## 4100

4100^{2} = 16810000, and 16 = 4^{2}, 810000 = 900^{2}.

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

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

by Yoshio Mimura, Kobe, Japan

## 4102

4102^{2} = 16826404, 1 * 682 * 6 + 40 / 4 = 4102.

by Yoshio Mimura, Kobe, Japan

## 4104

4104^{2} = (7^{2} + 8)(12^{2} + 8)(44^{2} + 8) = (7^{2} + 8)(8^{2} + 8)(64^{2} + 8).

228^{k} + 4104^{k} + 5700^{k} + 7657^{k} are squares for k = 1,2,3 (133^{2}, 10393^{2}, 838603^{2}).

Komachi equations:

4104^{2} = 9^{2} * 8^{2} * 76^{2} * 5^{2} / 4^{2} * 3^{2} * 2^{2} / 10^{2} = 9^{2} * 8^{2} * 76^{2} / 5^{2} / 4^{2} * 3^{2} / 2^{2} * 10^{2}

= 9^{2} / 8^{2} * 76^{2} / 5^{2} * 4^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 4106

4106^{2}± 3 are primes.

4106^{2} = 16859236, 1^{2} + 6^{2} + 8^{2} + 5^{2} + 9^{2} + 2^{2} + 3^{2} + 6^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 4107

4107^{2} = 16867449, 1 * 68 * 67 - 449 = 4107.

by Yoshio Mimura, Kobe, Japan

## 4108

2080^{k} + 3497^{k} + 4004^{k} + 4108^{k} are squares for k = 1,2,3 (117^{2}, 7033^{2}, 430443^{2}).

4108 = (1^{2} + 2^{2} + 3^{2} + ... + 39^{2}) / (1^{2} + 2^{2}),

4108 = (1^{2} + 2^{2} + 3^{2} + ... + 552^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 34^{2}).

by Yoshio Mimura, Kobe, Japan

## 4109

1 / 4109 = 0.00024336...., and 24336 = 156^{2}.

4109^{2} = S_{2}(286) + S_{2}(300), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

Komachi equation: 4109^{2} = 98^{2} * 7^{2} * 6^{2} + 5^{2} - 4^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 4112

4112^{2} = 63^{3} + 121^{3} + 246^{3} = 4^{4} + 16^{4} + 16^{4} + 64^{4} = 2^{8} + 4^{8} + 4^{8} + 8^{8}.

by Yoshio Mimura, Kobe, Japan

## 4113

4113^{2} = 16916769, 1 * 691 * 6 - 7 * 6 + 9 = 4113.

by Yoshio Mimura, Kobe, Japan

## 4115

4115^{2} = 16933225, 1^{2} + 6^{2} + 9^{2} + 3^{2} + 3^{2} + 2^{2} + 2^{2} + 5^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 4116

4116^{2} = 44^{3} + 150^{3} + 238^{3}.

4116^{2} = (2^{2} + 3)(3^{2} + 3)(12^{2} + 3)(37^{2} + 3) = (9^{2} + 3)(12^{2} + 3)(37^{2} + 3).

Komachi equations: 4116^{2} = 98^{2} * 7^{2} / 6^{2} * 54^{2} / 3^{2} * 2^{2} */ 1^{2}.

4116^{2} = (11 + 12 + 13 + ... + 38)^{2} + (39 + 40 + 41 + ... + 66)^{2} + (67 + 68 + 69 + ... + 94)^{2} + ... + (95 + 96 + 97 + ... + 122)^{2}.

by Yoshio Mimura, Kobe, Japan

## 4118

4118^{2} = S_{2}(229) + S_{2}(338), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 4119

4119^{2} = 16966161, a square with just 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 4122

786^{k} + 1614^{k} + 3882^{k} + 4122^{k} are squares for k = 1,2,3 (102^{2}, 5940^{2}, 365004^{2}).

by Yoshio Mimura, Kobe, Japan

## 4125

4125^{2} = 12^{3} + 34^{3} + 257^{3}.

by Yoshio Mimura, Kobe, Japan

## 4126

4126^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 4128

4128^{2} = 240^{3} + 20^{5} + 4^{7}.

by Yoshio Mimura, Kobe, Japan

## 4130

4130^{2} = (2^{2} + 6)(17^{2} + 6)(76^{2} + 6).

4130^{2} = 1^{5} + 9^{5} + 15^{5} + 18^{5} + 27^{5}.

by Yoshio Mimura, Kobe, Japan

## 4131

4131^{2} = 9^{3} + 162^{3} + 234^{3} = 65^{3} + 196^{3} + 210^{3}.

4131^{2} = 17065161, 170 * 6 + 51 * 61 = 4131.

by Yoshio Mimura, Kobe, Japan

## 4132

4132^{2} = 17073424, a zigzag square.

4132^{2} = 17073424, 1^{2} + 7^{2} + 0^{2} + 7^{2} + 3^{2} + 4^{2} + 2^{2} + 4^{2} = 12^{2}.

4132^{2} = 17073424, 1 + 707 + 3424 = 4132.

by Yoshio Mimura, Kobe, Japan

## 4137

4137^{2} = 72^{3} + 140^{3} + 241^{3}.

by Yoshio Mimura, Kobe, Japan

## 4138

4138^{5} = 1213255955047723168 : 1^{2} + 2^{2} + 1^{2} + 3^{2} + 2^{2} + 5^{2} + 5^{2} + 9^{2} + 55^{2} + 0^{2} + 4^{2} + 7^{2} + 7^{2} + 23^{2} + 16^{2} + 8^{2} = 1^{2} + 2^{2} + 1^{2} + 3^{2} + 2^{2} + 55^{2} + 9^{2} + 5^{2} + 5^{2} + 0^{2} + 4^{2} + 7^{2} + 7^{2} + 23^{2} + 16^{2} + 8^{2} =
12^{2} + 1^{2} + 3^{2} + 25^{2} + 5^{2} + 9^{2} + 55^{2} + 0^{2} + 4^{2} + 7^{2} + 7^{2} + 2^{2} + 3^{2} + 1^{2} + 6^{2} + 8^{2} = 4138.

by Yoshio Mimura, Kobe, Japan

## 4140

S_{2}(4140) = S_{2}(3170) + S_{2}(3394), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 4141

4141^{2} = 17147881, 1 * 7 - 1 + 47 * 88 - 1 = 4141.

by Yoshio Mimura, Kobe, Japan

## 4144

4144^{2} = 17172736, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 4148

4148 = (1^{2} + 2^{2} + 3^{2} + ... + 152^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 9^{2}).

by Yoshio Mimura, Kobe, Japan

## 4149

4149^{2} = 16^{5} + 20^{5} + 20^{5} + 25^{5}.

by Yoshio Mimura, Kobe, Japan

## 4150

4150^{2}± 3 are primes.

4150^{2} = 30^{3} + 85^{3} + 255^{3}.

by Yoshio Mimura, Kobe, Japan

## 4151

4151^{2} = 17230801, a zigzag square.

4151^{5} = 1232433988374745751 : 1^{2} + 23^{2} + 2^{2} + 43^{2} + 3^{2} + 9^{2} + 8^{2} + 8^{2} + 37^{2} + 4^{2} + 7^{2} + 4^{2} + 5^{2} + 7^{2} + 5^{2} + 1^{2} = 4151.

by Yoshio Mimura, Kobe, Japan

## 4152

4152^{2} = 106^{3} + 167^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 4153

4153^{2} = 17247409, 1 * 7 * 2 * 4 * 74 + 0 + 9 = 4153.

4153^{4} = 297473117213281, and 2^{2} + 9^{2} + 7^{2} + 47^{2} + 31^{2} + 1^{2} + 7^{2} + 2^{2} + 1^{2} + 3^{2} + 28^{2} + 1^{2} = 4153.

by Yoshio Mimura, Kobe, Japan

## 4155

4155^{2} = 56^{3} + 177^{3} + 226^{3}.

by Yoshio Mimura, Kobe, Japan

## 4157

4157^{2} = 17280649, a zigzag square with different digits.

by Yoshio Mimura, Kobe, Japan

## 4158

4158^{2} = 17288964, 1 * 7 / 2 * 88 * 9 * 6 / 4 = 4158.

The integral triangle of sides 11907, 17545, 29354 has square area 4158^{2}.

by Yoshio Mimura, Kobe, Japan

## 4159

4159^{2} = 60^{3} + 118^{3} + 249^{3}.

(4159^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(6^{2} - 1)(7^{2} - 1)(12^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(8^{2} - 1)(9^{2} - 1)(12^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 4160

4160^{2} = 4^{6} + 8^{6} + 8^{6} + 16^{6}.

Komachi equation: 4160^{2} = 12^{2} / 3^{2} * 4^{2} * 5^{2} * 6^{2} * 78^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 4163

Loop of length 56 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

4163 - 5650 - 5636 - 4432 - ... - 8450 - 9556 - 12161 - 4163

(Note f(4163) = 41^{2} + 63^{2} = 5650, f(5650) = 56^{2} + 50^{2} = 5636, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 4164

4164^{2} = 18^{3} + 187^{3} + 221^{3}.

1 / 4164 = 0.0002401...., and 2401 = 49^{2}.

4164^{2} = 17338896, 173 * 3 * 8 + 8 * 9 / 6 = 4164.

by Yoshio Mimura, Kobe, Japan

## 4165

4165^{k} + 7973^{k} + 17017^{k} + 27489^{k} are squares for k = 1,2,3 (238^{2}, 33558^{2}, 5126282^{2}).

by Yoshio Mimura, Kobe, Japan

## 4166

4166^{2} = 17355556, a square with odd digits except the last digit 6.

4166^{2} = 145^{3} + 170^{3} + 211^{3}.

1 / 4166 = 0.000240038406144983197....,

and 2^{2} + 400^{2} + 38^{2} + 4061^{2} + 4^{2} + 49^{2} + 831^{2} + 97^{2} = 4166^{2}.

4166^{2} = 17355556, 1 + 73 * 55 + 5 * 5 * 6 = 4166.

by Yoshio Mimura, Kobe, Japan

## 4168

4168^{2}± 3 are primes.

4168^{2} = 51^{3} + 133^{3} + 246^{3}.

1 / 4168 = 0.0002399232245...., 2^{2} + 39^{2} + 9^{2} + 23^{2} + 2^{2} + 2^{2} + 45^{2} = 4168.

by Yoshio Mimura, Kobe, Japan

## 4169

4169^{2} = (3^{2} + 2)(1257^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 4170

4170^{2}= 366 x 367 + 367 x 368 + 368 x 369 + 369 x 370 + ... + 465 x 466.

by Yoshio Mimura, Kobe, Japan

## 4172

4172^{2} = 17405584, 1^{2} + 7^{2} + 4^{2} + 0^{2} + 5^{2} + 5^{2} + 8^{2} + 4^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 4173

4173^{2} = 17413929, 17 * 41 / 3 * 9 * 2 - 9 = 4173.

by Yoshio Mimura, Kobe, Japan

## 4174

4174^{2} = 17422276, 1 - 7 + 4222 - 7 * 6 = 4174.

by Yoshio Mimura, Kobe, Japan

## 4175

4175^{2} = 17430625, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 4177

Loop of length 56 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

4177 - 7610 - 5876 - ... - 3878 - 7528 - 6409 - 4177

(Note f(4177) = 41^{2} + 77^{2} = 7610, f(7610) = 76^{2} + 10^{2} = 5876, etc. See 41)

4177^{2} = 9^{4} + 24^{4} + 24^{4} + 64^{4}.

4177^{2} = 17447329, 1^{2} + 7^{2} + 4^{2} + 4^{2} + 7^{2} + 3^{2} + 2^{2} + 9^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 4179

4179^{2} = 17464041, 174 * 6 * 4 + 0 + 4 - 1 = 4179.

4179^{2}± 2 are primes.

4179^{2} = 17464041, and 1^{2} + 7^{2} + 4^{2} + 64^{2} + 0^{2} + 4^{2} + 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 4180

4180 = (1^{2} + 2^{2} + 3^{2} + ... + 104^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 6^{2}).

by Yoshio Mimura, Kobe, Japan

## 4181

4181^{2} = 41^{3} + 62^{3} + 258^{3}.

4181^{4} = 305577005139121, and 30^{2} + 5^{2} + 5^{2} + 7^{2} + 7^{2} + 0^{2} + 0^{2} + 51^{2} + 3^{2} + 9^{2} + 1^{2} + 21^{2} = 4181.

by Yoshio Mimura, Kobe, Japan

## 4182

4182^{2} = (2^{2} + 2)(24^{2} + 2)(71^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 4184

4184^{2} = 160^{3} + 164^{3} + 208^{3}.

4184^{2} = 17505856, 1^{2} + 7^{2} + 5^{2} + 0^{2} + 5^{2} + 8^{2} + 5^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 4186

4186^{2} = 17522596, 1^{2} + 7^{2} + 5^{2} + 2^{2} + 2^{2} + 5^{2} + 9^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 4190

4190^{2} = 17556100, 1 * 75 * 56 - 10 + 0 = 4190.

730^{k} + 3970^{ k }+4190^{ k }+8010^{k} are squares for k = 1,2,3 (130^{2}, 9900^{2}, 806500^{2}).

by Yoshio Mimura, Kobe, Japan

## 4191

4191^{2} = 87^{3} + 103^{3} + 251^{3}.

by Yoshio Mimura, Kobe, Japan

## 4192

4192^{2} = 2^{5} + 14^{5} + 22^{5} + 26^{5}.

by Yoshio Mimura, Kobe, Japan

## 4193

4193^{2} = 17581249, 17 + 58 * 1 * 2 * 4 * 9 = 4193.

by Yoshio Mimura, Kobe, Japan

## 4194

4194^{2} = 17589636, 1 * 75 * 8 / 9 * 63 - 6 = 4194.

by Yoshio Mimura, Kobe, Japan

## 4196

4196^{2} = 17606416, 1 + 7 * 606 - 41 - 6 = 4196.

by Yoshio Mimura, Kobe, Japan

## 4199

408^{k} + 1156^{k} + 2601^{k} + 3060^{k} are squares for k = 1,2,3 (85^{2}, 4199^{2}, 218773^{2}).

4199^{2} = 9^{3} + 38^{3} + 260^{3} = 137^{3} + 195^{3} + 197^{3}.

by Yoshio Mimura, Kobe, Japan