## 4000

4000^{2} = 200^{3} + 200^{3} = 20^{5} + 20^{5} + 20^{5} + 20^{5} + 20^{5}.

400040001 = 20001^{2}.

by Yoshio Mimura, Kobe, Japan

## 4002

4002^{2}± 5 are primes.

4002^{2} = 7^{4} + 19^{4} + 19^{4} + 63^{4}.

by Yoshio Mimura, Kobe, Japan

## 4004

4004^{5} = 1029130250245121024 : 10^{2} + 2^{2} + 9^{2} + 1^{2} + 30^{2} + 25^{2} + 0^{2} + 2^{2} + 45^{2} + 12^{2} + 10^{2} + 2^{2} + 4^{2} = 4004.

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

by Yoshio Mimura, Kobe, Japan

## 4007

4007^{2} = 34^{4} + 46^{4} + 46^{4} + 49^{4}.

4007^{5} = 1032991414928036807 :

10^{2} + 32^{2} + 9^{2} + 9^{2} + 1^{2} + 41^{2} + 4^{2} + 9^{2} + 28^{2} + 0^{2} + 3^{2} + 6^{2} + 8^{2} + 0^{2} + 7^{2} = 4007.

by Yoshio Mimura, Kobe, Japan

## 4011

4011^{2} = 41^{3} + 196^{3} + 204^{3}.

The square root of 4011 is 63.33245..., and 63 = 3^{2} + 3^{2} + 2^{2} + 4^{2} + 5^{2}.

by Yoshio Mimura, Kobe, Japan

## 4016

4016^{3} = 64771076096, and 6^{2} + 4^{2} + 7^{2} + 7^{2} + 10^{2} + 7^{2} + 60^{2} + 9^{2} + 6^{2} = 4016.

by Yoshio Mimura, Kobe, Japan

## 4017

S_{2}(4017) = S_{2}(3076) + S_{2}(3293), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

4017^{2} = 27^{2} + 28^{2} + 29^{2} + 30^{2} + 31^{2} + ... + 364^{2}.

by Yoshio Mimura, Kobe, Japan

## 4018

4018^{2} = 7^{4} + 21^{4} + 21^{4} + 63^{4} = 17^{4} + 23^{4} + 53^{4} + 53^{4}.

by Yoshio Mimura, Kobe, Japan

## 4020

Komachi equations:

4020^{2} = 1^{2} * 2345^{2} / 6^{2} / 7^{2} * 8^{2} * 9^{2} = 1^{2} * 2^{2} / 3^{2} / 4^{2} * 5^{2} * 67^{2} * 8^{2} * 9^{2}

= 1^{2} / 2^{2} * 3^{2} * 45^{2} * 67^{2} * 8^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 4022

4022^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 4023

4023^{2} = 16184529, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 4024

4024^{2} = 100^{3} + 162^{3} + 222^{3}.

by Yoshio Mimura, Kobe, Japan

## 4026

330^{k} + 1353^{k} + 4026^{k} + 4092^{k} are squares for k = 1,2,3 (99^{2}, 5907^{2}, 369171^{2}).

by Yoshio Mimura, Kobe, Japan

## 4027

4027^{2} = 16216729, 1 - 6 + 2 * 16 * 7 * 2 * 9 = 4027.

by Yoshio Mimura, Kobe, Japan

## 4028

4028^{2} = 16224784, 1 * 6 / 2 * 24 * 7 * 8 - 4 = 4028.

by Yoshio Mimura, Kobe, Japan

## 4031

4031^{2} = 81^{3} + 94^{3} + 246^{3}.

4031^{2} = 16248961, 1 * 624 * 8 - 961 = 4031.

by Yoshio Mimura, Kobe, Japan

## 4032

4032^{2} = 120^{3} + 174^{3} + 210^{3} = 16^{4} + 32^{4} + 48^{4} + 56^{4} = 24^{4} + 48^{4} + 48^{4} + 48^{4} = 10^{5} + 14^{5} + 14^{5} + 20^{5} + 26^{5}.

4032^{2} = 16257024, 1 * 6 * 2 / 5 * 70 * 24 = 4032.

4032^{2} = (2^{2} - 1)(3^{2} - 1)(15^{2} - 1)(55^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(8^{2} - 1)(15^{2} - 1)

= (5^{2} - 1)(15^{2} - 1)(55^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(8^{2} - 1)(15^{2} - 1) = (4^{3} + 8)(6^{3} + 8)(10^{3} + 8).

Komachi equations:

4032^{2} = 12^{2} * 3456^{2} * 7^{2} / 8^{2} / 9^{2} = 98^{2} / 7^{2} / 6^{2} * 54^{2} * 32^{2} */ 1^{2}

= 9^{2} * 8^{2} * 7^{2} * 6^{2} * 5^{2} * 4^{2} / 3^{2} * 2^{2} / 10^{2} = 9^{2} * 8^{2} * 7^{2} * 6^{2} / 5^{2} * 4^{2} / 3^{2} / 2^{2} * 10^{2}

= 9^{2} * 8^{2} * 7^{2} / 6^{2} / 5^{2} * 4^{2} * 3^{2} * 2^{2} * 10^{2} = 98^{2} / 7^{2} * 6^{2} / 5^{2} * 4^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 4039

S_{2}(4039) = S_{2}(580) + S_{2}(4035), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

4039^{2} = S_{2}(265) + S_{2}(311).

by Yoshio Mimura, Kobe, Japan

## 4040

4040^{2} = 72^{3} + 160^{3} + 228^{3} = 74^{3} + 169^{3} + 223^{3} = 124^{3} + 177^{3} + 207^{3}.

by Yoshio Mimura, Kobe, Japan

## 4042

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

## 4044

4044^{2} = 16353936, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 4045

4045^{4} = 267715862100625, and 2^{2} + 6^{2} + 7^{2} + 7^{2} + 1^{2} + 58^{2} + 6^{2} + 21^{2} + 0^{2} + 0^{2} + 6^{2} + 2^{2} + 5^{2} = 4045.

by Yoshio Mimura, Kobe, Japan

## 4047

4047^{2} = 16378209, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 4048

4048^{2} = (19^{2} + 7)(211^{2} + 7) = (2^{2} + 7)(4^{2} + 7)(13^{2} + 7)(19^{2} + 7) = (2^{2} + 7)(4^{2} + 7)(9^{2} + 7)(27^{2} + 7)

= (3^{2} + 7)(4^{2} + 7)(211^{2} + 7) = (4^{2} + 7)(27^{2} + 7)(31^{2} + 7).

A cubic polynomial :

(X + 756^{2})(X + 4048^{2})(X + 28497^{2}) = X^{3} + 28793^{2}X^{2} + 117390252^{2}X + 87209027136^{2}.

by Yoshio Mimura, Kobe, Japan

## 4049

4049^{2} = 64^{3} + 168^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 4050

4050^{2} = 45^{4} + 45^{4} + 45^{4} + 45^{4}.

by Yoshio Mimura, Kobe, Japan

## 4054

4054^{2} = 16434916, and 1^{2} + 6^{2} + 4^{2} + 3^{2} + 4^{2} + 9^{2} + 1^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 4056

4056^{2} = (1^{2} + 3)(6^{2} + 3)(7^{2} + 3)(45^{2} + 3).

65^{2} + 4056 = 91^{2}, 65^{2} - 4056 = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 4058

4058^{2} = 16467364, 1 * 6 * 4 + 673 * 6 - 4 = 4058.

by Yoshio Mimura, Kobe, Japan

## 4060

4060 = (1^{2} + 2^{2} + 3^{2} + ... + 87^{2}) / (1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}).

4060^{2} = 2^{5} + 8^{5} + 12^{5} + 23^{5} + 25^{5}.

S_{2}(4060) = S_{2}(2650) + S_{2}(3642), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 4062

4062^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 4063

4063 = (1^{2} + 2^{2} + 3^{2} + ... + 119^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 7^{2}).

by Yoshio Mimura, Kobe, Japan

## 4065

4065 = (1^{2} + 2^{2} + 3^{2} + ... + 135^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 8^{2}).

by Yoshio Mimura, Kobe, Japan

## 4068

4068^{2} = 16^{3} + 54^{3} + 254^{3}.

1 / 4068 = 0.0002458210422, 24^{2} + 58^{2} + 2^{2} + 10^{2} + 4^{2} + 2^{2} + 2^{2} = 4068.

by Yoshio Mimura, Kobe, Japan

## 4069

4069^{2} = 16556761, 1 + 6 - 5 * 5 + 67 * 61 = 4069.

by Yoshio Mimura, Kobe, Japan

## 4070

4070 = (1^{2} + 2^{2} + 3^{2} + ... + 55^{2}) / (1^{2} + 2^{2} + 3^{2}).

4070^{2} = S_{2}(39) + S_{2}(367), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + n^{2}.

4070^{2} = 16564900, 1 * 65 * 64 - 90 + 0 = 4070.

by Yoshio Mimura, Kobe, Japan

## 4072

4072^{2} = 22^{4} + 22^{4} + 34^{4} + 62^{4}.

by Yoshio Mimura, Kobe, Japan

## 4073

4073^{2} = 27^{4} + 42^{4} + 42^{4} + 56^{4}.

by Yoshio Mimura, Kobe, Japan

## 4074

4074^{2} = 16597476, 1 + 65 * 9 * 7 - 4 * 7 + 6 = 4074.

4074^{2} = (1^{2} + 3)(12^{2} + 3)(168^{2} + 3) = 4074,(2^{2} + 3)(9^{2} + 3)(168^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 4076

1 / 4076 = 0.00024533856722, 2^{2} + 45^{2} + 3^{2} + 38^{2} + 5^{2} + 6^{2} + 7^{2} + 22^{2} = 4076.

by Yoshio Mimura, Kobe, Japan

## 4077

4077^{5} = 1126428311047604157 : 11^{2} + 26^{2} + 42^{2} + 8^{2} + 31^{2} + 10^{2} + 4^{2} + 7^{2} + 6^{2} + 0^{2} + 4^{2} + 15^{2} + 7^{2} = 4077.

by Yoshio Mimura, Kobe, Japan

## 4078

6948^{2} + 4078^{2} = 64904788, a mosaic square.

by Yoshio Mimura, Kobe, Japan

## 4082

4082^{2} = 16662724, 1 * 66 * 62 - 7 * 2 + 4 = 4082.

by Yoshio Mimura, Kobe, Japan

## 4086

4086 = (1^{2} + 2^{2} + 3^{2} + ... + 567^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 35^{2}).

4086^{2} = 87^{3} + 157^{3} + 230^{3}.

4086^{2} = 16662724, 16 * 6 * 9 * 5 - 39 * 6 = 4086.

by Yoshio Mimura, Kobe, Japan

## 4087

4087^{2} = 25^{3} + 136^{3} + 242^{3} = 74^{3} + 121^{3} + 244^{3}.

by Yoshio Mimura, Kobe, Japan

## 4088

4088^{2} = 16711744, 1^{2} + 6^{2} + 7^{2} + 1^{2} + 1^{2} + 7^{2} + 4^{2} + 4^{2} = 13^{2}.

4088^{2} = 442^{2} + 443^{2} + 444^{2} + 445^{2} + 446^{2} + ... + 514^{2}.

by Yoshio Mimura, Kobe, Japan

## 4089

4089^{k} + 9870^{k} + 15792^{k} + 25474^{k} are squares for k = 1,2,3 (235^{2}, 31819^{2}, 4636691^{2}).

by Yoshio Mimura, Kobe, Japan

## 4092

4092^{2} = 4^{4} + 36^{4} + 44^{4} + 58^{4}.

330^{k} + 1353^{k} + 4026^{k} + 4092^{k} are squares for k = 1,2,3 (99^{2}, 5907^{2}, 369171^{2}).

The integral triangle of sides 3330, 10057, 10571 has square area 4092^{2}.

by Yoshio Mimura, Kobe, Japan

## 4094

4094^{2}± 3 are primes.

4094^{2} = 18^{3} + 151^{3} + 237^{3} = 107^{3} + 145^{3} + 232^{3}.

Komachi equation: 4094^{2} = 1^{2} * 23^{2} / 4^{2} * 56^{2} / 7^{2} * 89^{2}.

4094^{3} = 68618862584, and 6^{2} + 8^{2} + 61^{2} + 8^{2} + 8^{2} + 6^{2} + 2^{2} + 5^{2} + 8^{2} + 4^{2} = 4084.

4094^{5} = 1150109502276960224, and 1^{2} + 1^{2} + 5^{2} + 0^{2} + 10^{2} + 9^{2} + 50^{2} + 2^{2} + 27^{2} + 6^{2} + 9^{2} + 6^{2} + 0^{2} + 22^{2} + 4^{2} = 1^{2} + 1^{2} + 50^{2} + 10^{2} + 9^{2} + 5^{2} + 0^{2} + 2^{2} + 27^{2} + 6^{2} + 9^{2} + 6^{2} + 0^{2} + 22^{2} + 4^{2} = 4094.

by Yoshio Mimura, Kobe, Japan

## 4095

4095^{2} = 16769025, 1 + 6 * 76 * 9 + 0 - 2 * 5 = 4095.

4095^{2} + 4096^{2} + 4097^{2} + ... + 4140^{2} = 4141^{2} + 4142^{2} + 4143^{2} + ... + 4185^{2}.

Komachi equations

4095^{2} = 9^{2} * 8^{2} * 7^{2} * 65^{2} * 4^{2} / 32^{2} */ 1^{2} = 9^{2} / 8^{2} * 7^{2} * 65^{2} / 4^{2} * 32^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 4096

The square of 64.

4096^{2} = 9^{3} + 58^{3} + 255^{3} = 9^{3} + 183^{3} + 220^{3} = 22^{3} + 57^{3} + 255^{3}.

1 / 4096 = 0.000244140625, 244140625 = 15625^{2}.

4096^{2} = (3^{2} + 7)(5^{2} + 7)(181^{2} + 7).

Komachi equations

4096^{2} = 12^{4} * 3^{4} / 4^{4} + 56^{4} / 7^{4} * 8^{4} - 9^{4} = 12^{4} * 3^{4} / 4^{4} * 56^{4} / 7^{4} * 8^{4} / 9^{4}

= - 12^{4} * 3^{4} / 4^{4} + 56^{4} / 7^{4} * 8^{4} + 9^{4},

4096^{2} = 12^{6} * 3^{6} * 4^{6} * 56^{6} / 7^{6} / 8^{6} / 9^{6} = 12^{6} * 3^{6} * 4^{6} / 56^{6} * 7^{6} * 8^{6} / 9^{6}.

4096^{2} = 16777216, 1^{2} + 6^{2} + 7^{2} + 7^{2} + 7^{2} + 2^{2} + 1^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 4097

4097^{2} = 1^{4} + 8^{4} + 8^{4} + 64^{4} = 1^{6} + 4^{6} + 4^{6} + 16^{6}.

4097^{2} = 16785409, a square with different digits.

85^{k} + 1649^{k} + 4097^{k} + 4573^{k} are squares for k = 1,2,3 (102^{2}, 6358^{2}, 410958^{2}).

by Yoshio Mimura, Kobe, Japan

## 4098

4098^{2} = (1^{2} + 5)(1673^{2} + 5).

by Yoshio Mimura, Kobe, Japan