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790 - 799

790

The smallest squares containing k 790's :
790321 = 8892,
790790641 = 281212,
790790359790025 = 281209952.

103490k + 117710k + 131930k + 270970k are squares for k = 1,2,3 (7902, 3397002, 1578973002).

(13 + 23 + ... + 5523)(5533 + 543 + ... + 7903) = 416108190122.

7902 = 624100 appears in the decimal expressions of π and e:
  π = 3.14159•••624100••• (from the 65891st digit),
  e = 2.71828•••62411••• (from the 102834 digit)

Page of Squares : First Upload October 24, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

791

The smallest squares containing k 791's :
579121 = 7612,
57779179129 = 2403732,
257917914791329 = 160598232.

3-by-3 magic squares consisting of different squares with constant 7912:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 314, 726, 414, 618, 269, 674, 381, 162),(6, 163, 774, 422, 654, 141, 669, 414, 82),
(6, 282, 739, 334, 669, 258, 717, 314, 114),(18, 366, 701, 531, 514, 282, 586, 477, 234),
(19, 162, 774, 522, 579, 134, 594, 514, 93),(28, 231, 756, 504, 588, 161, 609, 476, 168),
(51, 206, 762, 258, 726, 179, 746, 237, 114),(51, 442, 654, 486, 534, 323, 622, 381, 306),
(66, 190, 765, 390, 675, 134, 685, 366, 150),(66, 314, 723, 498, 579, 206, 611, 438, 246),
(90, 291, 730, 541, 510, 270, 570, 530, 141),(93, 174, 766, 246, 739, 138, 746, 222, 141),
(99, 402, 674, 522, 541, 246, 586, 414, 333),(118, 291, 726, 429, 638, 186, 654, 366, 253),
(125, 366,690, 534, 550, 195, 570, 435, 334),(134, 306, 717, 339, 678, 226, 702, 269, 246),
(174, 458,621, 534, 531, 242, 557, 366, 426),(179, 366, 678, 426, 627, 226, 642, 314, 339)

7912 = 625681, 6 + 2 + 5 + 6 + 81 = 102,
7912 = 625681, 63 + 23 + 563 + 813 = 8412,
7912 = 625681, 6 + 25 + 68 + 1 = 102.

7912 = 625681 appears in the decimal expression of e:
  e = 2.71828•••625681••• (from the 63852nd digit)

Page of Squares : First Upload October 24, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

792

The smallest squares containing k 792's :
7921 = 892,
47927929 = 69232,
224792357927929 = 149930772.

7922 = 627264, a zigzag square.

7922 = (42 + 8)(52 + 8)(282 + 8) = (52 + 8)(82 + 8)(162 + 8).

2973 + 7923 = 228692.

7922 = 627264, 6 + 2 + 7 + 2 + 64 = 92,
7922 = 627264, 62 + 7 + 2 + 6 + 4 = 92,
7922 = 627264, 6 + 2 + 72 + 64 = 122,
7922 = 627264, 62 + 72 + 6 + 4 = 122.

Kaprekar : 7922 = 627264, and 62 + 726 + 4 = 792.

A + B, A + C, A + D, B + C, B + D, and C + D are squares
for A = 792, B = 1512, C = 2457, and D = 6952.

Komachi equation: 7922 = 92 * 872 + 62 * 52 * 42 - 32 / 22 * 102.

(1 + 2 + ... + 11)(12 + 13 + ... + 20)(21 + 22 + 23) = 7922.

7922 = 627264 appears in the decimal expression of e:
  e = 2.71828•••627264••• (from the 87299th digit)

Page of Squares : First Upload October 24, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

793

The smallest squares containing k 793's :
207936 = 4562,
7930793025 = 890552,
272977937937936 = 165220442.

7932 = 2312 + 3522 + 6722 : 2762 + 2532 + 1322 = 3972.

793k + 1417k + 2977k + 11713k are squares for k = 1,2,3 (1302, 121942, 12793302).

3-by-3 magic squares consisting of different squares with constant 7932:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 143, 780, 305, 720, 132, 732, 300, 55),(8, 396, 687, 444, 567, 332, 657, 388, 216),
(15, 332, 720, 468, 585, 260, 640, 420, 207),(28, 228, 759, 528, 561, 188, 591, 512, 132),
(39, 312, 728, 552, 512, 249, 568, 519, 192),(48, 207, 764, 244, 732, 183, 753, 224, 108),
(48, 456, 647, 487, 528, 336, 624, 377, 312),(60, 440, 657, 532, 465, 360, 585, 468, 260),
(73, 228, 756, 468, 624, 143, 636, 433, 192),(84, 303, 728, 548, 504, 273, 567, 532, 156),
(108, 521, 588, 552, 372, 431, 559, 468, 312),(132, 393, 676, 424, 612, 273, 657, 316, 312),
(136, 273, 732, 372, 676, 183, 687, 312, 244) 

7932 = 46 + 66 + 66 + 96.

Page of Squares : First Upload March 6, 2006; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

794

The smallest squares containing k 794's :
379456 = 6162,
10794794404 = 1038982,
794794558794201 = 281921012.

7942 = 630436, 6 + 3 + 0 + 4 + 36 = 72,
7942 = 630436, 6 + 30 + 4 + 3 + 6 = 72.

Komachi Fraction : 5673924 / 81 = (794 / 3)2.

Page of Squares : First Upload October 24, 2005 ; Last Revised September 14, 2006
by Yoshio Mimura, Kobe, Japan

795

The smallest squares containing k 795's :
17956 = 1342,
21795797956 = 1476342,
1059795795795025 = 325545052.

7952 = 872 + 882 + 892 + ... + 1362.

Komachi Square Sum : 7952 = 382 + 422 + 562 + 7912 = 22 + 62 + 182 + 352 + 7942
  = 22 + 82 + 152 + 362 + 7942 = 43 + 83 + 93 + 153 + 623 + 733.

3-by-3 magic squares consisting of different squares with constant 7952:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 420, 675, 477, 540, 336, 636, 405, 252),(2, 89, 790, 505, 610, 70, 614, 502, 55),
(2, 230, 761, 386, 665, 202, 695, 370, 110),(7, 490, 626, 550, 455, 350, 574, 430, 343),
(12, 216, 765, 405, 660, 180, 684, 387, 120),(23, 86, 790, 314, 727, 70, 730, 310, 55),
(34, 313, 730, 350, 650, 295, 713, 334, 110),(50, 170, 775, 281, 730, 142, 742, 265, 106),
(50, 247, 754, 455, 610, 230, 650, 446, 103),(50, 329, 722, 425, 622, 254, 670, 370, 215),
(50, 425, 670, 518, 526, 295, 601, 418, 310),(55, 154, 778, 310, 722, 121, 730, 295, 110),
(55, 190, 770, 530, 583, 106, 590, 506, 167),(70, 185, 770, 430, 658, 119, 665, 406, 158),
(70, 274, 743, 505, 590, 170, 610, 457, 226),(70, 430, 665, 505, 490, 370, 610, 455, 230),
(71, 422, 670, 490, 505, 370, 622, 446, 215),(89, 302, 730, 370, 665, 230, 698, 314, 215),
(93, 324, 720, 540, 555, 180, 576, 468, 285),(106, 215, 758, 265, 730, 170, 742, 230, 169),
(106, 490, 617, 530, 505, 310, 583, 370, 394),(110, 295, 730, 370, 670, 215, 695, 310, 230),
(134, 362, 695, 455, 610, 230, 638, 359, 310),(167, 350, 694, 394, 650, 233, 670, 295, 310),
(170, 425, 650, 505, 566, 238, 590, 362, 391) 

7952 = 632025, 6 + 3 + 2 + 0 + 25 = 62,
7952 = 632025, 6 + 3 + 20 + 2 + 5 = 62,
7952 = 632025, 62 + 322 + 02 + 22 + 52 = 332,
7952 = 632025, 63 + 33 + 23 + 03 + 253 = 1262.

Page of Squares : First Upload October 24, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

796

The smallest squares containing k 796's :
45796 = 2142,
796029796 = 282142,
52479679649796 = 72442862.

The squares which begin with 796 and end in 796 are
796029796 = 282142,   79644741796 = 2822142,   79685385796 = 2822862,
796045821796 = 8922142,   796174305796 = 8922862,...

1 / 796 = 0.001256281407..., 12 + 252 + 62 + 22 + 82 + 12 + 42 + 072 = 796.

7962 = 633616, a square consisting of just 3 kinds of digits.

7962 = 633616, 6 + 3 + 3 + 6 + 1 + 6 = 52.

Page of Squares : First Upload October 24, 2005 ; Last Revised September 14, 2006
by Yoshio Mimura, Kobe, Japan

797

The smallest squares containing k 797's :
179776 = 4242,
79707970276 = 2823262,
734797797979716 = 271071542.

7972 = 635209, a square with different digits.

7972 = 635209, 6 + 3 + 5 + 2 + 0 + 9 = 52,
7972 = 635209, 65 + 35 + 55 + 25 + 05 + 95 = 2652.

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

(13 + 23 + ... + 6653)(6663 + 6673 + ... + 7973) = 505399494602

3-by-3 magic squares consisting of different squares with constant 7972:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 264, 752 464, 612, 213, 648, 437, 156),(4, 213, 768, 528, 576, 157, 597, 508, 144),
(4, 348, 717, 492, 563, 276, 627, 444, 212),(12, 333, 724, 563, 516, 228, 564, 508, 243),
(21, 212, 768, 428, 651, 168, 672, 408, 131),(32, 276, 747, 387, 648, 256, 696, 373, 108),
(36, 392, 693, 427, 576, 348, 672, 387, 184),(112, 483, 624, 516, 432, 427, 597, 464, 252),
(117, 392, 684, 492, 504, 373, 616, 477, 168) 

Page of Squares : First Upload October 24, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

798

The smallest squares containing k 798's :
279841 = 5292,
6597987984 = 812282,
10279897987984 = 32062282.

7982 = (42 + 3)(122 + 3)(152 + 3).

7982 = 636804, 6 + 3 + 68 + 0 + 4 = 92,
7982 = 636804, 63 + 6 + 8 + 0 + 4 = 92.

Komachi Square Sum : 7982 = 52 + 872 + 4912 + 6232.

The 4-by-4 magic squares consisting of different squares with constant 798:

02 52172222
102 12212162
132242 22 72
232142 82 32
      
12 62192202
102232122 52
112 82172182
242132 22 72
      
22 32162232
52222152 82
122172142132
252 42112 62

7982 + 7992 + 8002 + ... + 23112 = 628312.

7982 + 7992 + 8002 + ... + 2497982 = 720815482.

Page of Squares : First Upload October 24, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

799

The smallest squares containing k 799's :
799236 = 8942,
27997990276 = 1673262,
1719799799079961 = 414704692.

7992 = 638401, a square with diffrent digits.

7992 = 2342 + 3542 + 6772 : 7762 + 4532 + 4322 = 9972,
7992 = 2942 + 3582 + 6512 : 1562 + 8532 + 4922 = 9972.

92684k + 132634k + 151810k + 261273k are squares for k = 1,2,3 (7992, 3427712, 1564082452).

3-by-3 magic squares consisting of different squares with constant 7992:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 126, 789, 459, 646, 102, 654, 453, 74),(11, 198, 774, 306, 714,187, 738, 299, 66),
(21, 142, 786, 506, 606, 123, 618, 501, 74),(34, 357, 714, 462, 574, 309, 651, 426, 182),
(42, 331, 726, 366, 654, 277, 709, 318, 186),(54, 133, 786, 318, 726, 101, 731, 306, 102),
(54, 394, 693, 549, 522, 254, 578, 459, 306),(66, 411, 682, 507, 506, 354, 614, 462, 219),
(102, 306, 731, 354, 677, 234, 709, 294, 222),(102, 459, 646, 501, 542, 306, 614, 366, 357),
(110, 450, 651, 549, 430, 390, 570, 501, 250),(126, 219, 758, 254, 738, 171, 747, 214, 186),
(138, 331, 714, 486, 606, 187, 619, 402, 306),(214, 459, 618, 486, 578, 261, 597, 306, 434)

7992 = 638401, 6 + 38 + 4 + 0 + 1 = 72,
7992 = 638401, 63 + 8401 = 922.

7992 + 8002 + 8012 + ... + 30072 = 943292.

Page of Squares : First Upload October 24, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan