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750 - 759

750

The smallest squares containing k 750's :
75076 = 2742,
9750575025 = 987452,
750775015275076 = 274002742.

Komachi Cube Sum : 7502 = 93 + 183 + 353 + 423 + 763.

7502 = 562500, 5 * 6 / 2 * 50 + 0 = 5 * 6 * 25 + 0 + 0 = 5 * 6 * 25 + 0 * 0 = 750.

7502 = (5 x 6 x 7)2 + (8 x 9 x 10)2.

7502 = (92 + 9)(792 + 9).

7502 = 253 + 503 + 753.

The 4-by-4 magic squares consisting of different squares with constant 750:

02 22112252
102242 72 52
172132162 62
192 12182 82
      
02 52 72262
102152192 82
112222122 12
232 42142 32
      
12 32162222
52252 62 82
182 42172112
202102132 92

Page of Squares : First Upload September 26, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

751

The smallest squares containing k 751's :
751689 = 8672,
85751751556 = 2928342,
751277510751364 = 274094422.

(517 / 751)2 = 0.473915826... (Komachic).

3-by-3 magic squares consisting of different squares with constant 7512:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 122, 741, 507, 546, 94, 554, 501, 78),(6, 139, 738, 347, 654, 126, 666, 342, 59),
(18, 274, 699, 374, 603, 246, 651, 354, 122),(21, 354, 662, 402, 554, 309, 634, 363, 174),
(27, 346, 666, 414, 549, 302, 626, 378, 171),(42, 229, 714, 474, 546, 203, 581, 462, 114),
(42, 354, 661, 486, 517, 246, 571, 414, 258),(50, 270, 699, 510, 501, 230, 549, 490, 150),
(90, 374, 645, 405, 570, 274, 626, 315, 270),(94, 213, 714, 309, 666, 158, 678, 274, 171),
(94, 258, 699, 357, 634, 186, 654, 309, 202),(139, 318, 666, 438, 581, 186, 594, 354, 293),
(174, 482, 549, 507, 486, 266, 526, 309, 438) 

7512 = 564001, 5 + 6 + 4 + 0 + 0 + 1 = 42.

Page of Squares : First Upload September 26, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan

752

The smallest squares containing k 752's :
47524 = 2182,
13857527524 = 1177182,
5675267527524 = 23822822.

7522 = 565504, 5 + 6 + 5 + 5 + 0 + 4 = 52.

6572 + 6582 + 6592 + ... + 7522 = 69082,
2142 + 2152 + 2162 + ... + 7522 = 117812.

Page of Squares : First Upload September 26, 2005 ; Last Revised September 7, 2006
by Yoshio Mimura, Kobe, Japan

753

The smallest squares containing k 753's :
753424 = 8682,
975375361 = 312312,
67753575337536 = 82312562.

7532 = 2122 + 3522 + 6312 : 1362 + 2532 + 2122 = 3572.

3-by-3 magic squares consisting of different squares with constant 7532:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(8, 196, 727, 233, 692, 184, 716, 223, 68),(8, 433, 616, 464, 488, 337, 593, 376, 272),
(18, 294, 693, 378, 603, 246, 651, 342, 162),(20, 415, 628, 503, 460, 320, 560, 428, 265),
(23, 232, 716, 376, 617, 212, 652, 364, 97),(23, 244, 712, 296, 652, 233, 692, 287, 76),
(41, 188, 728, 488, 548, 169, 572, 481, 92),(47, 152, 736, 352, 656, 113, 664, 337, 112),
(55, 128, 740, 272, 695, 100, 700, 260, 97),(56, 188, 727, 428, 607, 124, 617, 404, 152),
(56, 343, 668, 508, 476, 287, 553, 472, 196),(68, 289, 692, 439, 548, 272, 608, 428, 119),
(79, 232, 712, 488, 559, 128, 568, 448, 209),(92, 343, 664, 436, 568, 233, 607, 356, 268),
(97, 404, 628, 524, 488, 233, 532, 407, 344),(112, 329, 668, 521, 448, 308, 532, 508, 161),
(117, 306, 678, 414, 597, 198, 618, 342, 261),(148, 308, 671, 352, 631, 212, 649, 272, 268),
(176, 457, 572, 503, 352, 436, 532, 484, 223) 

7532 = 567009, 5 + 67 + 0 + 0 + 9 = 92,
7532 = 567009, 567 + 0 + 0 + 9 = 242.

Page of Squares : First Upload September 26, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

754

The smallest squares containing k 754's :
6754801 = 25992,
18754754704 = 1369482,
587548754754025 = 242394052.

7542 = 568516, 56 * 8 + 51 * 6 = 754.

7542 = (52 + 4)(1402 + 4).

57681k + 69745k + 139113k + 301977k are squares for k = 1,2,3 (7542, 3445782, 1753871862).

(13 + 23 + ... + 643)(653 + 663 + ... + 1693)(1703 + 1713 + ... + 7543) = 84043159200002,
(13 + 23 + ... + 1803)(1813 + 1823 + ... + 7243)(7253 + 7263 + ... + 7543) = 4700960413380002.

7542 = 568516, 56 + 8 + 51 + 6 = 112,
7542 = 568516, 568 + 51 + 6 = 252,
7542 = 568516, 562 + 82 + 52 + 162 = 592.

7542 = 568516 appears in the decimal expression of π:
  π = 3.14159•••568516••• (from the 26681st digit)

Page of Squares : First Upload September 26, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

755

The smallest squares containing k 755's :
27556 = 1662,
755755081 = 274912,
755907553775529 = 274937732.

3-by-3 magic squares consisting of different squares with constant 7552:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(10, 159, 738, 270, 690, 145, 705, 262, 66),(10, 207, 726, 495, 550, 150, 570, 474, 143),
(10, 270, 705, 495, 530, 210, 570, 465, 170),(10, 402, 639, 495, 486, 298, 570, 415, 270),
(17, 306, 690, 450, 550, 255, 606, 417, 170),(30, 170, 735, 222, 705, 154, 721, 210, 78),
(30, 170, 735, 415, 618, 126, 630, 399, 118),(30, 255, 710, 305, 654, 222, 690, 278, 129),
(30, 255, 710, 486, 550, 177, 577, 450, 186),(30, 305, 690, 415, 570, 270, 630, 390, 145),
(30, 369, 658, 415, 558, 294, 630, 350, 225),(63, 350, 666, 530, 495, 210, 534, 450, 287),
(98, 225, 714, 390, 630, 145, 639, 350, 198),(102, 289, 690, 465, 570, 170, 586, 402, 255),
(129, 478, 570, 522, 354, 415, 530, 465, 270),(170, 438, 591, 465, 534, 262, 570, 305, 390)

7552 = 570025, 57 + 0 + 0 + 2 + 5 = 82,
7552 = 570025, 5 + 70 + 0 + 25 = 102.

7552 = 463 + 573 + 663.

Page of Squares : First Upload September 26, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan

756

The smallest squares containing k 756's :
7569 = 872,
977562756 = 312662,
3007567566756 = 17342342.

The squares which begin with 756 and end in 756 are
7561213054756 = 27497662,   7563787054756 = 27502342,..,
756437230756 = 8697342,   756492894756 = 8697662,   7561037070756 = 27497342.

7562 = 571536, a zigzag square.

7562± 5 are primes.

7562 = 571536, a square with odd digits except the last digit 6.

7562 = (12 + 5)(32 + 5)(72 + 5)(112 + 5) = (22 - 1)(82 - 1)(552 - 1).

Cubic Polynomial : (X + 7562)(X + 40482)(X + 284972) = X3 + 287932X2 + 1173902522X + 872090271362.

Kaprekar : 7562 = 571536, and 5 + 715 + 36 = 756.

Komachi equations:
7562 = 12 * 22 * 32 / 42 * 5672 * 82 / 92 = 92 * 82 * 72 / 62 * 542 / 32 / 22 */ 12
 = 982 / 72 * 62 * 542 / 32 / 22 */ 12 = 982 / 72 / 62 * 542 * 32 * 22 */ 12
 = 92 * 82 * 72 * 62 / 52 * 42 / 322 * 102 = 92 / 82 * 72 * 62 / 52 / 42 * 322 * 102.

7562 = 571536, 5 + 7 + 15 + 3 + 6 = 62,
7562 = 571536, 57 + 15 + 3 + 6 = 92,
7562 = 571536, 5 + 715 + 3 + 6 = 272.

7562 = (5 + 6 + 7 + ... + 16)2 + (17 + 18 + 19 + ... + 28)2 + (29 + 30 + 31 + ... + 40)2 + (41 + 42 + 43 + ... + 52)2.

(1 + 2)(3 + 4)(5 + 6 + 7)(8)(9)(10 + 11) = 7562,
(1 + 2)(3 + 4 + 5 + 6)(7)(8)(9)(10 + 11) = 7562,
(1)(2 + 3 + 4 + 5)(6)(7)(8 + 9 + 10)(11 + 12 + 13) = 7562,
(1)(2)(3 + 4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14) = 7562,
(1)(2)(3 + 4 + 5 + 6 + 7 + 8 + 9)(10 + 11)(12)(13 + 14) = 7562,
(1 + 2)(3)(4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12 + 13)(14) = 7562,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15) = 7562,
(1 + 2 + 3 + 4 + 5 + 6 + 7)(8 + ... + 16)(17 + ... + 25) = 7562,
(1 + 2 + 3 + 4 + 5 + 6)(7)(8 + ... + 88) = 7562.

(13 + 23 + ... + 1713)(1723 + 1733 + ... + 4763)(4773 + 4783 + ... + 7563) = 4348231044393602.

(12)(22 + 32 + 42 + 52)(62)(72 + 82 + 92 + 102) = 7562.

7562 = 35 + 105 + 105 + 135.

7562 = 571536 appears in the decimal expression of e:
  e = 2.71828•••571536••• (from the 114710th digit)

Page of Squares : First Upload September 26, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

757

The smallest squares containing k 757's :
375769 = 6132,
75775775076 = 2752742,
7617577757757696 = 872787362.

7572 = 573049, a square with different digits.

3-by-3 magic squares consisting of different squares with constant 7572:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(12, 224, 723, 424, 597, 192, 627, 408, 116),(19, 192, 732, 348, 648, 179, 672, 341, 72),
(45, 368, 660, 432, 555, 280, 620, 360, 243),(51, 252, 712, 388, 621, 192, 648, 352, 171),
(84, 208, 723, 243, 696, 172, 712, 213, 144),(96, 312, 683, 467, 564, 192, 588, 397, 264),
(144, 492, 557, 523, 336, 432, 528, 467, 276),(147, 348, 656, 432, 584, 213, 604, 333, 312),
(172, 411, 612, 492, 532, 219, 549, 348, 388) 

7572 = 573049, 5 + 7 + 3 + 0 + 49 = 82,
7572 = 573049, 57 + 30 + 4 + 9 = 102,
7572 = 573049, 5 + 730 + 49 = 282.

Page of Squares : First Upload September 26, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan

758

The smallest squares containing k 758's :
107584 = 3282,
7586758404 = 871022,
375801758758449 = 193856072.

1 / 758 = 0.001319261213720316622...,
and 132 + 192 + 2612 + 22 + 1372 + 202 + 3162 + 6222 = 7582.

7582 is the 6th square which is the sum of 5 fifth powers : 15 + 55 + 75 + 75 + 145.

7582 = 574564, 5 + 7 + 45 + 64 = 112,
7582 = 574564, 57 + 4 + 56 + 4 = 112,
7582 = 574564, 57 + 4 + 564 = 252.

Page of Squares : First Upload September 26, 2005 ; Last Revised September 7, 2006
by Yoshio Mimura, Kobe, Japan

759

The smallest squares containing k 759's :
1375929 = 11732,
13759759204 = 1173022,
759875975959104 = 275658482.

759 = (12 + 22 + 32 + ... + 222) / (12 + 22).

Komachi Fraction : 36 / 5184729 = (2 / 759)2, 72 / 10369458 = (2 / 759)2.

7592 = 2112 + 3142 + 6582 : 8562 + 4132 + 1122 = 9572,
7592 = 2182 + 3942 + 6112 : 1162 + 4932 + 8122 = 9572.

7592 = 443 + 493 + 723.

7592 = 576081, a square with different digits.

3-by-3 magic squares consisting of different squares with constant 7592:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 314, 691, 349, 614, 278, 674, 317, 146),(2, 314, 691, 349, 658, 146, 674, 211, 278),
(13, 86, 754, 446, 611, 62, 614, 442, 61),(14, 122, 749, 307, 686, 106, 694, 301, 62),
(14, 371, 662, 526, 482, 259, 547, 454, 266),(19, 278, 706, 482, 541, 226, 586, 454, 163),
(26, 386, 653, 478, 499, 314, 589, 422, 226),(29, 434, 622, 518, 466, 301, 554, 413, 314),
(34, 118, 749, 274, 701, 98, 707, 266, 74),(34, 194, 733, 509, 538, 166, 562, 499, 106),
(34, 205, 730, 355, 650, 166, 670, 334, 125),(34, 323, 686, 461, 554, 238, 602, 406, 221),
(36, 351, 672, 408, 576, 279, 639, 348, 216),(50, 310, 691, 530, 509, 190, 541, 470, 250),
(61, 226, 722, 502, 554, 131, 566, 467, 194),(72, 321, 684, 489, 504, 288, 576, 468, 159),
(74, 371, 658, 406, 538, 349, 637, 386, 146),(82, 499, 566, 526, 446, 317, 541, 358, 394),
(83, 266, 706, 314, 658, 211, 686, 269, 182),(86, 173, 734, 278, 694, 131, 701, 254, 142),
(86, 302, 691, 394, 611, 218, 643, 334, 226),(125, 434, 610, 490, 515, 266, 566, 350, 365),
(142, 461, 586, 499, 502, 274, 554, 334, 397),(146, 317, 674, 349, 614, 278, 658, 314, 211),
(166, 307, 674, 338, 646, 211, 659, 254, 278) 

7592 = 576081, 5 + 7 + 60 + 8 + 1 = 92,
7592 = 576081, 57 + 6 + 0 + 81 = 122.

7592 = 576081, and 576 = 242, 81 = 92.

7592 = 576081 appears in the decimal expression of π:
  π = 3.14159•••576081••• (from the 58637th digit)

Page of Squares : First Upload September 26, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan