---

630

The smallest squares containing k 630's :
63001 = 251²,
6305630464 = 79408²,
263048630063076 = 16218774².

630² is the 10th square which is the sum of 4 fifth powers : 4⁵ + 6⁵ + 7⁵ + 13⁵.

The integral triangle of sides 833, 1017, 1066 (or 314, 2555, 2619)(or 4201, 4375, 8574) has square area 630².

Komachi equations:
630² = 1² * 2² * 3² / 4² * 5² / 6² * 7² * 8² * 9² = 1² * 2² / 3² * 4² * 5² * 6² * 7² / 8² * 9²
 = 1² / 2² / 3² / 4² * 5² * 6² * 7² * 8² * 9² = 1² / 2² / 3² * 45² / 6² * 7² * 8² * 9²
 = 9² * 8² * 7² * 6² * 5² / 4² / 3² / 2² */ 1² = 9² * 8² * 7² / 6² * 5² / 4² * 3² * 2² */ 1²
 = 9² / 8² * 7² * 6² * 5² * 4² / 3² * 2² */ 1² = 98² / 7² * 6² * 5² / 4² * 3² * 2² */ 1².

630² = 4⁵ + 6⁵ + 7⁵ + 13⁵.

630² = 6 x 7 + 7 x 8 + 8 x 9 + 9 x 10 + ... + 105 x 106.

630² = (2 + 3 + 4)² + (5 + 6 + 7)² + (8 + 9 + 10)² + ... + (71 + 72 + 73)².

630² = (1² + 5)(10² + 5)(25² + 5) = (1² + 5)(4² + 5)(5² + 5)(10² + 5) = (5² + 5)(115² + 5)
= (2² + 5)(3² + 5)(5² + 5)(10² + 5) = (4² + 5)(5² + 5)(25² + 5) = (5² + 5)(10² + 5)(11² + 5).

(1)(2 + 3)(4)(5)(6 + 7 + 8)(9)(10 + 11) = 630²,
(1 + 2)(3 + 4)(5)(6)(7 + 8)(9 + 10 + 11 + 12) = 630²,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15) = 630²,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 39) = 630²,
(1 + 2)(3 + 4 + ... + 17)(18 + 19 + ... + 45) = 630²,
(1)(2 + 3 + ... + 26)(27 + 28 + ... + 54) = 630².

(1³ + 2³ + ... + 594³)(595³ + 596³ + ... + 629³)(630³) = 250393315095000².

Page of Squares : First Upload July 4, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

---

631

The smallest squares containing k 631's :
263169 = 513²,
63176319801 = 251349²,
316311631663104 = 17785152².

1² + 2² + 3² + ... + 631² = 83945716, the 6th (and biggest) 8-digit sum consisting of different digits.

631² = 398161, 39 + 8 + 16 + 1 = 8²,
631² = 398161, 3 + 9 + 81 + 6 + 1 = 10².

3-by-3 magic squares consisting of different squares with constant 631²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 234, 586, 366, 478, 189, 514, 339, 138),	(6, 189, 602, 406, 462, 141, 483, 386, 126),
(19, 126, 618, 402, 474, 109, 486, 397, 66),	(19, 294, 558, 354, 467, 234, 522, 306, 179),
(46, 291, 558, 333, 486, 226, 534, 278, 189),	(54, 413, 474, 438, 366, 269, 451, 306, 318),
(66, 163, 606, 318, 534, 109, 541, 294, 138),	(66, 242, 579, 387, 474, 154, 494, 339, 198),
(93, 354, 514, 406, 429, 222, 474, 298, 291)

280² + 281² + 282² + ... + 631² = 8756²,
225² + 226² + 227² + ... + 631² = 8954².

Page of Squares : First Upload July 4, 2005 ; Last Revised July 6, 2009
by Yoshio Mimura, Kobe, Japan

---

632

The smallest squares containing k 632's :
163216 = 404²,
12463266321 = 111639²,
163203563263225 = 12775115².

Komachi Fraction : 450 / 7189632 = (5 / 632)².

632² = 399424, 3 + 9 + 9 + 4 + 24 = 7².

Page of Squares : First Upload July 4, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

---

633

The smallest squares containing k 633's :
633616 = 796²,
263311633321 = 513139²,
16336333163329 = 4041823².

(328 / 633)² = 0.268497513... (Komachic).

633² = 211² + 422² + 422² : 224² + 224² + 112² = 336².

21^k + 45^k + 201^k + 633^k are squares for k = 1,2,3 (30², 666², 16182²).

Komachi equations:
633² = - 1³ + 2³ + 3³ + 4³ + 5³ * 6³ + 7³ + 8³ * 9³ = 9³ * 8³ + 7³ + 6³ * 5³ + 4³ + 3³ + 2³ - 1³
 = 9³ * 8³ - 7³ + 6³ * 5³ - 4³ * 3³ / 2³ + 10³ = 9³ * 8³ - 7³ - 6³ + 5³ * 4³ * 3³ / 2³ + 10³.

3-by-3 magic squares consisting of different squares with constant 633²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 112, 623, 343, 524, 92, 532, 337, 64),	(4, 308, 553, 392, 433, 244, 497, 344, 188),
(7, 176, 608, 224, 568, 167, 592, 217, 56),	(8, 79, 628, 271, 568, 68, 572, 268, 41),
(8, 295, 560, 440, 400, 217, 455, 392, 200),	(23, 76, 628, 212, 593, 64, 596, 208, 47),
(23, 188, 604, 316, 527, 152, 548, 296, 113),	(28, 143, 616, 407, 476, 92, 484, 392, 113),
(42, 342, 531, 414, 387, 282, 477, 366, 198),	(47, 352, 524, 428, 404, 233, 464, 337, 268),
(54, 243, 582, 387, 474, 162, 498, 342, 189),	(56, 388, 497, 428, 343, 316, 463, 364, 232),
(99, 282, 558, 378, 477, 174, 498, 306, 243),	(128, 239, 572, 268, 548, 169, 559, 208, 212),
(152, 404, 463, 433, 268, 376, 436, 407, 212)

633² = 400689, 4³ + 0³ + 0³ + 6³ + 8³ + 9³ = 39²,
633² = 400689, 4 + 0 + 0 + 68 + 9 = 9²,
633² = 400689, 40 + 0 + 689 = 27²,
633² = 400689, 400 + 689 = 33².

Page of Squares : First Upload July 4, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

634

The smallest squares containing k 634's :
2634129 = 1623²,
11063463489 = 105183²,
634563456634849 = 25190543².

634² = 401956, a square with different digits.

634²± 3 are primes.

The square root of 634 is 25. 1 7 9 3 5 6 6 2 4 0 2 8 3 4 3 1 11 4 8 8 ...,
and 25² = 1² + 7² + 9² + 3² + 5² + 6² + 6² + 2² + 4² + 0² + 2² + 8² + 3² + 4² + 3² + 1² + 11² + 4² + 8² + 8² = 25².

125^k + 241^k + 305^k + 485^k are squares for k = 1,2,3 (34², 634², 12586²).

634² = 401956, 4 + 0 + 1 + 9 + 5 + 6 = 5²,
634² = 401956, 4 + 0 + 1 + 956 = 31².

Page of Squares : First Upload July 4, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

---

635

The smallest squares containing k 635's :
63504 = 252²,
29263576356 = 171066²,
3363586356716356 = 57996434².

635² = 1! + 4! + 8! + 9!

635² = 403225, 40 * 32 / 2 - 5 = 635.

3-by-3 magic squares consisting of different squares with constant 635²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 158, 615, 342, 519, 130, 535, 330, 90),	(22, 246, 585, 279, 522, 230, 570, 265, 90),
(39, 210, 598, 402, 455, 186, 490, 390, 105),	(42, 185, 606, 231, 570, 158, 590, 210, 105),
(50, 150, 615, 183, 594, 130, 606, 167, 90),	(50, 249, 582, 375, 482, 174, 510, 330, 185),
(50, 375, 510, 402, 414, 265, 489, 302, 270),	(66, 238, 585, 438, 441, 130, 455, 390, 210),
(90, 265, 570, 330, 510, 185, 535, 270, 210),	(105, 346, 522, 390, 378, 329, 490, 375, 150),
(126, 318, 535, 345, 490, 210, 518, 249, 270),	(150, 375, 490, 409, 438, 210, 462, 266, 345)

635² = 403225, 4 + 0 + 3 + 2 + 2 + 5 = 4²,
635² = 403225, 40² + 32² + 25² = 57².

Page of Squares : First Upload July 4, 2005 ; Last Revised July 6, 2009
by Yoshio Mimura, Kobe, Japan

---

636

The smallest squares containing k 636's :
37636 = 194²,
63696361 = 7981²,
67363663606369 = 8207537².

The squares which begin with 636 and end in 636 are
6368997636 = 79806²,   63601813636 = 252194²,   63658317636 = 252306²,
636315717636 = 797694²,   636494413636 = 797806²,...

Komachi Square Sum : 636² = 1² + 3² + 9² + 247² + 586².

636² = 19³ + 29³ + 72³ = 22³ + 33³ + 71³.

636² = 404496, 4 + 0 + 44 + 96 = 12²,
636² = 404496, 40 + 4 + 4 + 96 = 12²,
636² = 404496, 404 + 496 = 30².

Page of Squares : First Upload July 4, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

---

637

The smallest squares containing k 637's :
463761 = 681²,
14263763761 = 119431²,
36374637637321 = 6031139².

637² = 405769, a square with different digits.

637² + 638² + 639² + 640² + ... + 190740² = 48095422².

3-by-3 magic squares consisting of different squares with constant 637²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 128, 624, 272, 564, 117, 576, 267, 52),	(20, 213, 600, 312, 520, 195, 555, 300, 88),
(24, 317, 552, 348, 456, 277, 533, 312, 156),	(36, 173, 612, 403, 468, 156, 492, 396, 83),
(52, 117, 624, 216, 592, 93, 597, 204, 88),	(52, 348, 531, 429, 412, 228, 468, 339, 268),
(60, 363, 520, 405, 380, 312, 488, 360, 195),	(63, 322, 546, 434, 378, 273, 462, 399, 182),
(72, 213, 596, 416, 468, 117, 477, 376, 192),	(93, 304, 552, 344, 492, 213, 528, 267, 236),
(96, 228, 587, 312, 533, 156, 547, 264, 192),	(128, 291, 552, 372, 488, 171, 501, 288, 268)

637² = 405769, 40 + 5 + 7 + 69 = 11²,
637² = 405769, 40 + 576 + 9 = 25².

Page of Squares : First Upload July 4, 2005 ; Last Revised July 6, 2009
by Yoshio Mimura, Kobe, Japan

---

638

The smallest squares containing k 638's :
16384 = 128²,
63820906384 = 252628²,
1566388638106384 = 39577628².

638² = 407044, a square with just 3 kinds of digits.

638² = (4² + 6)(136² + 6).

638^k + 4378^k + 7106^k + 11594^k are squares for k = 1,2,3 (154², 14300², 1414732²).
88^k + 638^k + 1122^k + 1177^k are squares for k = 1,2,3 (55², 1749², 57475²).
12122^k + 98890^k + 128238^k + 167794^k are squares for k = 1,2,3 (638², 233508², 88328548²).

Komachi Fraction : 638² = 21980376 / 54 = 32970564 / 81.

(1³ + 2³ + ... + 405³)(406³ + 407³ + ... + 594³)(595³ + 596³ + ... + 638³) = 1306653140331000².

638² = 407044, 40² + 7² + 0² + 4² + 4² = 41².

638² = 407044 appears in the decimal expression of e:
  e = 2.71828•••407044••• (from the 92484th digit).

Page of Squares : First Upload July 4, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

---

639

The smallest squares containing k 639's :
4363921 = 2089²,
51563963929 = 227077²,
11639639508639921 = 107887161².

639² = 408321, a square with different digits.

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

Komachi Square Sum : 639² = 2² + 6² + 9² + 374² + 518².

The square root of 639 is 25. 2 7 8 4 4 9 3 19 5 ...,
and 25² = 2² + 7² + 8² + 4² + 4² + 9² + 3² + 19² + 5².

639² = 12³ + 26³ + 73³.

3-by-3 magic squares consisting of different squares with constant 639²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 266, 581, 434, 427, 194, 469, 394, 182),	(11, 146, 622, 214, 587, 134, 602, 206, 59),
(11, 190, 610, 410, 470, 139, 490, 389, 130),	(14, 158, 619, 437, 454, 106, 466, 421, 118),
(22, 286, 571, 379, 454, 242, 514, 347, 154),	(24, 192, 609, 249, 564, 168, 588, 231, 96),
(24, 321, 552, 372, 456, 249, 519, 312, 204),	(27, 234, 594, 414, 459, 162, 486, 378, 171),
(38, 106, 629, 154, 613, 94, 619, 146, 62),	(46, 139, 622, 277, 566, 106, 574, 262, 101),
(46, 262, 581, 293, 526, 214, 566, 251, 158),	(53, 314, 554, 434, 389, 262, 466, 398, 181),
(59, 266, 578, 358, 466, 251, 526, 347, 106),	(85, 314, 550, 370, 475, 214, 514, 290, 245),
(91, 194, 602, 238, 574, 149, 586, 203, 154),	(91, 206, 598, 326, 533, 134, 542, 286, 181),
(94, 398, 491, 427, 326, 346, 466, 379, 218),	(101, 302, 554, 434, 374, 283, 458, 421, 146),
(146, 346, 517, 382, 469, 206, 491, 262, 314)

639² = 408321, 4 + 0 + 8 + 3 + 21 = 6²,
639² = 408321, 40 + 8 + 32 + 1 = 9²,
639² = 408321, 40 + 83 + 21 = 12²,
639² = 408321, 408 + 32 + 1 = 21²,
639² = 408321, 408 + 321 = 27².

Page of Squares : First Upload July 4, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---