## 3800

3800^{2} = 14440000, and 144 = 12^{2}, 40000 = 200^{2}.

by Yoshio Mimura, Kobe, Japan

## 3801

3801^{2} = 49^{3} + 54^{3} + 242^{3}.

The square root of 3801 is 61.65..., and 61 = 6^{2} + 5^{2}.

by Yoshio Mimura, Kobe, Japan

## 3807

3807^{2}± 2 are primes.

3807^{2} = 78^{3} + 138^{3} + 225^{3}.

A cubic polynomial :

(X + 1504^{2})(X + 2688^{2})(X + 3807^{2}) = X^{3} + 4897^{2}X^{2} + 12403488^{2}X + 15390756864^{2}.

by Yoshio Mimura, Kobe, Japan

## 3808

3808^{2}± 3 are primes.

3808^{2} = 81^{3} + 118^{3} + 231^{3}.

by Yoshio Mimura, Kobe, Japan

## 3809

A cubic polynomial :

(X + 628^{2})(X + 1104^{2})(X + 3591^{2}) = X^{3} + 3809^{2}X^{2} + 4613388^{2}X + 2489683392^{2}.

by Yoshio Mimura, Kobe, Japan

## 3811

3811^{2} = 14523721, 1 * 45 * 2 + 3721 = 3811.

by Yoshio Mimura, Kobe, Japan

## 3812

3812^{2} = 1^{5} + 4^{5} + 15^{5} + 18^{5} + 26^{5}.

1 / 3812 = 0.00026232948583, 2^{2} + 6^{2} + 23^{2} + 29^{2} + 48^{2} + 5^{2} + 8^{2} + 3^{2} = 3812.

by Yoshio Mimura, Kobe, Japan

## 3816

3816^{4} = 212047650164736,

and 2^{2} + 12^{2} + 0^{2} + 4^{2} + 7^{2} + 6^{2} + 5^{2} + 0^{2} + 1^{2} + 6^{2} + 47^{2} + 36^{2} = 3816,

and 2^{2} + 12^{2} + 0^{2} + 47^{2} + 6^{2} + 5^{2} + 0^{2} + 1^{2} + 6^{2} + 4^{2} + 7^{2} + 36^{2} = 3816.

by Yoshio Mimura, Kobe, Japan

## 3817

3817^{2} = 14569489, 1 + 456 * 9 - 4 * 8 * 9 = 3817.

1265^{k} + 1529^{k} + 3817^{k} + 5489^{k} are squares for k = 1,2,3 (110^{2}, 6974^{2}, 476014^{2}).

by Yoshio Mimura, Kobe, Japan

## 3819

3819^{2} = 14584761, 1 + 4 + 5 + 8 * 476 + 1 = 3819.

by Yoshio Mimura, Kobe, Japan

## 3821

1 / 3821 = 0.000261711593, 2^{2} + 6^{2} + 17^{2} + 1^{2} + 1^{2} + 59^{2} + 3^{2} = 3821.

by Yoshio Mimura, Kobe, Japan

## 3822

3822^{2} = (1^{2} + 3)(6^{2} + 3)(306^{2} + 3) = (2^{2} + 3)(6^{2} + 3)(12^{2} + 3)(19^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 3824

3824^{3} = 55918260224, and 5^{2} + 5^{2} + 9^{2} + 1^{2} + 8^{2} + 2^{2} + 60^{2} + 2^{2} + 2^{2} + 4^{2} = 3824.

by Yoshio Mimura, Kobe, Japan

## 3825

3825^{2} = (4^{2} + 9)(6^{2} + 9)(114^{2} + 9) = (6^{2} + 9)(12^{2} + 9)(46^{2} + 9).

3825^{2} = 10^{4} + 20^{4} + 35^{4} + 60^{4} = 15^{4} + 30^{4} + 30^{4} + 60^{4}.

by Yoshio Mimura, Kobe, Japan

## 3828

3828^{2} = 125^{3} + 130^{3} + 219^{3}.

by Yoshio Mimura, Kobe, Japan

## 3829

3829^{2} = 42^{3} + 113^{3} + 236^{3}.

by Yoshio Mimura, Kobe, Japan

## 3830

3830^{5} = 824126482214300000 : 8^{2} + 2^{2} + 4^{2} + 1^{2} + 2^{2} + 6^{2} + 48^{2} + 22^{2} + 1^{2} + 4^{2} + 30^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 3830.

by Yoshio Mimura, Kobe, Japan

## 3834

3834^{2} = 87^{3} + 90^{3} + 237^{3}.

3834^{2} = 14699556, 14 + 6 / 9 * 955 * 6 = 146 * 9 + 9 * 5 * 56 = 3834.

3834^{5} = 828439013253419424 : 8^{2} + 2^{2} + 8^{2} + 43^{2} + 9^{2} + 0^{2} + 13^{2} + 25^{2} + 3^{2} + 4^{2} + 19^{2} + 4^{2} + 24^{2} = 8^{2} + 28^{2} + 4^{2} + 3^{2} + 9^{2} + 0^{2} + 1^{2} + 32^{2} + 5^{2} + 34^{2} + 1^{2} + 9^{2} + 4^{2} + 24^{2} = 3834.

by Yoshio Mimura, Kobe, Japan

## 3835

1 / 3835 = 0.0002607561, 2^{2} + 6^{2} + 0^{2} + 7^{2} + 5^{2} + 61^{2} = 3835,

1 / 3835 = 0.0002607561, 2^{2} + 6^{2} + 07^{2} + 5^{2} + 61^{2} = 3835.

by Yoshio Mimura, Kobe, Japan

## 3837

3837^{4} = 216754037959761, and 2^{2} + 1^{2} + 6^{2} + 7^{2} + 5^{2} + 4^{2} + 0^{2} + 3^{2} + 7^{2} + 9^{2} + 59^{2} + 7^{2} + 6^{2} + 1^{2} = 3827.

by Yoshio Mimura, Kobe, Japan

## 3838

513^{k} + 1292^{k} + 3382^{k} + 3838^{k} are squares for k = 1,2,3 (95^{2}, 5301^{2}, 312265^{2}).

by Yoshio Mimura, Kobe, Japan

## 3839

3839^{2} = 98^{3} + 134^{3} + 225^{3}.

3839^{2} = 14737921, 1 + 47 + 3792 - 1 = 1 * 47 + 3792 * 1 = 3839.

by Yoshio Mimura, Kobe, Japan

## 3840

68^{2} + 3840 = 92^{2}, 68^{2} - 3840 = 28^{2}.

3840^{2} = (3^{2} - 1)(5^{2} - 1)(9^{2} - 1)(31^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 3841

3841^{2} = 14753281, 1^{2} + 4^{2} + 7^{2} + 5^{2} + 3^{2} + 2^{2} + 8^{2} + 1^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 3842

3842^{k} + 26216^{k} + 34465^{k} + 50398^{k} are squares for k = 1,2,3 (339^{2}, 66557^{2}, 13675599^{2}).

by Yoshio Mimura, Kobe, Japan

## 3844

The square of 62.

3844^{2} = 124^{3} + 186^{3} + 186^{3}.

1 / 3844 = 0.0002601..., and 2601 = 51^{2}.

3844^{2} = 14776336, 1 - 4 + 7 * 7 + 633 * 6 = 3844.

by Yoshio Mimura, Kobe, Japan

## 3846

3846^{2} = 14791716, 1 + 4 + 7 + 9 * 1 * 71 * 6 = 3846.

by Yoshio Mimura, Kobe, Japan

## 3847

3847^{2} = 41^{4} + 44^{4} + 44^{4} + 46^{4}.

by Yoshio Mimura, Kobe, Japan

## 3849

3849^{2} = 96^{3} + 113^{3} + 232^{3}.

by Yoshio Mimura, Kobe, Japan

## 3850

3850^{2}± 3 are primes.

3850^{2} = (1^{2} + 6)(2^{2} + 6)(4^{2} + 6)(7^{2} + 6)(13^{2} + 6) = (1^{2} + 6)(2^{2} + 6)(7^{2} + 6)(62^{2} + 6)

= (1^{2} + 6)(4^{2} + 6)(8^{2} + 6)(37^{2} + 6) = (2^{2} + 6)(13^{2} + 6)(92^{2} + 6) = (4^{2} + 6)(13^{2} + 6)(62^{2} + 6)

= (4^{2} + 6)(22^{2} + 6)(37^{2} + 6) = (4^{2} + 6)(7^{2} + 6)(8^{2} + 6)(13^{2} + 6) = (7^{2} + 6)(8^{2} + 6)(62^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 3852

3852^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 3854

3854 = (1^{2} + 2^{2} + 3^{2} + ... + 164^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 10^{2}).

3854^{2} = S_{2}(21) + S_{2}(354), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

3854^{2} = 539^{2} + 540^{2} + 541^{2} + 542^{2} + 543^{2} + 544^{2} + ... + 585^{2}.

by Yoshio Mimura, Kobe, Japan

## 3856

3856^{2} = 87^{3} + 116^{3} + 233^{3}.

by Yoshio Mimura, Kobe, Japan

## 3860

3860^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 3861

3861^{4} = 222228219397041, and 22^{2} + 22^{2} + 28^{2} + 21^{2} + 9^{2} + 39^{2} + 7^{2} + 0^{2} + 4^{2} + 1^{2} = 3861.

by Yoshio Mimura, Kobe, Japan

## 3862

3862^{2} = 1^{4} + 31^{4} + 37^{4} + 59^{4}.

1 / 3862 = 0.00025893319523, 25^{2} + 8^{2} + 9^{2} + 3^{2} + 3^{2} + 19^{2} + 52^{2} + 3^{2} = 3862.

by Yoshio Mimura, Kobe, Japan

## 3864

3864^{2} = 191^{3} + 24^{5} + 1^{7}.

by Yoshio Mimura, Kobe, Japan

## 3865

The quadratic polynomial 3865X^{2} - 15750X + 33201 takes the values 146^{2}, 131^{2}, 144^{2}, 179^{2}, 226^{2}, 279^{2} at X = 1, 2,..., 6,

by Yoshio Mimura, Kobe, Japan

## 3868

1 / 3868 = 0.000258531540847, 25^{2} + 8^{2} + 53^{2} + 15^{2} + 4^{2} + 08^{2} + 4^{2} + 7^{2} = 3868.

by Yoshio Mimura, Kobe, Japan

## 3869

3869^{2} = 14969161, 1 + 4 * 969 - 1 - 6 - 1 = 1 * 4 * 969 - 1 - 6 * 1 = 3869,

= 1 * 4 * 969 - 1 * 6 - 1 = 1 * 4 * 969 * 1 - 6 - 1 = 3869.

3869 = 62^{2}+ 05^{2} and 6205 = 38^{2} + 69^{2}.

by Yoshio Mimura, Kobe, Japan

## 3870

3870^{2} = (227 + 228 + 229)^{2} + (230 + 231 + 232)^{2} + (233 + 234 + 235)^{2} + ... + (296 + 297 + 298)^{2}.

by Yoshio Mimura, Kobe, Japan

## 3871

3871^{2} = 14984641, 1 * 4 * 984 - 64 - 1 = 1 + 49 * 84 - 6 * 41 = 3871.

by Yoshio Mimura, Kobe, Japan

## 3872

3872^{2} = 44^{4} + 44^{4} + 44^{4} + 44^{4}.

3872^{2} = 14992384, 1 * 4 * 992 - 3 * 8 * 4 = 3872.

3872^{2} = (1^{2} + 7)(2^{2} + 7)(13^{2} + 7)(31^{2} + 7) = (2^{2} + 7)(3^{2} + 7)(9^{2} + 7)(31^{2} + 7)

= (9^{2} + 7)(13^{2} + 7)(31^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 3873

3873^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 3875

1 / 3875 = 0.000258064..., and 258064 = 508^{2}.

by Yoshio Mimura, Kobe, Japan

## 3876

The integral triangle of sides 361, 104040, 104257 has square area 3876^{2}.

by Yoshio Mimura, Kobe, Japan

## 3878

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

3878 - 7528 - 6409 - 4177 - ... - 7251 - 7785 - 13154 - 3878

(Note f(3878) = 38^{2} + 78^{2} = 7528, f(7528) = 75^{2} + 28^{2} = 6409, etc. See 41)

The square root of 3878 is 62.273..., and 62 = 2^{2} + 7^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 3879

3879^{2}± 2 are primes.

3879^{2} = 46^{3} + 132^{3} + 233^{3}.

by Yoshio Mimura, Kobe, Japan

## 3880

3880^{2} = 99^{3} + 21^{5} + 10^{7}.

by Yoshio Mimura, Kobe, Japan

## 3882

3882^{2} = 15069924, 1506 + 99 * 24 = 3882.

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

## 3884

3884^{2} = 128^{3} + 138^{3} + 218^{3}.

by Yoshio Mimura, Kobe, Japan

## 3886

3886^{2} = 15100996, and 1^{2} + 5^{2} + 1^{2} + 0^{2} + 0^{2} + 9^{2} + 9^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 3887

3887^{2} = 13^{4} + 26^{4} + 52^{4} + 52^{4} = 1^{5} + 3^{5} + 3^{5} + 15^{5} + 27^{5}.

by Yoshio Mimura, Kobe, Japan

## 3888

3888^{2} = (1^{2} + 8)(2^{2} + 8)(8^{2} + 8)(44^{2} + 8) = (8^{2} + 8)(10^{2} + 8)(44^{2} + 8).

3888^{2} = 3^{5} + 3^{5} + 6^{5} + 15^{5} + 27^{5}.

A cubic polynomial :

(X + 3584^{2})(X + 3888^{2})(X + 16023^{2}) = X^{3} + 16873^{2}X^{2} + 85865808^{2}X + 223273967616.

Komachi equation: 3888^{2} = - 12^{2} * 3^{2} * 45^{2} + 6^{2} * 78^{2} * 9^{2}.

3888^{2} = 15116544, and 1^{2} + 5^{2} + 1^{2} + 1^{2} + 6^{2} + 5^{2} + 4^{2} + 4^{2} = 11^{2}.

by Yoshio Mimura, Kobe, Japan

## 3892

1 / x = 0.000256..., and 256 = 16^{2} for x = 3892, 3893,..., 3906.

by Yoshio Mimura, Kobe, Japan

## 3894

3894^{2} = 15163236, and 1^{2} + 5^{2} + 1^{2} + 6^{2} + 3^{2} + 2^{2} + 3^{2} + 6^{2} = 11^{2}.

by Yoshio Mimura, Kobe, Japan

## 3896

3896^{2} = 60^{3} + 114^{3} + 238^{3}.

3896^{2} = 15178816, and 15 - 1 * 7 + 8 * 81 * 6 = 15 * 1 - 7 + 8 * 81 * 6 = 3896.

by Yoshio Mimura, Kobe, Japan

## 3897

3897^{2} = 6^{4} + 42^{4} + 48^{4} + 51^{4}.

by Yoshio Mimura, Kobe, Japan

## 3898

1 / 3898 = 0.000256541816, 25^{2} + 6^{2} + 54^{2} + 1^{2} + 8^{2} + 16^{2} = 3898.

by Yoshio Mimura, Kobe, Japan