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6500 - 6599

6500

65002 = (12 + 1)(22 + 1)(32 + 1)(52 + 1)(72 + 1)(182 + 1)
= (12 + 1)(22 + 1)(52 + 1)(72 + 1)(572 + 1) = (32 + 1)(52 + 1)(72 + 1)(572 + 1)
= (12 + 4)(32 + 4)(42 + 4)(112 + 4)(162 + 4) = (112 + 4)(162 + 4)(362 + 4)
= (32 + 4)(42 + 4)(112 + 4)(362 + 4).

Page of Squares : First Upload November 2, 2013 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6501

65012 = 42263001, a reversible square (10036224 = 31682).

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6503

65032 = 474 + 584 + 584 + 624.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6505

65052 = 42315025, 42 * 31 * 5 + 0 * 2 - 5 = 6505.

Loop of length 14 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6505 - 4250 - 4264 - 5860 - ... - 12437 - 1946 - 2477 - 6505
(Note f(6505) = 652 + 052 = 4250,   f(4250) = 422 + 502 = 4264, etc. See 1946)

Page of Squares : First Upload February 25, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

6507

1 / 6507 = 0.0001536806, 12 + 52 + 32 + 62 + 802 + 62 = 6507.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6508

65085 = 11674484726453152768 :
12 + 12 + 62 + 72 + 442 + 82 + 42 + 72 + 262 + 42 + 532 + 12 + 52 + 272 + 62 + 82
= 12 + 12 + 672 + 42 + 42 + 82 + 42 + 72 + 262 + 42 + 52 + 312 + 52 + 22 + 72 + 62 + 82 = 6508.

65082 = 42354064 appears in the decimal expressions of π:
  π = 3.14159•••42354064••• (from the 44555th digit)
  (42354064 is the eighth 8-digit square in the expression of π.)

Page of Squares : First Upload November 4, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6509

65092 = 42367081, a square with different digits.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6510

65102 = (12 + 6)(22 + 6)(32 + 6)(52 + 6)(362 + 6) = (12 + 6)(32 + 6)(62 + 6)(982 + 6)
= (12 + 6)(52 + 6)(122 + 6)(362 + 6) = (22 + 6)(362 + 6)(572 + 6)
= (22 + 6)(52 + 6)(62 + 6)(572 + 6) = (32 + 6)(52 + 6)(82 + 6)(362 + 6).

Page of Squares : First Upload January 11, 2014 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6512

65122 = (22 + 7)(52 + 7)(172 + 7)(202 + 7) = (52 + 7)(202 + 7)(572 + 7).

65122 = 1323 + 1523 + 3323.

Page of Squares : First Upload October 2, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6515

65152 = 42445225, a square with just 3 kinds of digits.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6516

65162 = 303 + 1113 + 3453.

65162 = (62 + 63 + 64)2 + (65 + 66 + 67)2 + (68 + 69 + 70)2 + (71 + 72 + 73)2 + ... + (347 + 348 + 349)2.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6517

65172 = 103 + 1903 + 3293.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6518

65185 = 11764454127433609568 :
12 + 12 + 72 + 642 + 452 + 42 + 12 + 22 + 72 + 42 + 32 + 32 + 62 + 02 + 92 + 52 + 62 + 82 = 6518.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6519

65192 = 1533 + 2443 + 2903.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6522

65222 = 42536484, a zigzag square.

65225 = 11800596816398473632 :
112 + 82 + 02 + 02 + 592 + 62 + 82 + 12 + 62 + 392 + 82 + 42 + 72 + 32 + 62 + 322 = 6522.

1 / 6522 = 0.00015332720024, 152 + 332 + 22 + 722 + 02 + 02 + 22 + 42 = 6522,
1 / 6522 = 0.00015332720024, 152 + 332 + 22 + 722 + 02 + 022 + 42 = 6522,
1 / 6522 = 0.00015332720024, 152 + 332 + 22 + 722 + 002 + 22 + 42 = 6522,
1 / 6522 = 0.00015332720024, 152 + 332 + 22 + 722 + 0022 + 42 = 6522.

Page of Squares : First Upload February 25, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6525

65252 = 803 + 1003 + 3453.

65252 = (293 + 294 + 295 + ... + 307)2 + (308 + 309 + 310 + ... + 322)2 + (323 + 324 + 325 + ... + 337)2 + ... + (308 + 309 + 310 + ... + 322)2.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6526

65262 = (400 + 401)2 + (402 + 403)2 + (404 + 405)2 + (406 + 407)2 + ... + (502 + 503)2.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6528

65285 = 11854977354713530368 :
12 + 12 + 82 + 52 + 42 + 92 + 772 + 32 + 52 + 42 + 72 + 132 + 52 + 32 + 02 + 32 + 62 + 82 = 6528.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6532

65322 = 13512 + 13522 + 13532 + 13542 + 13552 + 13562 + ... + 13732.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6533

65332 = 42680089, and 422 + 682 + 02 + 02 + 82 + 92.

Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

6534

65342 = (12 + 2)(32 + 2)(42 + 2)(142 + 2)(192 + 2) = (12 + 2)(32 + 2)(82 + 2)(1402 + 2)
= (22 + 2)(192 + 2)(1402 + 2) = (22 + 2)(32 + 2)(52 + 2)(82 + 2)(192 + 2)
= (32 + 2)(142 + 2)(1402 + 2).

65342 = 363 + 2253 + 3153 = 2313 + 2423 + 2533.

Page of Squares : First Upload October 2, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6536

65362 = 42719296, a zigzag square.

The square root of 6536 is 80.84..., and 80 = 82 + 42.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6537

65372 = 42732369, 4 * 273 * 2 * 3 - 6 - 9 = 6537.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6541

1 / 6541 = 0.000152881..., and 152881 = 3912.

65413 = 279854598421, and 22 + 792 + 82 + 52 + 42 + 52 + 92 + 82 + 42 + 22 + 12 = 6541.

Page of Squares : First Upload February 25, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6543

Komachi equations:
65432 = 92 + 82 - 72 + 65432 + 22 - 102 = - 92 - 82 + 72 + 65432 - 22 + 102.

Page of Squares : First Upload November 2, 2010 ; Last Revised November 2, 2010
by Yoshio Mimura, Kobe, Japan

6544

65442 = 43 + 1843 + 3323.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6547

1 / 6547 = 0.000152741713762, 12 + 52 + 272 + 412 + 72 + 132 + 72 + 622 = 6547.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6548

67k + 209k + 233k + 275k are squares for k = 1,2,3 (282, 4222, 65482).

Page of Squares : First Upload June 28, 2011 ; Last Revised June 28, 2011
by Yoshio Mimura, Kobe, Japan

6551

65512 = 2283 + 2493 + 2503.

6551 is the 4th prime for which the Lendre Symbol (a/6551) = 1 for a = 1, 2, 3,..., 16.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6552

65522 = (32 + 3)(62 + 3)(92 + 3)(332 + 3) = (33 + 1)(53 + 1)(233 + 1).

The integral triangle of sides 2548, 47979, 49825 has square area 65522.

Komachi equations:
65522 = 12 * 22 / 32 / 42 * 562 * 782 * 92,
65522 = 92 / 82 * 72 * 652 * 42 * 322 / 102.

Page of Squares : First Upload November 2, 2010 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6555

65552 + 65562 + 65572 + ... + 66122 = 66132 + 66142 + 66152 + ... + 66692.

Page of Squares : First Upload September 16, 2011 ; Last Revised September 16, 2011
by Yoshio Mimura, Kobe, Japan

6556

65562 = 42981136, 4 * 298 * 11 * 3 / 6 = 6556.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6557

65572 = 1533 + 1883 + 3203.

65572 = 42994249, a square with just 3 kinds of digits.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6558

65582± 5 are primes.

Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

6561

The square of 81.

65612 = 2433 + 2433 + 2433 = 275 + 275 + 275.

Komachi equations:
65612 = 124 * 34 / 44 * 564 / 74 / 84 * 94 = 124 * 34 / 44 / 564 * 74 * 84 * 94.

Page of Squares : First Upload February 25, 2008 ; Last Revised November 2, 2010
by Yoshio Mimura, Kobe, Japan

6562

65622 = 14 + 94 + 94 + 814 = 18 + 38 + 38 + 98.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6563

65632 = 30 + 31 + 38 + 39 + 316.

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

6565

65652 = 1752 + 1762 + 1772 + 1782 + 1792 + 1802 + ... + 5122.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6566

65665 = 12204061624991252576 :
122 + 22 + 02 + 402 + 612 + 62 + 22 + 42 + 92 + 92 + 122 + 52 + 252 + 72 + 62 = 6566.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6569

65692 = 883 + 1903 + 3293.

65692 = 43151761, 43 * 151 + 76 * 1 = 6569.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6570

65702 = (82 + 9)(92 + 9)(812 + 9).

Komachi equation: 65702 = 92 * 8762 * 52 / 42 / 32 * 22 */ 12.

Page of Squares : First Upload November 2, 2010 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6571

1 / N = 0.0001521...., and 1521 = 392. (X = 6571, 6572, 6573, 6574)

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6573

The square root of 6573 is 81.07404..., and 81 = 02 + 72 + 42 + 02 + 42.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6577

65772 = 44 + 184 + 184 + 814.

65772 = 43256929, and 42 + 32 + 22 + 52 + 62 + 92 + 22 + 92 = 162.

65775 = 12306631859123116657 :
12 + 232 + 02 + 62 + 62 + 32 + 182 + 52 + 92 + 122 + 312 + 12 + 662 + 52 + 72
= 12 + 232 + 02 + 662 + 32 + 182 + 52 + 92 + 122 + 312 + 12 + 62 + 62 + 52 + 72 = 6577.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6578

The square root of 6578 is 81.1048..., and 81 = 12 + 02 + 42 + 82.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6579

65794 = 1873438951464081,
and 182 + 72 + 32 + 432 + 82 + 92 + 52 + 12 + 42 + 642 + 02 + 82 + 12 = 6579.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6580

65802 = 162 x 163 + 164 x 165 + 166 x 167 + 168 x 169 + ... + 640 x 641.

Komachi equations:
65802 = 9872 / 62 * 52 / 42 * 322 */ 12 = 9872 * 62 / 542 * 32 * 22 * 102.

Page of Squares : First Upload February 25, 2008 ; Last Revised November 2, 2010
by Yoshio Mimura, Kobe, Japan

6582

65825 = 12353482086083118432 :
122 + 32 + 52 + 32 + 42 + 82 + 202 + 82 + 602 + 82 + 32 + 12 + 182 + 432 + 22 = 6582.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6585

The square root of 6585 is 81.148...., and 81 = 12 + 42 + 82.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6587

The square root of 6587 is 81.1603351...., and 81 = 12 + 62 + 02 + 32 + 32 + 52 + 12.

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan

6588

65882 = 93 + 543 + 3513 = 1743 + 1983 + 3123.

1 / 6588 = 0.0001517911, 152 + 12 + 792 + 112 = 6588.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6589

65892 = 43414921, and 42 + 32 + 42 + 12 + 42 + 92 + 22 + 12 = 122.

704k + 1034k + 1870k + 6193k are squares for k = 1,2,3 (992, 65892, 4954952).

Page of Squares : First Upload February 25, 2008 ; Last Revised June 28, 2011
by Yoshio Mimura, Kobe, Japan

6591

65912 = 1693 + 3383.

The square root of 6591 is 81.184...., and 81 = 12 + 82 + 42.

Page of Squares : First Upload February 25, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6592

65922 = 164 + 564 + 644 + 644.

Page of Squares : First Upload October 2, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6596

65962 = 204 + 404 + 644 + 704 = 304 + 604 + 604 + 644.

65962 = 43507216, a square with different digits.

Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6596 - 13441 - 2838 - 2228 - ... - 5713 - 3418 - 1480 - 6596
(Note f(6596) = 652 + 962 = 13441,   f(13441) = 12 + 342 + 412 = 2838, etc. See 1268)

Page of Squares : First Upload February 25, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

6597

65975 = 12494889518881472757 :
12 + 242 + 92 + 482 + 82 + 92 + 52 + 182 + 82 + 82 + 12 + 472 + 272 + 52 + 72 = 6597.

1 / 6597 = 0.000151584053357, 152 + 152 + 82 + 42 + 02 + 532 + 32 + 572 = 6597,
1 / 6597 = 0.000151584053357, 152 + 152 + 82 + 42 + 0532 + 32 + 572 = 6597.

Page of Squares : First Upload February 25, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6599

65992 = 1! + 5! + 5! + 5! + 5! + 6! + 10! + 11!

Page of Squares : First Upload February 25, 2008 ; Last Revised February 25, 2008
by Yoshio Mimura, Kobe, Japan