + 9 = 312,
8332 = 693889, 69 + 3 + 889 = 312.

8332 = 693889 appears in the decimal expressions of π and e:
  π = 3.14159•••693889••• (from the 44267th digit).

Page of Squares : First Upload November 21, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

834

The smallest squares containing k 834's :
3583449 = 18932,
19583483481 = 1399412,
9183438348346564 = 958302582.

8342 = 695556, a square with 3 kinds of digits.

8342 = 253 + 593 + 783.

17514k + 159294k + 214338k + 304410k are squares for k = 1,2,3 (8342, 4053242, 2051890202).

Komachi equation: 8342 = 122 - 32 + 452 * 62 + 7892.

The 4-by-4 magic squares consisting of different squares with constant 834:

12 22102272
122252 72 42
172142182 52
202 32192 82
     
12 62112262
122252 42 72
172 22212102
202132162 32

8342 = 695556, 6 + 9 + 5 + 5 + 56 = 92,
8342 = 695556, 6 + 9 + 5 + 55 + 6 = 92,
8342 = 695556, 6 + 9 + 55 + 5 + 6 = 92,
8342 = 695556, 6 + 9 + 5 + 556 = 242,
8342 = 695556, 6 + 9 + 555 + 6 = 242,
8342 = 695556, 69 + 5556 = 752.

Page of Squares : First Upload November 21, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

835

The smallest squares containing k 835's :
83521 = 2892,
18358353049 = 1354932,
1835656835728356 = 428445662.

8354 = 4 8 6 1 2 2 7 0 0 6 25, and
  835 = 42 + 82 + 62 + 12 + 22 + 22 + 72 + 02 + 02 + 62 + 252.

22545k + 65130k + 213760k + 395790k are squares for k = 1,2,3 (8352, 4550752, 2684316252).

3-by-3 magic squares consisting of different squares with constant 8352:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 330, 767, 367, 690, 294, 750, 335, 150),(15, 90, 830, 258, 790, 81, 794, 255, 42),
(15, 90, 830, 470, 687, 66, 690, 466, 63),(15, 146, 822, 470, 678, 129, 690, 465, 70),
(15, 270, 790, 426, 682, 225, 718, 399, 150),(15, 270, 790, 570, 574, 207, 610, 543, 174),
(15, 470, 690, 570, 510, 335, 610, 465, 330),(66, 335, 762, 385, 690, 270, 738, 330, 209),
(70, 354, 753, 465, 610, 330, 690, 447, 146),(70, 354, 753, 465, 678, 146, 690, 335, 330),
(70, 390, 735, 465, 630, 290, 690, 385, 270),(74, 510, 657, 543, 470, 426, 630, 465, 290),
(81, 358, 750, 542, 594, 225, 630, 465, 290),(90, 218, 801, 255, 774, 182, 790, 225, 150),
(90, 270, 785, 510, 641, 162, 655, 462, 234),(90, 479, 678, 510, 570, 335, 655, 378, 354),
(97, 354, 750, 510, 570, 335, 654, 497, 150),(102, 465, 686, 561, 470, 402, 610, 510, 255),
(146, 447, 690, 465, 610, 330, 678, 354, 335),(178, 321, 750, 354, 722, 225, 735, 270, 290)

8352 = 697225, 6 + 9 + 7 + 2 + 25 = 72,
8352 = 697225, 6 + 9 + 7 + 22 + 5 = 72.

Page of Squares : First Upload November 21, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

836

The smallest squares containing k 836's :
8836 = 942,
966836836 = 310942,
183688362836224 = 135531682.

The squares which begin with 836 and end in 836 are
836138332836 = 9144062,   836482184836 = 9145942,   8360228656836 = 28914062,
8361315860836 = 28915942,   8363120312836 = 28919062,...

836 = (12 + 22 + 32 + ... + 5602) / (12 + 22 + 32 + ... + 592).

8362 = 698896, a palindromic square with just 3 kinds of digits.

8362 = 7 x 8 + 8 x 9 + 9 x 10 + 10 x 11 + ... + 127 x 128.

8362 = (62 + 8)(1262 + 8).

254k + 362k + 394k + 590k are squares for k = 1,2,3 (402, 8362, 181762).

8362 = 698896, 69 + 8 + 8 + 9 + 6 = 102.

(13 + 23 + ... + 433)(443 + 453 + ... + 993)(1003 + 1013 + ... + 8363) = 16079597614082.

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

837

The smallest squares containing k 837's :
837225 = 9152,
18837837001 = 1372512,
4883783798837476 = 698840742.

Komachi equation: 8372 = 122 + 32 + 42 * 52 * 62 * 72 - 82 * 92.

A cubic plynomial :
(X + 5122)(X + 8372)(X + 11042) = X3 + 14772X2 + 11649122 + 4731125762.

3-by-3 magic squares consisting of different squares with constant 8372:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(4, 413, 728, 448, 616, 347, 707, 388, 224),(5, 440, 712, 488, 580, 355, 680, 413, 260),
(6, 222, 807, 537, 618, 174, 642, 519, 138),(8, 91, 832, 572, 608, 61, 611, 568, 68),
(8, 236, 803, 344, 733, 212, 763, 328, 104),(8, 293, 784, 404, 688, 253, 733, 376, 148),
(16, 128, 827, 323, 764, 112, 772, 317, 64),(28, 356, 757, 587, 548, 236, 596, 523, 268),
(29, 412, 728, 548, 541, 328, 632, 488, 251),(40, 163, 820, 475, 680, 112, 688, 460, 125),
(42, 489, 678, 546, 498, 393, 633, 462, 294),(44, 232, 803, 547, 616, 148, 632, 517, 184),
(44, 373, 748, 523, 572, 316, 652, 484, 203),(57, 282, 786, 534, 618, 183, 642, 489, 222),
(64, 488, 677, 532, 547, 344, 643, 404, 352),(72, 477, 684, 576, 468, 387, 603, 504, 288),
(92, 181, 812, 292, 772, 139, 779, 268, 148),(92, 448, 701, 541, 568, 292, 632, 421, 352),
(102, 231, 798, 294, 762, 183, 777, 258, 174),(112, 293, 776, 524, 632, 163, 643, 464, 268),
(114, 327, 762, 498, 642, 201, 663, 426, 282),(128, 421, 712, 509, 608, 268, 652, 392, 349),
(138, 393, 726, 471, 642, 258, 678, 366, 327),(140, 412, 715, 580, 565, 212, 587, 460, 380),
(176, 412, 707, 443, 656, 272, 688, 317, 356) 

8372 = 700569, 7 + 0 + 0 + 5 + 69 = 92,
8372 = 700569, 70 + 0 + 5 + 69 = 122,
8372 = 700569, 7 + 0 + 0 + 569 = 242.

136392 = 3092 + 3102 + 3112 + ... + 8372.

Page of Squares : First Upload November 21, 2005 ; Last Revised July 6, 2010
by Yoshio Mimura, Kobe, Japan

838

The smallest squares containing k 838's :
138384 = 3722,
1838008384 = 428722,
838783889838756 = 289617662.

8382 is the 7th square which is the sum of 5 fifth powers : 15 + 35 + 55 + 115 + 145.

8382 = 702244, 74 + 04 + 224 + 44 + 44 = 4872,
8382 = 702244, 70 + 2 + 24 + 4 = 102,
8382 = 702244, 70 + 22 + 4 + 4 = 102.

Page of Squares : First Upload November 21, 2005 ; Last Revised September 21, 2006
by Yoshio Mimura, Kobe, Japan

839

The smallest squares containing k 839's :
683929 = 8272,
83968391529 = 2897732,
1118398398391296 = 334424642.

839 is the 4th prime for which the Legendre Symbol (a/p) = 1 for a = 1, 2, ..., 10.

8392 = 703921, a square with different digits.

3-by-3 magic squares consisting of different squares with constant 8392:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 147, 826, 259, 786, 138, 798, 254, 51),(6, 182, 819, 546, 621, 142, 637, 534, 114),
(6, 378, 749, 483, 614, 306, 686, 429, 222),(18, 306, 781, 506, 627, 234, 669, 466, 198),
(38, 291, 786, 579, 578, 186, 606, 534, 227),(42, 141, 826, 474, 686, 93, 691, 462, 114),
(45, 214, 810, 290, 765, 186, 786, 270, 115),(51, 486, 682, 578, 474, 381, 606, 493, 306),
(61, 318, 774, 522, 621, 214, 654, 466, 243),(66, 317, 774, 502, 606, 291, 669, 486, 142),
(70, 285, 786, 435, 686, 210, 714, 390, 205),(147, 254, 786, 394, 717, 186, 726, 354, 227),
(147, 394, 726, 574, 579, 198, 594, 462, 371),(178, 381, 726, 474, 654, 227, 669, 362, 354),
(186, 394, 717, 438, 669, 254, 691, 318, 354),(189, 506, 642, 574, 387, 474, 582, 546, 259)

8392 = 703921, 72 + 02 + 32 + 92 + 22 + 12 = 122,
8392 = 703921, 7 + 0 + 39 + 2 + 1 = 72,
8392 = 703921, 7 + 0 + 392 + 1 = 202.

Page of Squares : First Upload November 21, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan