logo
6700 - 6799

6700

67002 = 953 + 1053 + 3503.

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

6703

67035 = 13531504832293013743 :
12 + 32 + 52 + 312 + 52 + 02 + 42 + 82 + 32 + 22 + 22 + 92 + 32 + 02 + 12 + 32 + 742 + 32 = 6703.

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

6705

22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).

Page of Squares : First Upload July 1, 2011 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6707

1 / 6707 = 0.00014909795735, 142 + 92 + 092 + 792 + 52 + 72 + 32 + 52 = 6707.

67075 = 13571927521423150307 : 12 + 32 + 572 + 12 + 92 + 22 + 72 + 52 + 22 + 142 + 232 + 12 + 502 + 32 + 02 + 72 = 132 + 52 + 72 + 12 + 92 + 22 + 752 + 212 + 42 + 22 + 32 + 152 + 02 + 32 + 02 + 72 = 6707.

Page of Squares : First Upload March 10, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6708

67082 = 4! + 5! + 9! + 9! + 9! + 9! + 10! + 11!

Komachi equation: 67082 = 92 * 82 * 72 * 652 * 432 / 2102.

390k + 2262k + 4329k + 6708k are squares for k = 1,2,3 (1172, 83072, 6281732).

Page of Squares : First Upload March 10, 2008 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6709

67092 = 143 + 1583 + 3453.

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

6712

67125 = 13622591671114104832 : 12 + 32 + 62 + 222 + 592 + 12 + 62 + 72 + 12 + 12 + 142 + 102 + 482 + 32 + 22 = 132 + 62 + 222 + 592 + 12 + 62 + 72 + 12 + 112 + 42 + 12 + 02 + 482 + 32 + 22 = 132 + 62 + 222 + 592 + 12 + 62 + 72 + 112 + 12 + 42 + 12 + 02 + 482 + 32 + 22 = 6712.

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

6713

67132 = 663 + 2253 + 3223 = 1423 + 2523 + 2973 = 144 + 144 + 564 + 774.

67135 = 13632742633160464793 : 12 + 32 + 62 + 32 + 22 + 72 + 42 + 22 + 632 + 32 + 162 + 02 + 42 + 62 + 472 + 92 + 32 = 12 + 32 + 632 + 22 + 72 + 42 + 22 + 62 + 32 + 32 + 162 + 02 + 42 + 62 + 472 + 92 + 32 = 6713.

Page of Squares : First Upload October 2, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6714

67142 = 73 + 2453 + 3123.

1 / 6714 = 0.000148942508, 12 + 482 + 92 + 422 + 502 + 82 = 6714.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6718

67182 = 273 + 1043 + 3533.

67182 = 45131524, a zigzag square.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6719

67192 = 45144961, and 4 * 5 * 14 / 4 * 96 - 1 = 6719.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6720

67202 = (112 - 1)(152 - 1)(412 - 1) = (152 - 1)(4492 - 1) = (22 - 1)(52 - 1)(62 - 1)(92 - 1)(152 - 1)
= (22 - 1)(92 - 1)(152 - 1)(292 - 1) = (32 - 1)(42 - 1)(152 - 1)(412 - 1)
= (32 - 1)(42 - 1)(62 - 1)(72 - 1)(152 - 1) = (32 - 1)(62 - 1)(132 - 1)(312 - 1)
= (42 - 1)(92 - 1)(132 - 1)(152 - 1) = (62 - 1)(72 - 1)(112 - 1)(152 - 1) = (62 - 1)(92 - 1)(1272 - 1).

Komachi equations:
67202 = 122 / 32 * 42 * 52 / 62 * 72 * 82 * 92 = 122 / 32 * 452 * 62 * 72 * 82 / 92.

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

6721

67215 = 13714168369112097601 :
12 + 372 + 12 + 42 + 12 + 62 + 82 + 362 + 92 + 112 + 22 + 02 + 92 + 72 + 602 + 12 = 6721.

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

6722

67222 = 14 + 514 + 634 + 694.

56722567225 = 2381652.

Loop of length 6 by the function f(N) = b2 + a2 for N = 100b + a:
6722 - 4973 - 7730 - 6829 - 5465 - 7141 - 6722   (Note f(6722) = 672 + 222 = 4973.)

67225 = 13724373881659929632 :
132 + 72 + 22 + 42 + 32 + 72 + 32 + 82 + 82 + 12 + 652 + 92 + 92 + 292 + 62 + 322 = 6722.

Page of Squares : First Upload March 10, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6723

67232 = 93 + 2103 + 3303 = 1613 + 1723 + 3303.

67232 = (35 + 36 + 37 + ... + 52)2 + (53 + 54 + 55 + ... + 70)2 + (71 + 72 + 73 + ... + 88)2 + ... + (179 + 180 + 181 + ... + 196)2.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6724

The square of 82.

6724 = (6 + 72 + 4)2.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6724 - 5065 - 6725 - 5114 - ... - 3301 - 1090 - 8200 - 6724
(Note f(6724) = 672 + 242 = 5065,   f(5065) = 502 + 652 = 6725, etc. See 37)

Page of Squares : First Upload March 10, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

6725

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6725 - 5114 - 2797 - 10138 - ... - 8200 - 6724 - 5065 - 6725
(Note f(6725) = 672 + 252 = 5114,   f(5114) = 512 + 142 = 2797, etc. See 37)

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

6726

67262 = 45239076, a square with different digits.

62 + 132 + 202 + ... + (7x + 6)2 + ... + 67262 = 1204662.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6727

67272 = 314 + 624 + 624 + 624.

67272 = 45252529, a zigzag square.

S2(45) * S2(6727) = 564482102, and S2(6727) / S2(45) = 17982,
where S2(n) = 12 + 22 + 32 + ... + n2.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6728

67282 = 1163 + 1163 + 3483 = 584 + 584 + 584 + 584.

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

6732

67322 = (12 + 8)(32 + 8)(142 + 8)(382 + 8) = (12 + 8)(32 + 8)(52 + 8)(62 + 8)(142 + 8)
= (12 + 8)(32 + 8)(62 + 8)(822 + 8) = (22 + 2)(32 + 2)(42 + 2)(82 + 2)(242 + 2)
= (22 + 2)(82 + 2)(142 + 2)(242 + 2) = (22 + 8)(32 + 8)(52 + 8)(822 + 8)
= (22 + 8)(32 + 8)(62 + 8)(712 + 8) = (32 + 8)(62 + 8)(142 + 8)(172 + 8)
= (52 + 8)(142 + 8)(822 + 8) = (62 + 8)(142 + 8)(712 + 8).

The integral triangle of sides 6849, 13600, 16577 has square area 67322.

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

6734

67342± 3 are primes.

67342 = (72 + 3)(162 + 3)(582 + 3).

67342 = 272 x 273 + 274 x 275 + 276 x 277 + 278 x 279 + ... + 662 x 663.

Page of Squares : First Upload March 10, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

6735

67352 = 723 + 1003 + 3533 = 2113 + 2313 + 2873.

67352 = 45360225, and 45 * 3 + 60 * 22 * 5 = 6735.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6736

67364 = 2058772288700416,
and 22 + 02 + 52 + 82 + 772 + 222 + 82 + 82 + 72 + 02 + 02 + 42 + 12 + 62 = 6736.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6736 - 5785 - 10474 - 5493 - ... - 5365 - 7034 - 6056 - 6736
(Note f(6736) = 672 + 362 = 5785,   f(5785) = 572 + 852 = 10474, etc. See 37)

67362 = 45373696, and 4 + 53 * 7 * 3 * 6 + 9 * 6 = 6736.

Page of Squares : First Upload March 10, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6737

67372 = 45387169, a square with different digits.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6739

The square root of 6739 is 82.091...., and 82 = 02 + 92 + 12.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6742

67422 = 45454564, a square with just 3 kinds of digits.

The square root of 6742 is 82.109...., and 82 = 12 + 02 + 92.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6744

67442 = 45481536, a zigzag square.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6749

67492 = S2(327) + S2(466), where S2(n) = 12 + 22 + 32 + ... + n2.

67494 = 2074711492098001,
and 202 + 742 + 72 + 12 + 142 + 92 + 202 + 92 + 82 + 02 + 02 + 12 = 6749.

Page of Squares : First Upload March 10, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6750

67502 = (42 + 9)(62 + 9)(92 + 9)(212 + 9).

67502 = 693 + 2553 + 3063 = 1803 + 2553 + 2853 = 156 + 156 + 156 + 156.

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

6753

67532 = 364 + 494 + 644 + 684.

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

6755

67555 = 14064579971036721875 :
12 + 402 + 642 + 52 + 72 + 92 + 92 + 72 + 102 + 32 + 62 + 72 + 212 + 82 + 72 + 52 = 6755.

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

6756

67562± 5 are primes.

67562 = 863 + 1423 + 3483.

1 / 6756 = 0.000148016577, 142 + 802 + 12 + 62 + 52 + 72 + 72 = 6756.

The square root of 6756 is 82.19...., and 82 = 12 + 92.

Page of Squares : First Upload March 10, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

6759

67592 = 45684081, and 45 - 6 + 840 * 8 * 1 = 6759.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6760

67602 = (12 + 4)(22 + 4)(32 + 4)(102 + 4)(292 + 4) = (12 + 4)(62 + 4)(4782 + 4)
= (142 + 4)(4782 + 4) = (32 + 4)(62 + 4)(102 + 4)(292 + 4).

263 + 6760 = 1562, 263 - 6760 = 1042.

67602 = 523 + 763 + 3563.

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

6762

67622 = 45724644, and 452 + 72 + 242 + 642 + 42.

67622 = 1613 + 2113 + 3183.

67622 = (116 + 117 + 118 + ... + 122)2 + (123 + 124 + 125 + ... + 129)2 + (130 + 131 + 132 + ... + 136)2 + ... + (270 + 271 + 272 + ... + 276)2.

67625 = 14137604632082588832 :
12 + 42 + 12 + 32 + 762 + 02 + 42 + 62 + 32 + 22 + 02 + 82 + 252 + 82 + 82 + 82 + 32 + 22 = 6762.

Page of Squares : First Upload March 10, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6763

67632 = 45738169, a square with different digits.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6766

1 / 6766 = 0.0001477978125, 12+42 + 772 + 92 + 72 + 82 + 12 + 252 = 6766.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6767

67672 = 45792289, and 42 + 52 + 72 + 92 + 22 + 22 + 82 + 92 = 182.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6768

67685 = 14200438331374829568 :
12 + 42 + 202 + 02 + 42 + 382 + 32 + 32 + 12 + 32 + 72 + 42 + 82 + 22 + 92 + 52 + 682 = 6768.

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

6769

1 / 6769 = 0.00014773230905, 142 + 772 + 32 + 232 + 02 + 92 + 02 + 52 = 6769,
1 / 6769 = 0.00014773230905, 142 + 772 + 32 + 232 + 02 + 92 + 052 = 6769,
1 / 6769 = 0.00014773230905, 142 + 772 + 32 + 232 + 092 + 02 + 52 = 6769,
1 / 6769 = 0.00014773230905, 142 + 772 + 32 + 232 + 092 + 052 = 6769.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6775

67752 = 45900625, and 45 * 900 / 6 + 25 = 6775.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6776

67762 = (22 + 7)(72 + 7)(2732 + 7).

67762 = 1323 + 3523 = 353 + 2263 + 3253.

67762 = 45914176, and 42 + 52 + 92 + 12 + 42 + 12 + 72 + 62 = 152.

Page of Squares : First Upload March 10, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6777

67772 = 363 + 2703 + 2973.

67772 = (120 + 121 + 122)2 + (123 + 124 + 125)2 + ... + (360 + 361 + 362)2.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6778

67785 = 14305657479204350368 :
12 + 42 + 32 + 02 + 52 + 652 + 72 + 472 + 92 + 22 + 02 + 42 + 32 + 52 + 02 + 32 + 62 + 82 = 6778.

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

6779

67795 = 14316213601441869899 :
12 + 432 + 12 + 622 + 132 + 62 + 02 + 142 + 42 + 182 + 62 + 92 + 82 + 92 + 92 = 6779.

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

6781

67815 = 14337344540346837901 :
12 + 42 + 332 + 72 + 32 + 442 + 52 + 42 + 02 + 32 + 462 + 82 + 372 + 92 + 02 + 12
= 12 + 432 + 372 + 32 + 442 + 52 + 42 + 02 + 32 + 42 + 62 + 82 + 372 + 92 + 02 + 12 = 6781.

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

6784

67842 = 46022656, and 4 + 6 * 0 + 226 * 5 * 6 = 4 - 6 * 0 + 226 * 5 * 6 = 6784.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6785

67852 = 1383 + 2013 + 3283.

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

6786

67862 = (22 + 9)(152 + 9)(1232 + 9) = (32 + 9)(72 + 9)(2102 + 9).

67862 = 1023 + 2733 + 2913.

67862 + 67872 + 67882 + ... + 68442 = 68452 + 68462 + 68472 + ... + 69022.

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

6787

The square root of 6787 is 82.383...., and 82 = 32 + 82 + 32.

67872 = 46063369, and 460 + 6336 - 9 = 6787.

Page of Squares : First Upload March 10, 2008 ; Last Revised March 10, 2008
by Yoshio Mimura, Kobe, Japan

6791

67912 = 1403 + 1983 + 3293.

67913 = 313185171671, and 312 + 312 + 82 + 52 + 172 + 12 + 672 + 12 = 6791.

Page of Squares : First Upload October 2, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6792

67925 = 14454011175870431232 :
12 + 442 + 542 + 02 + 12 + 12 + 172 + 52 + 82 + 72 + 02 + 42 + 312 + 232 + 22 = 6792.

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

6794

67942 = 253 + 2463 + 3153.

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

6796

67962 = 853 + 2833 + 2843.

67962 = 46185616, a zigzag square.

Page of Squares : First Upload March 10, 2008 ; Last Revised October 2, 2008
by Yoshio Mimura, Kobe, Japan

6797

67974 = 2134366912225681,
and 212 + 342 + 32 + 662 + 92 + 122 + 222 + 52 + 62 + 82 + 12 = 6797.

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

6798

67982 = 2373 + 2393 + 2683.

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