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8500 - 8599

8500

85002 = (12 + 4)(42 + 4)(112 + 4)(762 + 4) = (12 + 4)(42 + 4)(82 + 4)(92 + 4)(112 + 4).

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

8501

1 / 8501 = 0.0001176332196212, 112 + 72 + 632 + 32 + 22 + 192 + 622 + 122 = 8501.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8503

85032 = 72301009, and 72 + 22 + 32 + 02 + 12 + 02 + 02 + 92 = 122.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8504

85042± 3 are primes.

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

8505

85052 = (102 + 5)(8302 + 5).

85052 = 93 + 703 + 4163 = 263 + 2173 + 3963.

Komachi equations: 85052 = 122 / 32 * 452 * 62 * 72 / 82 * 92.

Page of Squares : First Upload July 22, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8506

1 / 8506 = 0.000117564072, 112 + 72 + 562 + 42 + 02 + 722 = 8506,
1 / 8506 = 0.000117564072, 112 + 72 + 562 + 42 + 0722 = 8506.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8508

85085 = 44579727160094112768, and
442 + 52 + 72 + 92 + 722 + 72 + 162 + 02 + 02 + 92 + 42 + 12 + 12 + 272 + 62 + 82 = 8508.

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

8509

85092 = 72403081, a zigzag square.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8511

85112 = 72437121, a zigzag square.

85112 = 813 + 3263 + 3343 = 404 + 504 + 694 + 804.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8513

85134 = 5252070336226561,
and 52 + 22 + 522 + 02 + 72 + 02 + 32 + 362 + 22 + 262 + 52 + 612 = 8513,
85134 = 5252070336226561,
and 522 + 52 + 22 + 02 + 72 + 02 + 32 + 362 + 22 + 262 + 52 + 612 = 8513.

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

8514

85142 = (22 + 2)(32 + 2)(10482 + 2) = (52 + 2)(162 + 2)(1022 + 2) = (82 + 2)(10482 + 2).

85142 = 72488196, and 73 + 23 + 43 + 83 + 83 + 193 + 63.

Page of Squares : First Upload November 25, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8515

85152 + 85162 + 85172 + ... + 85802 = 85812 + 85822 + 85832 + ... + 86452.

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

8517

85175 = 44816014936943562357, and
442 + 82 + 12 + 602 + 142 + 92 + 32 + 62 + 92 + 432 + 52 + 62 + 232 + 52 + 72 = 8517.

Komachi equations: 85172 = 12 * 22 / 32 / 42 * 56782 * 92.

Page of Squares : First Upload December 8, 2008 ; Last Revised November 16, 2010
by Yoshio Mimura, Kobe, Japan

8518

85182± 3 are primes.

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

8520

85202 = 43 + 2963 + 3603 = 253 + 1673 + 4083 = 643 + 1643 + 4083.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8523

85232 = 233 + 493 + 4173.

8523, 8524, 8525 and 8526 are four consecutive integers having square factors (the 5th case).

Page of Squares : First Upload November 10, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

8524

85242 = 72658576, a zigzag square.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8525

85252 = (30 + 31 + 32 + ... + 39)2 + (40 + 41 + 42 + ... + 49)2 + (50 + 51 + 52 + ... + 59)2 + ... + (270 + 271 + 272 + ... + 279)2.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8526

85262± 5 are primes.

85262 = (32 + 5)(322 + 5)(712 + 5).

Page of Squares : First Upload January 11, 2014 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

8529

85292 = 703 + 1133 + 4143.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8530

Loop of length 14 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
8530 - 8125 - 7186 - 12437 - ... - 8857 - 10993 - 8731 - 8530
(Note f(8530) = 852 + 302 = 8125,   f(8125) = 812 + 252 = 7186, etc. See 1946)

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

8531

85312 = S2(121) + S2(600), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

85314 = 5296631607317521,
and 52 + 22 + 92 + 62 + 62 + 32 + 162 + 02 + 732 + 12 + 72 + 522 + 12 = 8531
and 522 + 92 + 62 + 62 + 32 + 162 + 02 + 732 + 12 + 72 + 52 + 22 + 12 = 8531.

85315 = 45185564242025771651 :
42 + 52 + 12 + 82 + 52 + 52 + 62 + 422 + 42 + 202 + 22 + 572 + 72 + 162 + 512
= 42 + 52 + 12 + 82 + 552 + 62 + 42 + 22 + 422 + 02 + 252 + 72 + 72 + 162 + 512
= 42 + 52 + 12 + 82 + 552 + 62 + 422 + 42 + 22 + 02 + 252 + 72 + 72 + 162 + 512
= 42 + 512 + 82 + 52 + 52 + 62 + 422 + 42 + 202 + 22 + 572 + 72 + 162 + 52 + 12
= 42 + 512 + 82 + 552 + 62 + 42 + 22 + 422 + 02 + 252 + 72 + 72 + 162 + 52 + 12
= 42 + 512 + 82 + 552 + 62 + 422 + 42 + 22 + 02 + 252 + 72 + 72 + 162 + 52 + 12 = 8531.

Page of Squares : First Upload July 22, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8532

85322 = 1903 + 2963 + 3423.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8533

85332 = 1683 + 2763 + 3613 = 2243 + 2443 + 3613.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8535

85352± 2 are primes.

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

8540

The square root of 8540 is 92.412120417...., and
92 = 42 + 12 + 22 + 12 + 22 + 02 + 42 + 12 + 72.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8541

85412 = 72948681, a zigzag square.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8546

85462 = 73034116, and 72 + 32 + 02 + 32 + 42 + 12 + 12 + 62 = 112.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8547

85472 = 1833 + 3013 + 3413.

85472 = 73051209, and 72 + 32 + 02 + 52 + 12 + 22 + 02 + 92 = 132.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8548

85485 = 45637575818582483968, and
42 + 52 + 62 + 372 + 52 + 72 + 52 + 82 + 12 + 82 + 52 + 82 + 242 + 82 + 392 + 682 = 8548.

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

8549

85492 = 73085401, and 7 + 3 + 0 + 8540 - 1 = 7 + 3 + 8540 - 1 = 8549.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8550

The square root of 8550 is 92.4662...., and 92 = 42 + 62 + 62 + 22.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8552

85522 = 73136704, and 72 + 32 + 12 + 32 + 62 + 72 + 02 + 42 = 132.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8553

85532 = 163 + 2843 + 3693 = 274 + 344 + 584 + 884.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8555

85552 = 5532 + 5542 + 5552 + 5562 + ... + 7292.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8558

2090k + 8074k + 8558k + 12254k are squares for k = 1,2,3 (1762, 171162, 17327202).

Page of Squares : First Upload July 12, 2011 ; Last Revised July 12, 2011
by Yoshio Mimura, Kobe, Japan

8559

85592 = 73256481, a square with different digits.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8560

85602 = 373 + 1563 + 4113.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8568

85682 = (22 - 1)(32 - 1)(352 - 1)(502 - 1) = (52 - 1)(352 - 1)(502 - 1).

85682 = 143 + 723 + 4183.

85682 = 108 x 109 x 110 + 110 x 111 x 112 + 112 x 113 x 114 + ... + 162 x 163 x 164.

Komachi equations: 85682 = 12 / 22 * 342 * 5672 * 82 / 92.

The integral triangle of sides 6689, 76960, 83393 has square area 85682.

Page of Squares : First Upload October 18, 2011 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8569

8569 = (12 + 22 + 32 + ... + 57192) / (12 + 22 + 32 + ... + 2792).

85692 = 153 + 973 + 4173.

Page of Squares : First Upload July 22, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

8573

1 / 8573 = 0.00011664...., and 11664 = 1082.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8574

85745 = 46335877910988486624, and
42 + 632 + 32 + 582 + 72 + 72 + 92 + 102 + 92 + 82 + 82 + 42 + 82 + 62 + 62 + 242 = 8574.

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

8575

85752 = 354 + 704 + 704 + 704.

Komachi equations:
85752 = 982 * 72 * 62 * 52 / 42 / 32 / 22 * 102 = 982 * 72 / 62 * 52 / 42 * 32 * 22 * 102.

Page of Squares : First Upload July 22, 2008 ; Last Revised November 16, 2010
by Yoshio Mimura, Kobe, Japan

8576

85762 = (32 + 7)(52 + 7)(3792 + 7).

85762 = 1603 + 1763 + 4003.

Page of Squares : First Upload July 22, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8577

85772 = 45 + 65 + 285 + 285 + 335.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8578

85782 = 73582084, and 7 * 3 * 5 * 82 + 0 - 8 * 4 = 8578.

85783 = 631187116552, and 632 + 112 + 82 + 72 + 112 + 652 + 52 + 22 = 8578.

Page of Squares : First Upload July 22, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

8579

85792 = (241 + 242 + 243 + ... + 263)2 + (264 + 265 + 266 + ... + 286)2.

85795 = 46471141357882998899, and
42 + 62 + 42 + 72 + 12 + 12 + 42 + 12 + 32 + 52 + 72 + 82 + 82 + 22 + 92 + 92 + 82 + 892 + 92 = 8579.

Page of Squares : First Upload July 22, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8581

85812 = 2133 + 2813 + 3473.

85813 = 631849586941, and 632 + 182 + 492 + 52 + 82 + 62 + 92 + 412 = 8581.

Page of Squares : First Upload July 22, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

8582

1 / 8582 = 0.0001165229550221, 12 + 12 + 652 + 22 + 292 + 552 + 0222 + 12 = 8582.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8583

85832 = 73667889, and 7 + 3 * 6 * 67 * 8 * 8 / 9 = 8583.

85832 = 103 + 2813 + 3723.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8585

8585 = (12 + 22 + 32 + ... + 502) / (12 + 22).

85852 = 73702225, and 72 + 32 + 72 + 02 + 22 + 22 + 22 + 52 = 122.

Page of Squares : First Upload July 22, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

8586

85862 = 73719396, a zigzag square with odd digits except the last digit 6.

85862 = (77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85)2 + (86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94)2 + (95 + 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103)2 + ... + (284 + 285 + 286 + 287 + 288 + 289 + 290 + 291 + 292)2.

Page of Squares : First Upload July 22, 2008 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

8587

85875 = 46688219766966551707, and
42 + 62 + 62 + 82 + 82 + 212 + 92 + 72 + 62 + 62 + 92 + 62 + 62 + 52 + 512 + 702 + 72 = 8587.

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

8591

85912 = 73805281, a zigzag square.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8594

85945 = 46878828132462120224, and
42 + 62 + 82 + 782 + 82 + 282 + 12 + 322 + 42 + 62 + 212 + 22 + 02 + 22 + 22 + 42 = 8594.

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

8595

1 / 8595 = 0.0001163467, 112 + 632 + 42 + 672 = 8595.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8596

85962 = 2283 + 3063 + 3223.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8598

85982 = 73925604, a square with different digits.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan

8599

85992 = 73942801, a square with different digits.

Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008
by Yoshio Mimura, Kobe, Japan