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940

The smallest squares containing k 940's :
9409 = 97²,
9409194001 = 97001²,
894004199409409 = 29899903².

Page of Squares : First Upload February 13, 2006 ; Last Revised October 13, 2006
by Yoshio Mimura, Kobe, Japan

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941

The smallest squares containing k 941's :
1094116 = 1046²,
13941941776 = 118076²,
1081941094194129 = 32892873².

3-by-3 magic squares consisting of different squares with constant 941²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 261, 904, 384, 824, 243, 859, 372, 96),	(13, 96, 936, 576, 741, 68, 744, 572, 69),
(36, 132, 931, 364, 861, 108, 867, 356, 84),	(36, 229, 912, 453, 804, 184, 824, 432, 141),
(40, 216, 915, 645, 660, 184, 684, 635, 120),	(69, 252, 904, 616, 696, 147, 708, 581, 216),
(76, 372, 861, 483, 756, 284, 804, 419, 252),	(84, 480, 805, 645, 616, 300, 680, 525, 384),
(96, 499, 792, 603, 576, 436, 716, 552, 261),	(99, 436, 828, 492, 684, 419, 796, 477, 156),
(176, 411, 828, 576, 708, 229, 723, 464, 384),	(216, 453, 796, 499, 744, 288, 768, 356, 411)

Page of Squares : First Upload February 13, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

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942

The smallest squares containing k 942's :
94249 = 307²,
19423439424 = 139368²,
1362942194294209 = 36918047².

(1³ + 2³ + ... + 552³)(553³ + 554³ + ... + 942³) = 63661901940².

1 / 942 = 0.0010615711252...,
and 1² + 06² + 15² + 7² + 1² + 1² + 25² + 2² = 942.

498^k + 942^k + 3138^k + 3522^k are squares for k = 1,2,3 (90², 4836², 274860²).

The 4-by-4 magic squares consisting of different squares with constant 942:

942² = 887364, 8 + 8 + 7 + 3 + 6 + 4 = 6²,
942² = 887364, 88 + 73 + 64 = 15²,
942² = 887364, 887 + 3 + 6 + 4 = 30².

Page of Squares : First Upload February 13, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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943

The smallest squares containing k 943's :
394384 = 628²,
18943943769 = 137637²,
999439431943744 = 31613912².

943² = 11² + 462² + 822² : 228² + 264² + 11² = 349².

3-by-3 magic squares consisting of different squares with constant 943²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 402, 853, 587, 666, 318, 738, 533, 246),	(27, 538, 774, 606, 603, 398, 722, 486, 363),
(42, 331, 882, 389, 798, 318, 858, 378, 101),	(42, 453, 826, 507, 686, 402, 794, 462, 213),
(43, 438, 834, 654, 587, 342, 678, 594, 277),	(66, 267, 902, 603, 682, 246, 722, 594, 123),
(70, 318, 885, 357, 830, 270, 870, 315, 182),	(90, 507, 790, 645, 610, 318, 682, 510, 405),
(110, 315, 882, 630, 682, 165, 693, 570, 290),	(123, 246, 902, 398, 837, 174, 846, 358, 213),
(123, 438, 826, 574, 693, 282, 738, 466, 357),	(133, 306, 882, 522, 763, 186, 774, 462, 277),
(258, 549, 722, 578, 678, 309, 699, 358, 522)

943² = 889249, 8 + 8 + 92 + 4 + 9 = 11².

Page of Squares : First Upload February 13, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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944

The smallest squares containing k 944's :
44944 = 212²,
11494412944 = 107212²,
165944944969444 = 12881962².

The squares which begin with 944 and end in 944 are
94425914944 = 307288²,   944224210944 = 971712²,   944371916944 = 971788²,
9441559034944 = 3072712²,   9442026092944 = 3072788²,...

(1³ + 2³ + ... + 604³)(605³ + 606³ + ... + 904³)(905³ + 906³ + ... + 944³) = 11891070281310000².

944² = 891136, 8 + 9 + 11 + 36 = 8²,
944² = 891136, 89 + 1 + 1 + 3 + 6 = 10²,
944² = 891136, 89 + 1136 = 35².

Page of Squares : First Upload February 13, 2006 ; Last Revised October 13, 2006
by Yoshio Mimura, Kobe, Japan

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945

The smallest squares containing k 945's :
379456 = 616²,
594579456 = 24384²,
94569459450489 = 9724683².

945² is the ninth square which is the sum of 5 fifth powers : 6⁵ + 6⁵ + 9⁵ + 9⁵ + 15⁵.

945² = 893025 : 8 + 930 + 2 + 5 = 945.

945² = 893025, a square with different digits.

945² = (4² + 5)(10² + 5)(20² + 5) = (4² - 1)(244² - 1).

Komachi equations:
945² = 12² * 3² * 4² * 5² / 6² * 7² / 8² * 9² = 12² * 3² * 45² * 6² * 7² / 8² / 9².

945² + 946² + 947² + ... + 11528² = 714462².

945² = 42³ + 66³ + 81³ = 33³ + 56³ + 88³.

(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + 22 + ... + 29) = 945²,
(1)(2 + 3 + ... + 43)(44 + 45 + ... + 61) = 945²,
(1 + 2)(3 + 4 + 5 + 6 + 7 )(8 + 9 + ... + 154) = 945².

3-by-3 magic squares consisting of different squares with constant 945²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 97, 940, 503, 796, 80, 800, 500, 55),	(4, 272, 905, 545, 740, 220, 772, 521, 160),
(8, 220, 919, 425, 820, 200, 844, 415, 92),	(9, 288, 900, 612, 684, 225, 720, 585, 180),
(17, 344, 880, 400, 800, 305, 856, 367, 160),	(20, 95, 940, 215, 916, 88, 920, 212, 41),
(20, 95, 940, 464, 820, 73, 823, 460, 64),	(20, 215, 920, 640, 680, 145, 695, 620, 160),
(20, 380, 865, 488, 745, 316, 809, 440, 212),	(20, 380, 865, 640, 631, 292, 695, 592, 244),
(20, 607, 724, 640, 524, 457, 695, 500, 400),	(30, 165, 930, 267, 894, 150, 906, 258, 75),
(30, 165, 930, 510, 786, 123, 795, 498, 114),	(30, 330, 885, 426, 795, 282, 843, 390, 174),
(30, 330, 885, 645, 654, 222, 690, 597, 246),	(30, 510, 795, 645, 570, 390, 690, 555, 330),
(31, 92, 940, 260, 905, 80, 908, 256, 55),	(40, 151, 932, 580, 740, 95, 745, 568, 124),
(40, 248, 911, 655, 664, 152, 680, 625, 200),	(40, 580, 745, 655, 520, 440, 680, 535, 380),
(41, 520, 788, 620, 580, 415, 712, 535, 316),	(55, 340, 880, 500, 737, 316, 800, 484, 137),
(55, 340, 880, 628, 671, 220, 704, 572, 265),	(66, 363, 870, 555, 690, 330, 762, 534, 165),
(68, 500, 799, 535, 680, 380, 776, 425, 332),	(70, 245, 910, 350, 854, 203, 875, 322, 154),
(70, 350, 875, 490, 763, 266, 805, 434, 238),	(72, 279, 900, 396, 828, 225, 855, 360, 180),
(75, 438, 834, 570, 645, 390, 750, 534, 213),	(75, 570, 750, 654, 510, 453, 678, 555, 354),
(80, 260, 905, 316, 863, 220, 887, 284, 160),	(80, 335, 880, 520, 752, 239, 785, 464, 248),
(80, 335, 880, 608, 656, 305, 719, 592, 160),	(80, 520, 785, 568, 655, 376, 751, 440, 368),
(95, 244, 908, 460, 808, 169, 820, 425, 200),	(95, 380, 860, 548, 680, 361, 764, 535, 152),
(95, 380, 860, 580, 704, 247, 740, 503, 304),	(95, 460, 820, 580, 620, 415, 740, 545, 220),
(104, 535, 772, 572, 580, 479, 745, 520, 260),	(145, 424, 832, 640, 568, 401, 680, 625, 200),
(148, 311, 880, 415, 820, 220, 836, 352, 265),	(150, 498, 789, 555, 690, 330, 750, 411, 402),
(150, 555, 750, 618, 510, 501, 699, 570, 282),	(160, 436, 823, 620, 673, 236, 695, 500, 400),
(160, 620, 695, 649, 580, 368, 668, 415, 524),	(215, 536, 748, 620, 652, 289, 680, 425, 500),
(215, 568, 724, 620, 460, 545, 680, 599, 268)

The 4-by-4 magic squares consisting of different squares with constant 945:

945² = 893025, 8 + 9 + 302 + 5 = 18²,
945² = 893025, 89 + 30 + 25 = 12²,
945² = 893025, 893 + 0 + 2 + 5 = 30².

945² = 893025 appears in the decimal expression of e:
  e = 2.71828•••893025••• (from the 59442nd digit)

Page of Squares : First Upload February 13, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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946

The smallest squares containing k 946's :
946729 = 973²,
3694694656 = 60784²,
2946946344494656 = 54285784².

946² = 894916, a zigzag square.

946² = 402² + 424² + 744² : 447² + 424² + 204² = 649².

946² = 51³ + 72³ + 73³.

Page of Squares : First Upload February 13, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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947

The smallest squares containing k 947's :
329476 = 574²,
2947947025 = 54295²,
394700947194724 = 19867082².

947² = 896809, a zigzag square.

Komachi square sum : 947² = 2² + 7² + 8² + 156² + 934².

1 / 947 = 0.00 1 0 5 5 9 6 6 2 0 9 0 8 1 3 0 9 3 9 8 0 9 9 2 6 0 8 2 3 6 5 3,
and the sum of the squares of its digits is 947.

3-by-3 magic squares consisting of different squares with constant 947²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 97, 942, 657, 678, 74, 682, 654, 63),	(15, 178, 930, 578, 735, 150, 750, 570, 97),
(15, 470, 822, 522, 690, 385, 790, 447, 270),	(18, 574, 753, 666, 543, 398, 673, 522, 414),
(38, 273, 906, 447, 794, 258, 834, 438, 97),	(38, 399, 858, 561, 682, 342, 762, 522, 209),
(38, 447, 834, 561, 718, 258, 762, 426, 367),	(63, 466, 822, 542, 657, 414, 774, 498, 223),
(66, 182, 927, 223, 906, 162, 918, 207, 106),	(70, 465, 822, 570, 678, 335, 753, 470, 330),
(90, 447, 830, 655, 570, 378, 678, 610, 255),	(102, 286, 897, 546, 753, 178, 767, 498, 246),
(102, 498, 799, 609, 578, 438, 718, 561, 258),	(146, 447, 822, 543, 718, 294, 762, 426, 367),
(162, 591, 722, 641, 462, 522, 678, 578, 321),	(207, 398, 834, 466, 783, 258, 798, 354, 367)

947² = 896809, 8 + 96 + 8 + 0 + 9 = 11².

Page of Squares : First Upload February 13, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

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948

The smallest squares containing k 948's :
94864 = 308²,
948948025 = 30805²,
859489948409481 = 29317059².

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

948² + 078² = 904788.

948² = 898704, 8 + 9 + 8 + 7 + 0 + 4 = 6²,
948² = 898704, 8 + 9 + 8 + 704 = 27².

Page of Squares : First Upload February 13, 2006 ; Last Revised October 13, 2006
by Yoshio Mimura, Kobe, Japan

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949

The smallest squares containing k 949's :
894916 = 946²,
68794994944 = 262288²,
1949294994974041 = 44150821².

16817² = 197² + 198² + 199² + ... + 949².

3-by-3 magic squares consisting of different squares with constant 949²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 365, 876, 624, 660, 275, 715, 576, 240),	(13, 156, 936, 216, 912, 149, 924, 211, 48),
(13, 156, 936, 504, 792, 139, 804, 499, 72),	(21, 288, 904, 456, 796, 243, 832, 429, 156),
(21, 312, 896, 392, 819, 276, 864, 364, 147),	(27, 364, 876, 516, 741, 292, 796, 468, 219),
(48, 309, 896, 621, 688, 204, 716, 576, 237),	(51, 148, 936, 468, 819, 104, 824, 456, 117),
(68, 141, 936, 264, 904, 117, 909, 252, 104),	(76, 240, 915, 435, 824, 180, 840, 405, 176),
(76, 357, 876, 411, 804, 292, 852, 356, 219),	(84, 472, 819, 616, 651, 312, 717, 504, 364),
(104, 468, 819, 576, 621, 428, 747, 544, 216),	(141, 572, 744, 616, 624, 363, 708, 429, 464),
(156, 429, 832, 464, 768, 309, 813, 356, 336),	(156, 572, 741, 603, 636, 364, 716, 411, 468),
(211, 516, 768, 636, 653, 264, 672, 456, 491)

949² = 900601, 9 + 0 + 0 + 6 + 0 + 1 = 4²,
949² = 900601, 900 + 60 + 1 = 31².

Page of Squares : First Upload February 13, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

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