## 6100

6100^{2} = (4^{2} + 4)(1364^{2} + 4).

a 4-cycle : 6100^{2} = 37210000 -- 2100^{2} = 04410000 -- 4100^{2} = 16810000 -- 8100^{2} = 65610000,

(Other examples: 2916 -- 5030 -- 3009 -- 0540 -- 2916, and 1600 -- 5600 -- 3600 -- 9600 -- 1600)

by Yoshio Mimura, Kobe, Japan

## 6101

6101^{4} = 1385492247284401 :

and 1^{2} + 3^{2} + 8^{2} + 54^{2} + 9^{2} + 2^{2} + 24^{2} + 7^{2} + 28^{2} + 4^{2} + 40^{2} + 1^{2} = 6101.

by Yoshio Mimura, Kobe, Japan

## 6102

6102^{2} = 153^{3} + 171^{3} + 306^{3}.

1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + 6102^{2} = 75753395955, which consists of odd digits,

(the third 11-digit sum with odd digits. There are 3 such sums.)

by Yoshio Mimura, Kobe, Japan

## 6103

6103^{2} = 9^{3} + 172^{3} + 318^{3}.

by Yoshio Mimura, Kobe, Japan

## 6104

Komachi equation: 6104^{2} = 98^{2} / 7^{2} * 654^{2} / 3^{2} * 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 6105

6105^{2} + 6106^{2} + 6107^{2} + ... + 6160^{2} = 6161^{2} + 6162^{2} + 6163^{2} + ... + 6215^{2}.

by Yoshio Mimura, Kobe, Japan

## 6106

6106^{2} = 37283236, 37 * 2 * 83 - 2 * 3 * 6 = 6106.

1505^{k} + 4042^{k} + 4988^{k} + 6106^{k} are squares for k = 1,2,3 (129^{2}, 8987^{2}, 648999^{2}).

by Yoshio Mimura, Kobe, Japan

## 6108

6108^{2} = 50^{3} + 156^{3} + 322^{3}.

by Yoshio Mimura, Kobe, Japan

## 6110

6110^{2} = S_{2}(120) + S_{2}(479), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 6111

6111^{2} = (185 + 186 + 187 + ... + 205)^{2} + (206 + 207 + 208 + ... + 226)^{2}.

6111^{2} = (44 + 45 + 46)^{2} + (47 + 48 + 49)^{2} + (50 + 51 + 52)^{2} + ... + (332 + 333 + 334)^{2}.

by Yoshio Mimura, Kobe, Japan

## 6112

6112^{2} = 58^{3} + 136^{3} + 326^{3} = 66^{3} + 93^{3} + 331^{3} = 196^{3} + 240^{3} + 252^{3}.

by Yoshio Mimura, Kobe, Japan

## 6115

6115^{2} = 37393225, 37 + 3 + 9 * 3 * 225 = 6115.

by Yoshio Mimura, Kobe, Japan

## 6118

6118^{2} = 37429924, 3742 + 99 * 24 = 6118.

6118^{2} = 57^{2} + 59^{2} + 61^{2} + 63^{2} + 65^{2} + 67^{2} + 69^{2} + ... + 607^{2}.

6118^{5} = 8571313170730857568 :

8^{2} + 5^{2} + 7^{2} + 13^{2} + 13^{2} + 1^{2} + 7^{2} + 0^{2} + 73^{2} + 0^{2} + 8^{2} + 5^{2} + 7^{2} + 5^{2} + 6^{2} + 8^{2} = 6118.

by Yoshio Mimura, Kobe, Japan

## 6119

6119^{2} = 421^{3} - 420^{2} + 419^{2} - 418^{2} + 417^{2} - 416^{2} + ... + 1^{3}.

by Yoshio Mimura, Kobe, Japan

## 6120

6120^{2} = (11^{2} - 1)(16^{2} - 1)(35^{2} - 1) = (2^{2} - 1)(35^{2} - 1)(101^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(16^{2} - 1)(35^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 6122

6122^{2} = 9^{4} + 51^{4} + 57^{4} + 67^{4}.

by Yoshio Mimura, Kobe, Japan

## 6124

6124^{2} = 126^{3} + 245^{3} + 275^{3}.

by Yoshio Mimura, Kobe, Japan

## 6125

6125^{2} = (13^{2} + 6)(463^{2} + 6).

6125^{2} = (88 + 89 + 90 + ... + 122)^{2} + (123 + 124 + 125 + ... + 157)^{2}.

6125^{2} = (33 + 34 + 35 + 36 + 37)^{2} + (38 + 39 + 40 + 41 + 42)^{2} + (43 + 44 + 45 + 46 + 47)^{2} + ... + (278 + 279 + 280 + 281 + 282)^{2}.

by Yoshio Mimura, Kobe, Japan

## 6126

6126^{2} = 37527876, 375 * 2 * 7 + 876 = 6126.

by Yoshio Mimura, Kobe, Japan

## 6127

6127^{2} = 37540129, a square with different digits.

The square root of 6127 is 78.275..., and 78 = 2^{2} + 7^{2} + 5^{2}.

by Yoshio Mimura, Kobe, Japan

## 6128

6128^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 6129

6129^{2} = 37564641, 3 * 75 + 6 * 4 * 6 * 41 = 6129.

by Yoshio Mimura, Kobe, Japan

## 6131

The square root of 6131 is 78.3007024..., and 78 = 3^{2} + 0^{2} + 0^{2} + 7^{2} + 0^{2} + 2^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 6133

6133^{2} = 98^{3} + 102^{3} + 329^{3}.

6133^{2} = 37613689, 3 - 7 + 6136 - 8 + 9 = 6133.

6133^{5} = 8676904617957425893 : 8^{2} + 6^{2} + 7^{2} + 6^{2} + 9^{2} + 0^{2} + 4^{2} + 6^{2} + 1^{2} + 7^{2} + 9^{2} + 5^{2} + 74^{2} + 2^{2} + 5^{2} + 8^{2} + 9^{2} + 3^{2} = 6133.

by Yoshio Mimura, Kobe, Japan

## 6135

6135^{2} = 843^{2} + 844^{2} + 845^{2} + 846^{2} + 847^{2} + 848^{2} + ... + 892^{2}.

by Yoshio Mimura, Kobe, Japan

## 6136

The square root of 6136 is 78.332624..., and 78 = 3^{2} + 3^{2} + 2^{2} + 6^{2} + 2^{2} + 4^{2}.

6136^{2} = S_{2}(375) + S_{2}(391), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 6137

6137^{2} = 76^{3} + 209^{3} + 304^{3} = 19^{4} + 22^{4} + 52^{4} + 74^{4} = 19^{4} + 38^{4} + 38^{4} + 76^{4}.

by Yoshio Mimura, Kobe, Japan

## 6144

6144^{2} = 96^{3} + 160^{3} + 320^{3} = 160^{3} + 248^{3} + 264^{3} = 216^{3} + 236^{3} + 244^{3} = 16^{5} + 16^{5} + 16^{5} + 16^{5} + 32^{5}.

80^{2} + 6144 = 112^{2}, 80^{2} - 6144 = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 6146

6146^{2} = 37773316, a square with odd digits except the last digit 6.

6146^{2} = 168^{3} + 241^{3} + 267^{3} = 27^{4} + 35^{4} + 45^{4} + 75^{4}.

by Yoshio Mimura, Kobe, Japan

## 6147

6147^{2} = 13^{3} + 95^{3} + 333^{3}.

Komachi equation: 6147^{2} = 9^{3} + 8^{3} * 7^{3} * 6^{3} - 54^{3} + 3^{3} + 21^{3}.

by Yoshio Mimura, Kobe, Japan

## 6148

6148^{2} = 37797904, 37 + 7 * 97 * 9 + 0 * 4 = 6148.

by Yoshio Mimura, Kobe, Japan

## 6150

6150^{2} = (1^{2} + 9)(4^{2} + 9)(14^{2} + 9)(27^{2} + 9).

6150^{2} = 1^{3} + 147^{3} + 326^{3}.

by Yoshio Mimura, Kobe, Japan

## 6156

6156^{2} = (1^{2} + 8)(7^{2} + 8)(10^{2} + 8)(26^{2} + 8).

6156^{2} = 99^{3} + 333^{3} = 212^{3} + 238^{3} + 246^{3}.

by Yoshio Mimura, Kobe, Japan

## 6157

6157^{2} = 49^{3} + 200^{3} + 310^{3}.

by Yoshio Mimura, Kobe, Japan

## 6160

6160^{2} = 2^{7} + 2^{7} + 4^{7} + 8^{7} + 12^{7}.

6160^{2} = 100 x 101 + 102 x 103 + 104 x 105 + 106 x 107 + 108 x 109 + ... + 610 x 611.

by Yoshio Mimura, Kobe, Japan

## 6162

6162^{2}± 5 are primes.

6162^{2} = 13^{4} + 49^{4} + 51^{4} + 71^{4}.

by Yoshio Mimura, Kobe, Japan

## 6163

6163^{2} = 1^{5} + 10^{5} + 11^{5} + 28^{5} + 29^{5}.

6163^{5} = 8891209401336447043 :

8^{2} + 8^{2} + 9^{2} + 12^{2} + 0^{2} + 9^{2} + 40^{2} + 1^{2} + 3^{2} + 3^{2} + 6^{2} + 4^{2} + 47^{2} + 0^{2} + 43^{2} = 6163.

by Yoshio Mimura, Kobe, Japan

## 6164

6164^{2} = 581^{2} + 582^{2} + 583^{2} + 584^{2} + 585^{2} + 586^{2} + ... + 676^{2}.

6164^{4} = 1443612122050816,

and 1^{2} + 44^{2} + 3^{2} + 61^{2} + 2^{2} + 12^{2} + 2^{2} + 0^{2} + 5^{2} + 0^{2} + 8^{2} + 16^{2} = 6164.

by Yoshio Mimura, Kobe, Japan

## 6167

6167^{2} = 1^{4} + 46^{4} + 64^{4} + 64^{4} = 2^{4} + 37^{4} + 62^{4} + 68^{4}.

6167^{5} = 8920100390461615607 :

8^{2} + 9^{2} + 2^{2} + 0^{2} + 10^{2} + 0^{2} + 3^{2} + 9^{2} + 0^{2} + 46^{2} + 1^{2} + 6^{2} + 1^{2} + 5^{2} + 60^{2} + 7^{2} = 6167.

by Yoshio Mimura, Kobe, Japan

## 6168

6168^{2} = 18^{3} + 92^{3} + 334^{3} = 38^{3} + 186^{3} + 316^{3}.

6168^{2} = 38044224, 380 * 4 * 4 + 22 * 4 = 6168.

by Yoshio Mimura, Kobe, Japan

## 6169

6169^{2} = 38056561, 3^{2} + 8^{2} + 0^{2} + 5^{2} + 6^{2} + 5^{2} + 6^{2} + 1^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 6170

6170^{2} = (190 + 191 + 192 + 193 + 194)^{2} + (195 + 196 + 197 + 198 + 199)^{2} + (200 + 201 + 202 + 203 + 204)^{2} + ... + (305 + 306 + 307 + 308 + 309)^{2}.

by Yoshio Mimura, Kobe, Japan

## 6174

6174^{2} = (1^{2} + 5)(4^{2} + 5)(17^{2} + 5)(32^{2} + 5) = (11^{2} + 5)(17^{2} + 5)(32^{2} + 5)

= (2^{2} + 5)(3^{2} + 5)(17^{2} + 5)(32^{2} + 5) = (3^{2} + 5)(4^{2} + 5)(11^{2} + 5)(32^{2} + 5).

6174^{2} = 3^{3} + 57^{3} + 336^{3}.

by Yoshio Mimura, Kobe, Japan

## 6175

6175^{2} = 38130625, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 6176

6176^{2} = 38142976, a square with different digits.

6176 = (1^{2} + 2^{2} + 3^{2} + ... + 192^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 10^{2}).

by Yoshio Mimura, Kobe, Japan

## 6180

1065^{k} + 3360^{k} + 6180^{k} + 7620^{k} are squares for k = 1,2,3 (135^{2}, 10425^{2}, 847125^{2}).

by Yoshio Mimura, Kobe, Japan

## 6181

6181^{2} = 38204761, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 6183

6183^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 6184

6184^{2} = 36^{3} + 64^{3} + 336^{3}.

6184^{2} = 38241856, a zigzag square.

6184^{3} = 236487637504, and 23^{2} + 64^{2} + 8^{2} + 7^{2} + 6^{2} + 37^{2} + 5^{2} + 0^{2} + 4^{2} = 6184.

by Yoshio Mimura, Kobe, Japan

## 6185

6185^{4} = 1463385730350625,

and 1^{2} + 46^{2} + 3^{2} + 3^{2} + 8^{2} + 5^{2} + 7^{2} + 3^{2} + 0^{2} + 3^{2} + 5^{2} + 0^{2} + 62^{2} + 5^{2} = 6185.

by Yoshio Mimura, Kobe, Japan

## 6188

6188^{4} = 1466227025326336,

and 1^{2} + 46^{2} + 6^{2} + 2^{2} + 2^{2} + 7^{2} + 0^{2} + 2^{2} + 53^{2} + 2^{2} + 6^{2} + 33^{2} + 6^{2} = 6188,

and 14^{2} + 6^{2} + 62^{2} + 27^{2} + 0^{2} + 2^{2} + 5^{2} + 3^{2} + 2^{2} + 6^{2} + 3^{2} + 36^{2} = 6188.

by Yoshio Mimura, Kobe, Japan

## 6191

6191^{2} = 38328481, 3 * 8 * 32 * 8 + 48 - 1 = 6191.

by Yoshio Mimura, Kobe, Japan

## 6192

6192^{2} = (2^{2} + 8)(4^{2} + 8)(11^{2} + 8)(32^{2} + 8).

6192^{2} = 2^{5} + 14^{5} + 16^{5} + 20^{5} + 32^{5}.

by Yoshio Mimura, Kobe, Japan

## 6193

6193^{2} = 129^{3} + 234^{3} + 286^{3}.

704^{k} + 1034^{k} + 1870^{k} + 6193^{k} are squares for k = 1,2,3 (99^{2}, 6589^{2}, 495495^{2}).

by Yoshio Mimura, Kobe, Japan

## 6194

6194^{2} = 38365636, a zigzag square.

6194^{2} = (4^{2} + 3)(1421^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 6195

6195^{2} = 38378025, and 3^{2} + 8^{2} + 3^{2} + 78^{2} + 0^{2} + 2^{2} + 5^{2}.

by Yoshio Mimura, Kobe, Japan

## 6196

6196^{2} = 38390416, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 6199

6199^{2} = 38427601, a square with different digits.

by Yoshio Mimura, Kobe, Japan