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6100 - 6199

6100

61002 = (42 + 4)(13642 + 4).

a 4-cycle : 61002 = 37210000 -- 21002 = 04410000 -- 41002 = 16810000 -- 81002 = 65610000,
(Other examples: 2916 -- 5030 -- 3009 -- 0540 -- 2916, and 1600 -- 5600 -- 3600 -- 9600 -- 1600)

Page of Squares : First Upload January 28, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6101

61014 = 1385492247284401 :
and 12 + 32 + 82 + 542 + 92 + 22 + 242 + 72 + 282 + 42 + 402 + 12 = 6101.

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

6102

61022 = 1533 + 1713 + 3063.

12 + 22 + 32 + 42 + ... + 61022 = 75753395955, which consists of odd digits,
(the third 11-digit sum with odd digits. There are 3 such sums.)

Page of Squares : First Upload January 28, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6103

61032 = 93 + 1723 + 3183.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6104

Komachi equation: 61042 = 982 / 72 * 6542 / 32 * 22 */ 12.

Page of Squares : First Upload September 26, 2008 ; Last Revised October 29, 2010
by Yoshio Mimura, Kobe, Japan

6105

61052 + 61062 + 61072 + ... + 61602 = 61612 + 61622 + 61632 + ... + 62152.

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

6106

61062 = 37283236, 37 * 2 * 83 - 2 * 3 * 6 = 6106.

1505k + 4042k + 4988k + 6106k are squares for k = 1,2,3 (1292, 89872, 6489992).

Page of Squares : First Upload January 28, 2008 ; Last Revised June 24, 2011
by Yoshio Mimura, Kobe, Japan

6108

61082 = 503 + 1563 + 3223.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6110

61102 = S2(120) + S2(479), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6111

61112 = (185 + 186 + 187 + ... + 205)2 + (206 + 207 + 208 + ... + 226)2.

61112 = (44 + 45 + 46)2 + (47 + 48 + 49)2 + (50 + 51 + 52)2 + ... + (332 + 333 + 334)2.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6112

61122 = 583 + 1363 + 3263 = 663 + 933 + 3313 = 1963 + 2403 + 2523.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6115

61152 = 37393225, 37 + 3 + 9 * 3 * 225 = 6115.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6118

61182 = 37429924, 3742 + 99 * 24 = 6118.

61182 = 572 + 592 + 612 + 632 + 652 + 672 + 692 + ... + 6072.

61185 = 8571313170730857568 :
82 + 52 + 72 + 132 + 132 + 12 + 72 + 02 + 732 + 02 + 82 + 52 + 72 + 52 + 62 + 82 = 6118.

Page of Squares : First Upload January 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6119

61192 = 4213 - 4202 + 4192 - 4182 + 4172 - 4162 + ... + 13.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6120

61202 = (112 - 1)(162 - 1)(352 - 1) = (22 - 1)(352 - 1)(1012 - 1) = (32 - 1)(42 - 1)(162 - 1)(352 - 1).

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

6122

61222 = 94 + 514 + 574 + 674.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6124

61242 = 1263 + 2453 + 2753.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6125

61252 = (132 + 6)(4632 + 6).

61252 = (88 + 89 + 90 + ... + 122)2 + (123 + 124 + 125 + ... + 157)2.

61252 = (33 + 34 + 35 + 36 + 37)2 + (38 + 39 + 40 + 41 + 42)2 + (43 + 44 + 45 + 46 + 47)2 + ... + (278 + 279 + 280 + 281 + 282)2.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6126

61262 = 37527876, 375 * 2 * 7 + 876 = 6126.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6127

61272 = 37540129, a square with different digits.

The square root of 6127 is 78.275..., and 78 = 22 + 72 + 52.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6128

61282± 3 are primes.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

6129

61292 = 37564641, 3 * 75 + 6 * 4 * 6 * 41 = 6129.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6131

The square root of 6131 is 78.3007024..., and 78 = 32 + 02 + 02 + 72 + 02 + 22 + 42.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6133

61332 = 983 + 1023 + 3293.

61332 = 37613689, 3 - 7 + 6136 - 8 + 9 = 6133.

61335 = 8676904617957425893 : 82 + 62 + 72 + 62 + 92 + 02 + 42 + 62 + 12 + 72 + 92 + 52 + 742 + 22 + 52 + 82 + 92 + 32 = 6133.

Page of Squares : First Upload January 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6135

61352 = 8432 + 8442 + 8452 + 8462 + 8472 + 8482 + ... + 8922.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6136

The square root of 6136 is 78.332624..., and 78 = 32 + 32 + 22 + 62 + 22 + 42.

61362 = S2(375) + S2(391), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6137

61372 = 763 + 2093 + 3043 = 194 + 224 + 524 + 744 = 194 + 384 + 384 + 764.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6144

61442 = 963 + 1603 + 3203 = 1603 + 2483 + 2643 = 2163 + 2363 + 2443 = 165 + 165 + 165 + 165 + 325.

802 + 6144 = 1122, 802 - 6144 = 162.

Page of Squares : First Upload September 26, 2008 ; Last Revised August 2, 2011
by Yoshio Mimura, Kobe, Japan

6146

61462 = 37773316, a square with odd digits except the last digit 6.

61462 = 1683 + 2413 + 2673 = 274 + 354 + 454 + 754.

Page of Squares : First Upload September 26, 2008 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

6147

61472 = 133 + 953 + 3333.

Komachi equation: 61472 = 93 + 83 * 73 * 63 - 543 + 33 + 213.

Page of Squares : First Upload September 26, 2008 ; Last Revised October 29, 2010
by Yoshio Mimura, Kobe, Japan

6148

61482 = 37797904, 37 + 7 * 97 * 9 + 0 * 4 = 6148.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6150

61502 = (12 + 9)(42 + 9)(142 + 9)(272 + 9).

61502 = 13 + 1473 + 3263.

Page of Squares : First Upload September 26, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6156

61562 = (12 + 8)(72 + 8)(102 + 8)(262 + 8).

61562 = 993 + 3333 = 2123 + 2383 + 2463.

Page of Squares : First Upload September 26, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6157

61572 = 493 + 2003 + 3103.

Page of Squares : First Upload September 26, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6160

61602 = 27 + 27 + 47 + 87 + 127.

61602 = 100 x 101 + 102 x 103 + 104 x 105 + 106 x 107 + 108 x 109 + ... + 610 x 611.

Page of Squares : First Upload January 28, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6162

61622± 5 are primes.

61622 = 134 + 494 + 514 + 714.

Page of Squares : First Upload September 26, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

6163

61632 = 15 + 105 + 115 + 285 + 295.

61635 = 8891209401336447043 :
82 + 82 + 92 + 122 + 02 + 92 + 402 + 12 + 32 + 32 + 62 + 42 + 472 + 02 + 432 = 6163.

Page of Squares : First Upload September 26, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6164

61642 = 5812 + 5822 + 5832 + 5842 + 5852 + 5862 + ... + 6762.

61644 = 1443612122050816,
and 12 + 442 + 32 + 612 + 22 + 122 + 22 + 02 + 52 + 02 + 82 + 162 = 6164.

Page of Squares : First Upload January 28, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6167

61672 = 14 + 464 + 644 + 644 = 24 + 374 + 624 + 684.

61675 = 8920100390461615607 :
82 + 92 + 22 + 02 + 102 + 02 + 32 + 92 + 02 + 462 + 12 + 62 + 12 + 52 + 602 + 72 = 6167.

Page of Squares : First Upload September 26, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6168

61682 = 183 + 923 + 3343 = 383 + 1863 + 3163.

61682 = 38044224, 380 * 4 * 4 + 22 * 4 = 6168.

Page of Squares : First Upload January 28, 2008 ; Last Revised September 26, 2008
by Yoshio Mimura, Kobe, Japan

6169

61692 = 38056561, 32 + 82 + 02 + 52 + 62 + 52 + 62 + 12 = 142.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6170

61702 = (190 + 191 + 192 + 193 + 194)2 + (195 + 196 + 197 + 198 + 199)2 + (200 + 201 + 202 + 203 + 204)2 + ... + (305 + 306 + 307 + 308 + 309)2.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6174

61742 = (12 + 5)(42 + 5)(172 + 5)(322 + 5) = (112 + 5)(172 + 5)(322 + 5)
= (22 + 5)(32 + 5)(172 + 5)(322 + 5) = (32 + 5)(42 + 5)(112 + 5)(322 + 5).

61742 = 33 + 573 + 3363.

Page of Squares : First Upload September 26, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6175

61752 = 38130625, a zigzag square.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6176

61762 = 38142976, a square with different digits.

6176 = (12 + 22 + 32 + ... + 1922) / (12 + 22 + 32 + ... + 102).

Page of Squares : First Upload January 28, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

6180

1065k + 3360k + 6180k + 7620k are squares for k = 1,2,3 (1352, 104252, 8471252).

Page of Squares : First Upload June 24, 2011 ; Last Revised June 24, 2011
by Yoshio Mimura, Kobe, Japan

6181

61812 = 38204761, a square with different digits.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6183

61832± 2 are primes.

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

6184

61842 = 363 + 643 + 3363.

61842 = 38241856, a zigzag square.

61843 = 236487637504, and 232 + 642 + 82 + 72 + 62 + 372 + 52 + 02 + 42 = 6184.

Page of Squares : First Upload January 28, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

6185

61854 = 1463385730350625,
and 12 + 462 + 32 + 32 + 82 + 52 + 72 + 32 + 02 + 32 + 52 + 02 + 622 + 52 = 6185.

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

6188

61884 = 1466227025326336,
and 12 + 462 + 62 + 22 + 22 + 72 + 02 + 22 + 532 + 22 + 62 + 332 + 62 = 6188,
and 142 + 62 + 622 + 272 + 02 + 22 + 52 + 32 + 22 + 62 + 32 + 362 = 6188.

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

6191

61912 = 38328481, 3 * 8 * 32 * 8 + 48 - 1 = 6191.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6192

61922 = (22 + 8)(42 + 8)(112 + 8)(322 + 8).

61922 = 25 + 145 + 165 + 205 + 325.

Page of Squares : First Upload September 26, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6193

61932 = 1293 + 2343 + 2863.

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

Page of Squares : First Upload September 26, 2008 ; Last Revised January 4, 2014
by Yoshio Mimura, Kobe, Japan

6194

61942 = 38365636, a zigzag square.

61942 = (42 + 3)(14212 + 3).

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6195

61952 = 38378025, and 32 + 82 + 32 + 782 + 02 + 22 + 52.

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

6196

61962 = 38390416, a zigzag square.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan

6199

61992 = 38427601, a square with different digits.

Page of Squares : First Upload January 28, 2008 ; Last Revised January 28, 2008
by Yoshio Mimura, Kobe, Japan