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930

The smallest squares containing k 930's :
93025 = 305²,
7930793025 = 89055²,
6859308930693025 = 82820945².

930² = 11³ + 73³ + 78³.

930² = 18*19*20 + 19*20*21 + 20*21*22 + 21*22*23 + ... + 42*43*44.

The 4-by-4 magic squares consisting of different squares with constant 930:

Page of Squares : First Upload February 6, 2006 ; Last Revised October 26, 2013
by Yoshio Mimura, Kobe, Japan

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931

The square root of 931 is 30.512..., and 30 = 5² + 1² + 2².

The smallest squares containing k 931's :
393129 = 627²,
2931789316 = 54146²,
5931931598293156 = 77019034².

931² = 5³ + 21³ + 95³.

3-by-3 magic squares consisting of different squares with constant 931²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 177, 914, 382, 834, 159, 849, 374, 78),	(6, 450, 815, 625, 606, 330, 690, 545, 306),
(21, 252, 896, 644, 651, 168, 672, 616, 189),	(22, 306, 879, 474, 753, 274, 801, 454, 138),
(31, 258, 894, 498, 751, 234, 786, 486, 113),	(31, 534, 762, 582, 606, 401, 726, 463, 354),
(33, 174, 914, 606, 698, 111, 706, 591, 138),	(50, 255, 894, 606, 670, 225, 705, 594, 130),
(54, 302, 879, 626, 639, 258, 687, 606, 166),	(66, 383, 846, 642, 594, 319, 671, 606, 222),
(86, 234, 897, 351, 842, 186, 858, 321, 166),	(86, 561, 738, 654, 562, 351, 657, 486, 446),
(102, 239, 894, 401, 822, 174, 834, 366, 193),	(122, 369, 846, 639, 582, 346, 666, 626, 177),
(129, 482, 786, 558, 591, 454, 734, 534, 207),	(159, 438, 806, 486, 734, 303, 778, 369, 354),
(162, 346, 849, 446, 783, 234, 801, 366, 302),	(174, 382, 831, 417, 786, 274, 814, 321, 318),
(186, 463, 786, 618, 654, 239, 671, 474, 438)

931² = 866761, 86 + 6 + 76 + 1 = 13².

931² = 866761 appears in the decimal expression of π:
  π = 3.14159•••866761••• (from the 98847th digit).

Page of Squares : First Upload February 6, 2006 ; Last Revised October 23, 2009
by Yoshio Mimura, Kobe, Japan

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932

The smallest squares containing k 932's :
19321 = 139²,
3932293264 = 62708²,
93259329325569 = 9657087².

932² = 868624, a square with even digits.

932² = 868624, 86 + 86 + 24 = 14².

932² = 868624 appears in the decimal expression of e:
  e = 2.71828•••868624••• (from the 71944th digit).

Page of Squares : First Upload February 6, 2006 ; Last Revised October 10, 2006
by Yoshio Mimura, Kobe, Japan

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933

The smallest squares containing k 933's :
933156 = 966²,
293393388964 = 541658²,
933193317493369 = 30548213².

933² = 870489 is an exchangeable square (898704 = 948²).

933² = 311² + 622² + 622² : 226² + 226² + 113² = 339².

210^k + 241^k + 538^k + 692^k are squares for k = 1,2,3 (41², 933², 22591²).

Komachi equation: 933² = 9² + 876² - 5² + 4² + 321².

3-by-3 magic squares consisting of different squares with constant 933²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(8, 136, 923, 643, 668, 104, 676, 637, 88),	(8, 155, 920, 445, 808, 140, 820, 440, 67),
(8, 389, 848, 428, 752, 349, 829, 392, 172),	(18, 234, 903, 441, 798, 198, 822, 423, 126),
(18, 393, 846, 582, 666, 297, 729, 522, 258),	(29, 272, 892, 388, 808, 259, 848, 379, 88),
(32, 413, 836, 517, 704, 328, 776, 452, 253),	(32, 484, 797, 536, 643, 412, 763, 472, 256),
(43, 412, 836, 556, 683, 308, 748, 484, 277),	(66, 297, 882, 567, 714, 198, 738, 522, 231),
(67, 392, 844, 652, 584, 323, 664, 613, 232),	(76, 323, 872, 547, 692, 304, 752, 536, 133),
(85, 260, 892, 620, 683, 140, 692, 580, 235),	(92, 452, 811, 491, 668, 428, 788, 469, 172),
(92, 619, 692, 652, 452, 491, 661, 532, 388),	(104, 253, 892, 512, 764, 157, 773, 472, 224),
(133, 488, 784, 616, 637, 292, 688, 476, 413),	(153, 414, 822, 498, 738, 279, 774, 393, 342),
(157, 304, 868, 356, 832, 227, 848, 293, 256),	(160, 517, 760, 605, 640, 308, 692, 440, 445),
(162, 582, 711, 633, 594, 342, 666, 423, 498),	(172, 547, 736, 584, 512, 517, 707, 556, 248),
(176, 347, 848, 452, 784, 227, 797, 368, 316),	(221, 448, 788, 512, 731, 272, 748, 368, 419),
(224, 488, 763, 532, 707, 296, 733, 364, 448)

The 4-by-4 magic square consisting of different squares with constant 933:

933² = 870489, 8 + 7 + 0 + 4 + 8 + 9 = 6²,
933² = 870489, 87 + 0 + 48 + 9 = 12²,
933² = 870489, 87 + 0 + 489 = 24²,
933² = 870489, 8 + 704 + 8 + 9 = 27².

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

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934

The smallest squares containing k 934's :
659344 = 812²,
9348569344 = 96688²,
5293493449934404 = 72756398².

934² = 872356, a square with different digits.

(1² + 2² + ... + 399²) + (1² + 2² + ... + 909²) = (1² + 2² + ... 934²).

934² = 872356, 8 + 7 + 23 + 5 + 6 = 7²,
934² = 872356, 8 + 72 + 35 + 6 = 11²,
934² = 872356, 87 + 23 + 5 + 6 = 11²,
934² = 872356, 8 + 7 + 235 + 6 = 16².

Page of Squares : First Upload February 6, 2006 ; Last Revised October 10, 2006
by Yoshio Mimura, Kobe, Japan

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935

The smallest squares containing k 935's :
935089 = 967²,
23935493521 = 154711²,
4935719359355625 = 70254675².

36465^k + 108460^k + 241230^k + 488070^k are squares for k = 1,2,3 (935², 556325², 362803375²).

3-by-3 magic squares consisting of different squares with constant 935²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 440, 825, 561, 660, 352, 748, 495, 264),	(6, 350, 867, 467, 750, 306, 810, 435, 170),
(19, 342, 870, 558, 694, 285, 750, 525, 190),	(30, 190, 915, 306, 867, 170, 883, 294, 90),
(30, 190, 915, 541, 750, 138, 762, 525, 134),	(30, 285, 890, 397, 810, 246, 846, 370, 147),
(30, 285, 890, 618, 674, 195, 701, 582, 210),	(35, 90, 930, 174, 915, 82, 918, 170, 51),
(35, 486, 798, 630, 602, 339, 690, 525, 350),	(45, 210, 910, 602, 690, 189, 714, 595, 102),
(51, 170, 918, 450, 810, 125, 818, 435, 126),	(51, 282, 890, 510, 755, 210, 782, 474, 195),
(62, 210, 909, 534, 755, 138, 765, 510, 170),	(90, 195, 910, 390, 838, 141, 845, 366, 162),
(90, 390, 845, 442, 765, 306, 819, 370, 258),	(90, 413, 834, 530, 666, 387, 765, 510, 170),
(90, 530, 765, 630, 531, 442, 685, 558, 306),	(102, 350, 861, 595, 690, 210, 714, 525, 298),
(114, 323, 870, 573, 714, 190, 730, 510, 285),	(125, 378, 846, 450, 771, 278, 810, 370, 285),
(163, 390, 834, 534, 730, 237, 750, 435, 350),	(170, 435, 810, 510, 730, 285, 765, 390, 370),
(170, 477, 786, 510, 714, 323, 765, 370, 390)

935² = 874225, 8 + 7 + 42 + 2 + 5 = 8²,
935² = 874225, 87 + 4 + 2 + 2 + 5 = 10²,
935² = 874225, 8³ + 7³ + 42³ + 2³ + 5³ = 274².

Page of Squares : First Upload February 6, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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936

The smallest squares containing k 936's :
1936 = 44²,
5593693681 = 74791²,
903936936015936 = 30065544².

The squares which begin with 936 and end in 936 are
93609073936 = 305956²,   93662929936 = 306044²,   936141391936 = 967544²,
936938817936 = 967956²,   9360271015936 = 3059456²,...

936² = 312² + 624² + 624² : 426² + 426² + 213² = 639².

10³ + 936 = 44², 10³ - 936 = 8².

936² + 937² + 938² + ... + 58559² = 8181530².

(1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + ... + 106) = 936².

936² = (1³ + 8)(46³ + 8).

The square root of 936 is 30. 5 9 4 11 7 0 8 15 5 6 7 0 8 9 8 ...,
and 30² = 5² + 9² + 4² + 11² + 7² + 0² + 8² + 15² + 5² + 6² + 7² + 0² + 8² + 9² + 8².

936² = 876096, 8 + 7 + 6 + 0 + 9 + 6 = 6².

936² = 876096 appears in the decimal expression of e:
  e = 2.71828•••876096••• (from the 108506th digit).

Page of Squares : First Upload February 6, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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937

The smallest squares containing k 937's :
293764 = 542²,
937829376 = 30624²,
288893793759376 = 16996876².

937² = 877969, a square every digit of which is greater than 5.

937² = 877969, 877 - 9 + 69 = 937.

937² = 9³ + 36³ + 94³.

937² + 938² + 939² + ... + 8457² = 448753².

3-by-3 magic squares consisting of different squares with constant 937²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 136, 927, 423, 828, 116, 836, 417, 72),	(12, 231, 908, 609, 692, 168, 712, 588, 159),
(12, 415, 840, 660, 600, 287, 665, 588, 300),	(12, 476, 807, 513, 672, 404, 784, 447, 252),
(28, 432, 831, 588, 639, 352, 729, 532, 252),	(33, 584, 732, 648, 543, 404, 676, 492, 423),
(39, 228, 908, 388, 831, 192, 852, 368, 129),	(80, 513, 780, 612, 620, 345, 705, 480, 388),
(96, 577, 732, 612, 516, 487, 703, 528, 324),	(168, 287, 876, 324, 852, 217, 863, 264, 252),
(172, 444, 807, 543, 712, 276, 744, 417, 388)

937² = 877969, 8 + 7 + 7 + 9 + 69 = 10².

Page of Squares : First Upload February 6, 2006 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

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938

The smallest squares containing k 938's :
693889 = 833²,
93893829241 = 306421²,
593893829383969 = 24369937².

938² + 939² + 940² + ... + 2724² = 80415²,
938² + 939² + 940² + ... + 293618² = 91857472².

938^k + 1274^k + 3430^k + 6902^k are squares for k = 1,2,3 (112², 7868², 609952²).

The 4-by-4 magic square consisting of different squares with constant 938:

938² = 879844, 8 + 7 + 98 + 4 + 4 = 11².

Page of Squares : First Upload February 6, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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939

The smallest squares containing k 939's :
1179396 = 1086²,
5939939041 = 77071²,
2939939939191369 = 54221213².

Komachi square sum : 939² = 15² + 74² + 86² + 932².

1 / 939 = 0.001064962726...,
and 1² + 06² + 4² + 9² + 6² + 27² + 2² + 6² = 939.

3-by-3 magic squares consisting of different squares with constant 939²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 311, 886, 506, 746, 263, 791, 478, 166),	(10, 439, 830, 550, 670, 361, 761, 490, 250),
(14, 250, 905, 530, 745, 214, 775, 514, 130),	(14, 302, 889, 343, 826, 286, 874, 329, 98),
(23, 94, 934, 154, 922, 89, 926, 151, 38),	(23, 214, 914, 446, 802, 199, 826, 439, 82),
(25, 310, 886, 614, 665, 250, 710, 586, 185,	(38, 361, 866, 569, 698, 266, 746, 514, 247),
(58, 346, 871, 646, 647, 214, 679, 586, 278),	(74, 137, 926, 178, 914, 121, 919, 166, 98),
(74, 409, 842, 446, 758, 329, 823, 374, 254),	(86, 374, 857, 569, 662, 346, 742, 551, 166),
(86, 553, 754, 599, 614, 382, 718, 446, 409),	(89, 298, 886, 406, 814, 233, 842, 361, 206),
(89, 298, 886, 430, 814, 185, 830, 361, 250),	(89, 430, 830, 530, 710, 311, 770, 439, 310),
(94, 338, 871, 494, 761, 242, 793, 434, 254),	(94, 542, 761, 647, 514, 446, 674, 569, 322),
(121, 406, 838, 634, 583, 374, 682, 614, 199),	(151, 478, 794, 586, 574, 457, 718, 569, 206),
(154, 482, 791, 574, 679, 302, 727, 434, 406),	(156, 333, 864, 396, 816, 243, 837, 324, 276),
(166, 473, 794, 649, 634, 242, 658, 506, 439),	(168, 531, 756, 621, 504, 492, 684, 588, 261)

939² = 881721, 8 + 8 + 17 + 2 + 1 = 6².

939² = 881721 appears in the decimal expression of e:
  e = 2.71828•••881721••• (from the 61015th digit).

Page of Squares : First Upload February 6, 2006 ; Last Revised October 23, 2009
by Yoshio Mimura, Kobe, Japan

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