## 3901

517^{k} + 893^{k} + 3525^{k} + 3901^{k} are squares for k = 1,2,3 (94^{2}, 5358^{2}, 322514^{2}).

by Yoshio Mimura, Kobe, Japan

## 3903

3903^{2} = 2^{4} + 3^{4} + 26^{4} + 62^{4} = 4^{4} + 8^{4} + 42^{4} + 59^{4}.

by Yoshio Mimura, Kobe, Japan

## 3904

3904^{2} = 15241216, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3907

3907^{2} = 15264649, a zigzag square.

(3907^{2} + 8) = (3^{2} + 8)(7^{2} + 8)(9^{2} + 8)(13^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 3908

3908^{4} = 233248156631296, and 2^{2} + 3^{2} + 32^{2} + 48^{2} + 15^{2} + 6^{2} + 6^{2} + 3^{2} + 12^{2} + 9^{2} + 6^{2} = 3908.

by Yoshio Mimura, Kobe, Japan

## 3911

3911 is the 10th prime for which the Legendre symbol (a/3911) = 1 for a = 1,2,...,12.

Page of Squares : First Upload July 30, 2007 ; Last Revised July 30, 2007by Yoshio Mimura, Kobe, Japan

## 3912

3912^{2} = 16^{3} + 36^{3} + 248^{3}.

by Yoshio Mimura, Kobe, Japan

## 3914

3914^{2} = 15319396, a square with odd digits except the last digit 6.

The square root of 3914 is 62.561..., and 62 = 5^{2} + 6^{2} + 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 3915

3915^{2} = 15327225, and 1^{2} + 5^{2} + 3^{2} + 2^{2} + 7^{2} + 2^{2} + 2^{2} + 5^{2} = 11^{2}.

Komachi equations:

3915^{2} = 9^{2} * 87^{2} * 6^{2} * 5^{2} / 4^{2} / 3^{2} * 2^{2} */ 1^{2} = 9^{2} * 87^{2} / 6^{2} * 5^{2} * 4^{2} * 3^{2} / 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 3916

3916^{2}± 3 are primes.

3916^{2} + 3917^{2} + 3918^{2} + ... + 3960^{2} = 3961^{2} + 3962^{2} + 3963^{2} + ... + 4004^{2}.

by Yoshio Mimura, Kobe, Japan

## 3918

3918^{2} = 15350724, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3920

Komachi equation: 3920^{2} = 9^{2} * 8^{2} * 7^{2} * 6^{2} / 54^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 3921

3921^{2} = 34^{3} + 158^{3} + 225^{3} = 108^{3} + 137^{3} + 226^{3}.

3921^{2} = 15374241, and 1^{2} + 5^{2} + 3^{2} + 7^{2} + 4^{2} + 2^{2} + 4^{2} + 1^{2} = 11^{2}.

3921^{2} = 15374241 , 1 + 53 * 74 + 2 - 4 * 1 = 1 * 53 * 74 + 2 - 4 + 1 = 3921.

by Yoshio Mimura, Kobe, Japan

## 3923

3923^{2} = 15389929, 1 + 5 + 3899 + 2 * 9 = 3923.

by Yoshio Mimura, Kobe, Japan

## 3924

3924^{2} = 15397776, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 3925

3925^{2} = 1^{3} - 2^{3} + 3^{3} - 4^{3} + 5^{3} - 6^{3} + ... + 311^{3} - 312^{3} + 313^{3}.

3925^{2} = 15405625, and 1 = 1^{2}, 5405625 = 2325^{2}.

by Yoshio Mimura, Kobe, Japan

## 3926

1 / 3926 = 0.00025471217524, 25^{2} + 47^{2} + 1^{2} + 21^{2} + 7^{2} + 5^{2} + 24^{2} = 3926.

by Yoshio Mimura, Kobe, Japan

## 3928

3928^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 3930

2460^{k} + 2490^{k} + 3930^{k} + 9345^{k} are squares for k = 1,2,3 (135^{2}, 10725^{2}, 952425^{2}).

by Yoshio Mimura, Kobe, Japan

## 3932

3932^{2} = 15460624, a zigzag square.

3932^{2} = 61^{3} + 96^{3} + 243^{3} = 62^{3} + 166^{3} + 220^{3} = 8^{4} + 16^{4} + 28^{4} + 62^{4} = 12^{4} + 24^{4} + 24^{4} + 62^{4}.

by Yoshio Mimura, Kobe, Japan

## 3933

3933^{5} = 941065239592294893 : 9^{2} + 4^{2} + 10^{2} + 6^{2} + 5^{2} + 23^{2} + 9^{2} + 5^{2} + 9^{2} + 22^{2} + 9^{2} + 48^{2} + 9^{2} + 3^{2} = 3933.

by Yoshio Mimura, Kobe, Japan

## 3934

3934^{2} = 77^{3} + 79^{3} + 244^{3}.

3934^{2} = 15476356, and 154 + 7 * 6 * 3 * 5 * 6 = 3934.

by Yoshio Mimura, Kobe, Japan

## 3937

49^{k} + 98^{k} + 170^{k} + 212^{k} are squares for k = 1,2,3 (23^{2}, 293^{2}, 3937^{2}).

by Yoshio Mimura, Kobe, Japan

## 3938

3938^{2} = 15507844, and 1^{2} + 5^{2} + 5^{2} + 0^{2} + 7^{2} + 8^{2} + 4^{2} + 4^{2} = 14^{2}.

3938^{2} = 15507844, 1 * 550 * 7 + 84 + 4 = 3938.

by Yoshio Mimura, Kobe, Japan

## 3942

3942^{2} = 15539364, 1 + 5 / 5 + 3936 + 4 = 1 * 5 + 5 + 3936 - 4 = 15 - 5 + 3936 - 4 = 3942.

3942^{4} = 241471833524496, and 24^{2} + 14^{2} + 7^{2} + 1^{2} + 8^{2} + 3^{2} + 3^{2} + 5^{2} + 24^{2} + 49^{2} + 6^{2} = 3942.

3942^{5} = 951881967753563232, and 9^{2} + 51^{2} + 8^{2} + 8^{2} + 19^{2} + 6^{2} + 7^{2} + 7^{2} + 5^{2} + 3^{2} + 5^{2} + 6^{2} + 3^{2} + 23^{2} + 2^{2} = 3942.

by Yoshio Mimura, Kobe, Japan

## 3943

3943^{2} = 15547249, 1 + 5 + 54 * 72 + 49 = 3943.

by Yoshio Mimura, Kobe, Japan

## 3944

3944^{2} = 15555136, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 3945

Komachi equations:

3945^{2} = 1^{2} * 2^{2} / 3^{2} / 4^{2} * 5^{2} * 6^{2} * 789^{2} = 1^{2} / 2^{2} * 3^{2} * 4^{2} * 5^{2} / 6^{2} * 789^{2}

= 1^{2} * 2^{2} / 3^{2} * 45^{2} / 6^{2} * 789^{2}.

by Yoshio Mimura, Kobe, Japan

## 3950

3950^{2} = (98 + 99 + 100 + 101 + 102)^{2} + (103 + 104 + 105 + 106 + 107)^{2} + (108 + 109 + 110 + 111 + 112)^{2} + ... + (213 + 214 + 215 + 216 + 217)^{2}.

by Yoshio Mimura, Kobe, Japan

## 3951

3951^{2} = 134^{3} + 144^{3} + 217^{3}.

by Yoshio Mimura, Kobe, Japan

## 3952

3952^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 3953

3953^{2} = 29^{4} + 34^{4} + 44^{4} + 56^{4}.

3953^{2} = 103^{2} + 105^{2} + 107^{2} + 109^{2} + 111^{2} + 113^{2} + ... + 455^{2}.

by Yoshio Mimura, Kobe, Japan

## 3954

3954^{2} = 15634116, 1 * 5 * 6 * 3 * 4 * 11 - 6 = 3954.

by Yoshio Mimura, Kobe, Japan

## 3955

1 / 3955 = 0.00025284450063211, 2^{2} + 5^{2} + 28^{2} + 44^{2} + 5^{2} + 006^{2} + 32^{2} + 11^{2} = 3955.

3955^{2} = (393 + 394 + 395 + 396 + 397 + 398 + 399)^{2} + (400 + 401 + 402 + 403 + 404 + 405 + 406)^{2} + (407 + 408 + 409 + 410 + 411 + 412 + 413)^{2} + ... + (400 + 401 + 402 + 403 + 404 + 405 + 406)^{2}.

by Yoshio Mimura, Kobe, Japan

## 3956

3956^{2} = 15649936, 1 - 5 - 6 + 4 * 993 - 6 = 3956.

by Yoshio Mimura, Kobe, Japan

## 3958

3958^{2} = 15665764, 1 + 5 + 6 + 657 * 6 + 4 = 1 + 5 + 6 * 657 + 6 + 4 = 3958.

by Yoshio Mimura, Kobe, Japan

## 3960

3960^{2} = 12^{3} + 38^{3} + 250^{3}.

3960^{2}= 9 x 10 + 10 x 11 + 11 x 12 +...+ 360 x 361.

34^{3} + 3960 = 208^{2}, 34^{3} - 3960 = 188^{2},

101^{2} + 3960 = 119^{2}, 101^{2} - 3960 = 79^{2}.

3960^{2} = (179 + 180 + 181)^{2} + (182 + 183 + 184)^{2} + (185 + 186 + 187)^{2} + (188 + 189 + 190)^{2} + ... + (275 + 276 + 277)^{2}.

3960^{2} = (10^{2} - 1)(19^{2} - 1)(21^{2} - 1) = (2^{2} - 1)(10^{2} - 1)(11^{2} - 1)(21^{2} - 1) = (2^{2} - 1)(21^{2} - 1)(109^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(10^{2} - 1)(21^{2} - 1) = (4^{2} - 1)(5^{2} - 1)(10^{2} - 1)(21^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 3963

1 / 3963 = 0.0002523340903, 2^{2} + 52^{2} + 3^{2} + 34^{2} + 0^{2} + 9^{2} + 0^{2} + 3^{2} = 3963,

1 / 3963 = 0.0002523340903, 2^{2} + 52^{2} + 3^{2} + 34^{2} + 0^{2} + 9^{2} + 03^{2} = 3963,

1 / 3963 = 0.0002523340903, 2^{2} + 52^{2} + 3^{2} + 34^{2} + 09^{2} + 0^{2} + 3^{2} = 3963,

1 / 3963 = 0.0002523340903, 2^{2} + 52^{2} + 3^{2} + 34^{2} + 09^{2} + 03^{2} = 3963.

by Yoshio Mimura, Kobe, Japan

## 3969

The square of 63.

3969^{2} = 50^{3} + 129^{3} + 238^{3} = 63^{3} + 147^{3} + 231^{3}.

Komachi equations:

3969^{2} = 12^{2} * 3^{2} / 4^{2} * 56^{2} * 7^{2} / 8^{2} * 9^{2},

3969^{2} = 9^{4} * 8^{4} * 7^{4} * 6^{4} * 5^{4} / 4^{4} / 3^{4} / 2^{4} / 10^{4} = 9^{4} * 8^{4} * 7^{4} / 6^{4} * 5^{4} / 4^{4} * 3^{4} * 2^{4} / 10^{4}

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

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

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

by Yoshio Mimura, Kobe, Japan

## 3970

3970^{2} = 142^{3} + 181^{3} + 191^{3}.

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

## 3971

3971^{2} = 15768841,and 1^{2} + 5^{2} + 7^{2} + 6^{2} + 8^{2} + 8^{2} + 4^{2} + 1^{2} = 16^{2}.

3971^{2} = 659^{2} + 661^{2} + 663^{2} + 665^{2} + 667^{2} + 669^{2} + ... + 723^{2}.

by Yoshio Mimura, Kobe, Japan

## 3972

3972^{2} = 18^{4} + 32^{4} + 52^{4} + 52^{4} = 1^{5} + 3^{5} + 6^{5} + 17^{5} + 27^{5}.

3972^{2} = 15776784, and 1^{2} + 5^{2} + 7^{2} + 7^{2} + 6^{2} + 7^{2} + 8^{2} + 4^{2} = 17^{2}.

3972^{2} = 15776784, 1 + 5 * 776 + 7 + 84 = 3972.

by Yoshio Mimura, Kobe, Japan

## 3973

3973^{2} = 15784729, and 1^{2} + 5^{2} + 7^{2} + 8^{2} + 4^{2} + 7^{2} + 2^{2} + 9^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 3975

3975^{2} = 32^{3} + 169^{3} + 222^{3}.

by Yoshio Mimura, Kobe, Japan

## 3976

3976^{2} = (7^{2} + 7)(8^{2} + 7)(63^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 3978

3978^{2} = (2^{2} + 9)(5^{2} + 9)(12^{2} + 9)(15^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 3984

3984^{2} = 74^{3} + 118^{3} + 240^{3} = 100^{3} + 120^{3} + 236^{3}.

by Yoshio Mimura, Kobe, Japan

## 3988

3988^{2} = 138^{3} + 138^{3} + 220^{3}.

The square root of 3988 is 63.15061..., and 63 = 1^{2} + 5^{2} + 0^{2} + 6^{2} + 1^{2}.

3988^{5} = 1008731883934471168 : 1^{2} + 0^{2} + 0^{2} + 8^{2} + 7^{2} + 3^{2} + 1^{2} + 8^{2} + 8^{2} + 39^{2} + 3^{2} + 44^{2} + 7^{2} + 11^{2} + 6^{2} + 8^{2} = 3988.

by Yoshio Mimura, Kobe, Japan

## 3989

Loop of length 35 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

3989 - 9442 - 10600 - 37 - ... - 8685 - 14621 - 2558 - 3989

(Note f(3989) = 39^{2} + 89^{2} = 9442, f(9442) = 94^{2} + 42^{2} = 10600, etc. See 37)

by Yoshio Mimura, Kobe, Japan

## 3990

3990^{2} = 35^{3} + 76^{3} + 249^{3}.

The integral triangle of sides 861, 37405, 37544 (or 9386, 10955, 20091) has square area 3990^{2}.

by Yoshio Mimura, Kobe, Japan

## 3993

3993^{2} = 121^{3} + 242^{3}.

by Yoshio Mimura, Kobe, Japan

## 3996

3996^{2}± 5 are primes.

3996^{2} = 15968016, 1 * 59 * 68 + 0 - 16 = 1 * 59 * 68 - 16 = 3996.

by Yoshio Mimura, Kobe, Japan

## 3997

3997^{2} = 35^{3} + 187^{3} + 211^{3}.

154^{k} + 490^{k} + 1708^{k} + 3577^{k} are squares for k = 1,2,3 (77^{2}, 3997^{2}, 225547^{2}).

by Yoshio Mimura, Kobe, Japan