Squares:920s

920

The smallest squares containing k 920's :
39204 = 198²,
9792092025 = 98955²,
592060920619204 = 24332302².

920²± 3 are primes.

920² = 846400 appears in the decimal expression of π:
π = 3.14159•••846400••• (from the 79090th digit).

Page of Squares : First Upload January 30, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

921

The smallest squares containing k 921's :
7921 = 89²,
1921857921 = 43839²,
64921759219216 = 8057404².

The squares which begin with 921 and end in 921 are
92166280921 = 303589², 921290905921 = 959839², 921429127921 = 959911²,
921770887921 = 960089², 921909145921 = 960161²,...

921² = 848241, a zigzag square.

(2² + 9)(4² + 9)(6² + 9)(7² + 9) = 921² + 9.

18^k + 282^k + 921^k + 1380^k are squares for k = 1,2,3 (51², 1683², 58581²).
6^k + 408^k + 690^k + 921^k are squares for k = 1,2,3 (45², 1221², 34317²).

3-by-3 magic squares consisting of different squares with constant 921²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(7, 176, 904, 496, 761, 152, 776, 488, 89),	(16, 268, 881, 463, 764, 224, 796, 439, 148),
(16, 359, 848, 544, 688, 281, 743, 496, 224),	(20, 175, 904, 604, 680, 145, 695, 596, 100),
(20, 271, 880, 329, 820, 260, 860, 320, 79),	(44, 128, 911, 244, 881, 112, 887, 236, 76),
(44, 337, 856, 649, 596, 268, 652, 616, 209),	(49, 224, 892, 304, 847, 196, 868, 284, 119),
(49, 356, 848, 512, 716, 271, 764, 457, 236),	(54, 387, 834, 477, 726, 306, 786, 414, 243),
(55, 460, 796, 596, 625, 320, 700, 496, 335),	(56, 167, 904, 391, 824, 128, 832, 376, 121),
(56, 431, 812, 623, 616, 284, 676, 532, 329),	(64, 356, 847, 572, 649, 316, 719, 548, 176),
(76, 247, 884, 559, 716, 152, 728, 524, 209),	(76, 457, 796, 568, 604, 401, 721, 524, 232),
(79, 496, 772, 548, 596, 439, 736, 497, 244),	(89, 328, 856, 632, 601, 296, 664, 616, 167),
(92, 364, 841, 401, 776, 292, 824, 337, 236),	(99, 246, 882, 378, 819, 186, 834, 342, 189),
(119, 524, 748, 596, 527, 464, 692, 544, 271),	(145, 296, 860, 440, 785, 196, 796, 380, 265),
(167, 376, 824, 616, 596, 337, 664, 593, 236),	(188, 544, 719, 631, 604, 292, 644, 433, 496),
(236, 593, 664, 628, 376, 559, 631, 596, 308)

921² = 848241, 8 + 48 + 24 + 1 = 9²,
921² = 848241, 848 + 241 = 33².

Page of Squares : First Upload January 30, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

922

The smallest squares containing k 922's :
99225 = 315²,
19792269225 = 140685²,
595729229229225 = 24407565².

(473 / 922)² = 0.263184579... (Komachic).

922² = 850084, 8² + 5² + 0² + 0² + 8² + 4² = 13².

922² = 850084, 8 + 5 + 0 + 0 + 8 + 4 = 5²,
922² = 850084, 8² + 5² + 0² + 0² + 8² + 4² = 13²,
922² = 850084, 85 + 0 + 0 + 84 = 13².

Page of Squares : First Upload January 30, 2006 ; Last Revised October 10, 2006
by Yoshio Mimura, Kobe, Japan

923

The smallest squares containing k 923's :
799236 = 894²,
5092392321 = 71361²,
192392379231364 = 13870558².

923 is the second integer which is the sum of a square and a prime in 12 ways:
2² + 919, 4² + 907, 6² + 887, 8² + 859, 10² + 823, 14² + 727, 18² + 599, 20² + 523, 22² + 439, 24² + 347, 28² + 139, 30² + 23.

923² = 121² + 402² + 822² : 228² + 204² + 121² = 329².

(1³ + 2³ + ... + 33³)(34³ + 35³ + ... + 714³)(715³ + 716³ + ... + 923³) = 48914966328228².

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

3-by-3 magic squares consisting of different squares with constant 923²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(14, 498, 777, 642, 553, 366, 663, 546, 338),	(39, 282, 878, 338, 822, 249, 858, 311, 138),
(39, 338, 858, 418, 759, 318, 822, 402, 121),	(39, 338, 858, 598, 663, 234, 702, 546, 247),
(39, 598, 702, 642, 522, 409, 662, 471, 438),	(42, 446, 807, 546, 663, 338, 743, 462, 294),
(57, 266, 882, 546, 702, 247, 742, 537, 114),	(58, 294, 873, 567, 678, 266, 726, 553, 138),
(90, 455, 798, 490, 702, 345, 777, 390, 310),	(94, 327, 858, 438, 774, 247, 807, 382, 234),
(102, 423, 814, 473, 726, 318, 786, 382, 297),	(105, 498, 770, 590, 630, 327, 702, 455, 390),
(114, 402, 823, 633, 634, 222, 662, 537, 354),	(130, 423, 810, 585, 590, 402, 702, 570, 185),
(231, 498, 742, 558, 679, 282, 698, 378, 471),	(234, 558, 697, 598, 633, 306, 663, 374, 522)

923² = 851929, 8² + 5² + 1² + 9² + 2² + 9² = 16².

Page of Squares : First Upload January 30, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

924

The smallest squares containing k 924's :
13924 = 118²,
8069249241 = 89829²,
392492402039241 = 19811421².

The squares which begin with 924 and end in 924 are
92487757924 = 304118², 924255349924 = 961382², 924709177924 = 961618²,
9240882573924 = 3039882², 9242317453924 = 3040118²,...

924² = 10³ + 42³ + 92³.

924² = 853776 , 853 + 77 - 6 = 924.

The integral triangle of sides 1617, 2425, 3944 (or 693, 11986, 12665) has square area 924².

924^k + 1078^k + 1617^k + 2310^k are squares for k = 1,2,3 (77², 3157², 136367²).

(1 + 2)(3 + 4 + ... + 30)(31 + 32 + ... + 46) = 924²,
(1 + 2 + ... + 32)(33 + 34 + ... + 65) = 924².

924² = 853776, 8 + 5 + 3 + 7 + 7 + 6 = 6²,
924² = 853776, 8 + 53 + 7 + 7 + 6 = 9²,
924² = 853776, 8 + 53 + 7 + 76 = 12²,
924² = 853776, 8 + 53 + 77 + 6 = 12².

Page of Squares : First Upload January 30, 2006 ; Last Revised September 30, 2011
by Yoshio Mimura, Kobe, Japan

925

The smallest squares containing k 925's :
925444 = 962²,
9259250625 = 96225²,
154925792592561 = 12446919².

925² = (2² + 1)(6² + 1)(68² + 1).

(388 / 925)² = 0.175946238... (Komachic).

1³ - 2³ + 3³ - 4³ + 5³ - ... - 924³ + 925³ = 19909².

301² + 302² + 303² + ... + 925² = 15975²,
204² + 205² + 206² + ... + 925² = 16169².

3-by-3 magic squares consisting of different squares with constant 925²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 259, 888, 300, 840, 245, 875, 288, 84),	(0, 285, 880, 555, 704, 228, 740, 528, 171),
(0, 300, 875, 555, 700, 240, 740, 525, 180),	(0, 520, 765, 555, 612, 416, 740, 459, 312),
(3, 96, 920, 600, 700, 75, 704, 597, 60),	(11, 360, 852, 648, 605, 264, 660, 600, 245),
(16, 213, 900, 387, 816, 200, 840, 380, 75),	(24, 93, 920, 380, 840, 75, 843, 376, 60),
(24, 268, 885, 632, 651, 180, 675, 600, 200),	(24, 515, 768, 565, 600, 420, 732, 480, 299),
(52, 264, 885, 480, 765, 200, 789, 448, 180),	(60, 205, 900, 612, 684, 115, 691, 588, 180),
(72, 420, 821, 605, 600, 360, 696, 565, 228),	(75, 200, 900, 504, 765, 128, 772, 480, 171),
(75, 380, 840, 600, 660, 245, 700, 525, 300),	(75, 444, 808, 600, 592, 381, 700, 555, 240),
(92, 420, 819, 525, 700, 300, 756, 435, 308),	(124, 360, 843, 600, 675, 200, 693, 520, 324),
(180, 525, 740, 556, 660, 333, 717, 380, 444),	(192, 515, 744, 619, 480, 492, 660, 600, 245)

925² = 855625, 8 + 5 + 5 + 6 + 25 = 7²,
925² = 855625, 85 + 5 + 6 + 25 = 11².

Page of Squares : First Upload January 30, 2006 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

926

The smallest squares containing k 926's :
292681 = 541²,
9269260729 = 96277²,
392692641792676 = 19816474².

926² = 857476, a zigzag square.

(471 / 926)² = 0.258713946... (Komachic).

926² = 857476, 8 + 5 + 7 + 4 + 76 = 10²,
926² = 857476, 8 + 5 + 74 + 7 + 6 = 10².

Page of Squares : First Upload January 30, 2006 ; Last Revised October 10, 2006
by Yoshio Mimura, Kobe, Japan

927

The smallest squares containing k 927's :
192721 = 439²,
392792761 = 19819²,
19279273927489 = 4390817².

927² = 6⁴ + 21⁴ + 24⁴ + 24⁴.

927² + 928² + 929² + ... + 5135² = 211853².

3-by-3 magic squares consisting of different squares with constant 927²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 218, 901, 341, 838, 202, 862, 331, 82),	(2, 485, 790, 590, 610, 373, 715, 502, 310),
(5, 230, 898, 610, 677, 170, 698, 590, 155),	(12, 303, 876, 372, 804, 273, 849, 348, 132),
(12, 456, 807, 519, 672, 372, 768, 447, 264),	(22, 134, 917, 526, 757, 98, 763, 518, 94),
(22, 229, 898, 502, 758, 181, 779, 482, 142),	(22, 322, 869, 581, 682, 238, 722, 539, 218),
(26, 362, 853, 418, 757, 334, 827, 394, 142),	(33, 264, 888, 552, 708, 231, 744, 537, 132),
(37, 386, 842, 538, 677, 334, 754, 502, 197),	(43, 226, 898, 394, 818, 187, 838, 373, 134),
(43, 358, 854, 434, 763, 298, 818, 386, 203),	(43, 554, 742, 622, 533, 434, 686, 518, 347),
(48, 420, 825, 600, 615, 348, 705, 552, 240),	(57, 516, 768, 636, 537, 408, 672, 552, 321),
(82, 379, 842, 491, 698, 362, 782, 478, 139),	(98, 290, 875, 475, 770, 202, 790, 427, 230),
(98, 293, 874, 347, 826, 238, 854, 302, 197),	(106, 373, 842, 523, 722, 254, 758, 446, 293),
(111, 348, 852, 588, 687, 204, 708, 516, 303),	(133, 238, 886, 314, 853, 182, 862, 274, 203),
(139, 538, 742, 638, 482, 469, 658, 581, 298),	(149, 358, 842, 562, 709, 202, 722, 478, 331),
(166, 587, 698, 622, 446, 523, 667, 562, 314),	(182, 494, 763, 587, 658, 286, 694, 427, 442),
(230, 523, 730, 565, 670, 302, 698, 370, 485)

927² = 859329, 8 + 5 + 9 + 3 + 2 + 9 = 6²,
927² = 859329, 8 + 59 + 3 + 2 + 9 = 9²,
927² = 859329, 859 + 32 + 9 = 30².

Page of Squares : First Upload January 30, 2006 ; Last Revised October 23, 2009
by Yoshio Mimura, Kobe, Japan

928

The smallest squares containing k 928's :
49284 = 222²,
2319289281 = 48159²,
40749289289289 = 6383517².

928² = 16³ + 56³ + 88³.

65² + 66² + 67² + ... + 928² = 16332².

928² = 861184, 8 + 6 + 1 + 1 + 84 = 10²,
928² = 861184, 86 + 1 + 1 + 8 + 4 = 10²,
928² = 861184, 8² + 6² + 1² + 18² + 4² = 21².

Page of Squares : First Upload January 30, 2006 ; Last Revised October 10, 2006
by Yoshio Mimura, Kobe, Japan

929

The smallest squares containing k 929's :
5929 = 77²,
36929929 = 6077²,
19296929051929 = 4392827².

The squares which begin with 929 and end in 929 are
9297394929 = 96423², 92919499929 = 304827², 92978035929 = 304923²,
929147549929 = 963923², 929444461929 = 964077²,...

929² = 863041, a square with different digits.

Komachi square sum : 929² = 14² + 38² + 76² + 925².

the square root of 929 is 30. 4 7 9 5 0 13 0 8 2 5 6 3 4 0 9 5 8 10 8 6 6 ...,
and 30² = 4² + 7² + 9² + 5² + 0² + 13² + 0² + 8² + 2² + 5² + 6² + 3² + 4² + 0² + 9² + 5² + 8² + 10² + 8² + 6² + 6².

3-by-3 magic squares consisting of different squares with constant 929²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(7, 96, 924, 444, 812, 81, 816, 441, 52),	(20, 465, 804, 615, 596, 360, 696, 540, 295),
(24, 153, 916, 601, 696, 132, 708, 596, 81),	(25, 396, 840, 504, 700, 345, 780, 465, 196),
(48, 396, 839, 596, 657, 276, 711, 524, 288),	(57, 144, 916, 384, 839, 108, 844, 372, 111),
(57, 304, 876, 564, 708, 209, 736, 519, 228),	(84, 273, 884, 312, 844, 231, 871, 276, 168),
(84, 592, 711, 623, 564, 396, 684, 441, 448),	(96, 196, 903, 252, 879, 164, 889, 228, 144),
(96, 385, 840, 560, 696, 255, 735, 480, 304),	(111, 452, 804, 636, 624, 263, 668, 519, 384),
(164, 552, 729, 636, 601, 312, 657, 444, 484)

929² = 863041, 8 + 6 + 30 + 4 + 1 = 7²,
929² = 863041, 86 + 30 + 4 + 1 = 11².

Page of Squares : First Upload January 30, 2006 ; Last Revised October 23, 2009
by Yoshio Mimura, Kobe, Japan