## 5600

4-cycle : 5600^{2} = 31360000 - 3600^{2} = 12960000 - 9600^{2} = 92160000 - 1600^{2} = 02560000,

(Other examples : 2916 - 5030 - 3009 - 0540 - 2916, 2100 - 4100 - 8100 - 6100 - 2100.)

5600^{2} = 751 x 752 + 753 x 754 + 755 x 756 + 757 x 758 + 759 x 760+ ... + 847 x 848.

by Yoshio Mimura, Kobe, Japan

## 5601

5601^{4} = 984152252182401, and 9^{2} + 8^{2} + 4^{2} + 1^{2} + 52^{2} + 25^{2} + 21^{2} + 8^{2} + 2^{2} + 40^{2} + 1^{2} = 5601.

by Yoshio Mimura, Kobe, Japan

## 5604

5604^{2} = 4^{3} + 53^{3} + 315^{3}.

by Yoshio Mimura, Kobe, Japan

## 5605

5605^{2} = S_{2}(352) + S_{2}(369), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 5607

5607^{2} = (351 + 352)^{2} + (353 + 354)^{2} + (355 + 356)^{2} + (357 + 358)^{2} + ... + (447 + 448)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5610

5610^{2} = 342 x 343 + 344 x 345 + 346 x 347 + 348 x 349 + 350 x 351 + ... + 610 x 611.

The integral triangle of sides 2245, 663256, 665499 has square area 5610^{2}.

by Yoshio Mimura, Kobe, Japan

## 5612

1 / 5612 = 0.00017818959372, 17^{2} + 8^{2} + 18^{2} + 9^{2} + 59^{2} + 37^{2} + 2^{2} = 5612.

by Yoshio Mimura, Kobe, Japan

## 5617

Loop of length 5 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

5617 - 3425 - 1781 - 6850 - ... - 1781 - 6850 - 7124 - 5617

(Note f(5617) = 56^{2} + 17^{2} = 3425, f(3425) = 34^{2} + 25^{2} = 1781, etc. See 1781)

by Yoshio Mimura, Kobe, Japan

## 5618

5618^{2} = 53^{4} + 53^{4} + 53^{4} + 53^{4}.

5618^{3} = 177314889032, and 1^{2} + 7^{2} + 73^{2} + 1^{2} + 4^{2} + 8^{2} + 8^{2} + 9^{2} + 0^{2} + 3^{2} + 2^{2} = 5618.

by Yoshio Mimura, Kobe, Japan

## 5620

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

5620 - 3536 - 2521 - 1066 - ... - 6260 - 7444 - 7412 - 5620

(Note f(5620) = 56^{2} + 20^{2} = 3536, f(3536) = 35^{2} + 36^{2} = 2521, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 5622

5622^{2} = 38^{3} + 127^{3} + 309^{3}.

1 / 5622 = 0.0001778726, 17^{2} + 7^{2}+8^{2} + 72^{2} + 6^{2} = 5622.

by Yoshio Mimura, Kobe, Japan

## 5624

5624^{2} = 31629376, a zigzag square.

5624^{2} = 31629376, 3^{2} + 1^{2} + 6^{2} + 2^{2} + 9^{2} + 3^{2} + 7^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 5625

The square of 75.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007by Yoshio Mimura, Kobe, Japan

## 5628

5628^{2} = (2^{2} + 3)(3^{2} + 3)(8^{2} + 3)(75^{2} + 3) = (8^{2} + 3)(9^{2} + 3)(75^{2} + 3).

5628^{2} = 8^{4} + 8^{4} + 36^{4} + 74^{4}.

5628^{2} = (40 + 41 + 42 + ... + 67)^{2} + (68 + 69 + 70 + ... + 95)^{2} + (96 + 97 + 98 + ... + 123)^{2} + ... + (124 + 125 + 126 + ... + 151)^{2}.

994^{k} + 1470^{k} + 2338^{k} + 4802^{k} are squares for k = 1,2,3 (98^{2}, 5628^{2}, 357308^{2}).

by Yoshio Mimura, Kobe, Japan

## 5629

5629^{2} = 31685641, 3 - 1 - 6 - 8 + 5641 = 5629.

by Yoshio Mimura, Kobe, Japan

## 5631

5631^{2} = 31708161, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5632

5632^{2} = (1^{2} + 7)(2^{2} + 7)(11^{2} + 7)(53^{2} + 7) = (1^{2} + 7)(2^{2} + 7)(3^{2} + 7)(11^{2} + 7)(13^{2} + 7)

= (1^{2} + 7)(2^{2} + 7)(5^{2} + 7)(9^{2} + 7)(11^{2} + 7) = (1^{2} + 7)(3^{2} + 7)(9^{2} + 7)(53^{2} + 7)

= (1^{2} + 7)(5^{2} + 7)(11^{2} + 7)(31^{2} + 7) = (2^{2} + 7)(3^{2} + 7)(5^{2} + 7)(75^{2} + 7)

= (2^{2} + 7)(9^{2} + 7)(181^{2} + 7) = (3^{2} + 7)(9^{2} + 7)(11^{2} + 7)(13^{2} + 7) = (31^{2} + 7)(181^{2} + 7)

= (5^{2} + 7)(13^{2} + 7)(75^{2} + 7) = (9^{2} + 7)(11^{2} + 7)(53^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 5634

5634^{2} = 3^{4} + 15^{4} + 15^{4} + 75^{4}.

by Yoshio Mimura, Kobe, Japan

## 5635

5635^{5} = 5681586223670021875 : 5^{2} + 6^{2} + 8^{2} + 1^{2} + 5^{2} + 8^{2} + 6^{2} + 22^{2} + 3^{2} + 67^{2} + 0^{2} + 0^{2} + 2^{2} + 18^{2} + 7^{2} + 5^{2} = 5635.

by Yoshio Mimura, Kobe, Japan

## 5636

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

5636 - 4432 - 2960 - 4441 - ... - 12161 - 4163 - 5650 - 5636

(Note f(5636) = 56^{2} + 36^{2} = 4432, f(4432) = 44^{2} + 32^{2} = 2960, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 5638

5638^{2} = 5^{5} + 13^{5} + 25^{5} + 25^{5} + 26^{5}.

5638^{2} = 31787044, 3 + 1 * 7 + 8 * 704 - 4 = 3 * 1 + 7 + 8 * 704 - 4 = 5638.

by Yoshio Mimura, Kobe, Japan

## 5640

5640^{2} = 164^{3} + 208^{3} + 264^{3}.

5640^{2} = 31809600, 3 * 1 * 80 + 9 * 600 = 5640.

by Yoshio Mimura, Kobe, Japan

## 5643

5643^{2} = (3^{2} + 2)(5^{2} + 2)(13^{2} + 2)(25^{2} + 2).

5643^{2} = 102^{3} + 188^{3} + 289^{3}.

by Yoshio Mimura, Kobe, Japan

## 5650

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

5650 - 5636 - 4432 - 2960 - ... - 9556 - 12161 - 4163 - 5650

(Note f(5650) = 56^{2} + 50^{2} = 5636, f(5636) = 56^{2} + 36^{2} = 4432, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 5653

1 / 5653 = 0.00017689...., 17689 = 133^{2}.

5653^{2} = 31956409, 3 + 19 + 5640 - 9 = 31 - 9 + 5640 - 9 = 5653.

by Yoshio Mimura, Kobe, Japan

## 5655

5655^{2} = 26^{3} + 225^{3} + 274^{3}.

by Yoshio Mimura, Kobe, Japan

## 5657

5657^{5} = 5793365032534520057 : 5^{2} + 7^{2} + 9^{2} + 33^{2} + 6^{2} + 5^{2} + 0^{2} + 32^{2} + 5^{2} + 3^{2} + 4^{2} + 5^{2} + 2^{2} + 0^{2} + 0^{2} + 57^{2} = 57^{2} + 9^{2} + 33^{2} + 6^{2} + 5^{2} + 0^{2} + 32^{2} + 5^{2} + 3^{2} + 4^{2} + 5^{2} + 2^{2} + 0^{2} + 0^{2} + 5^{2} + 7^{2} = 5657.

by Yoshio Mimura, Kobe, Japan

## 5662

5662^{2} = 100^{3} + 139^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 5663

5663^{4} = 1028457255845761,

and 1^{2} + 0^{2} + 2^{2} + 8^{2} + 4^{2} + 5^{2} + 7^{2} + 2^{2} + 5^{2} + 58^{2} + 45^{2} + 7^{2} + 6^{2} + 1^{2} = 5663,

5663^{4} = 1028457255845761,

and 1^{2} + 0^{2} + 2^{2} + 8^{2} + 45^{2} + 7^{2} + 2^{2} + 5^{2} + 58^{2} + 4^{2} + 5^{2} + 7^{2} + 6^{2} + 1^{2} = 5663.

by Yoshio Mimura, Kobe, Japan

## 5664

5664^{2} = 90^{3} + 116^{3} + 310^{3} = 112^{3} + 168^{3} + 296^{3}.

by Yoshio Mimura, Kobe, Japan

## 5666

1 / N = 0.0001764...., where 1764 = 42^{2} (for N = 5666, 5667, 5668).

by Yoshio Mimura, Kobe, Japan

## 5670

5670^{2} = (1^{2} + 5)(10^{2} + 5)(11^{2} + 5)(20^{2} + 5) = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(10^{2} + 5)(20^{2} + 5)

= (1^{2} + 5)(20^{2} + 5)(115^{2} + 5) = (1^{2} + 5)(4^{2} + 5)(20^{2} + 5)(25^{2} + 5) = (11^{2} + 5)(20^{2} + 5)(25^{2} + 5)

= (2^{2} + 5)(3^{2} + 5)(20^{2} + 5)(25^{2} + 5) = (2^{2} + 5)(3^{2} + 5)(4^{2} + 5)(5^{2} + 5)(20^{2} + 5)

= (2^{2} + 5)(4^{2} + 5)(5^{2} + 5)(7^{2} + 5)(10^{2} + 5) = (2^{2} + 5)(5^{2} + 5)(17^{2} + 5)(20^{2} + 5)

= (2^{2} + 5)(7^{2} + 5)(10^{2} + 5)(25^{2} + 5) = (3^{2} + 5)(7^{2} + 5)(10^{2} + 5)(20^{2} + 5)

= (4^{2} + 5)(5^{2} + 5)(11^{2} + 5)(20^{2} + 5) = (5^{2} + 5)(10^{2} + 5)(101^{2} + 5).

5670^{2} = (5 + 6 + 7 + ... + 13)^{2} + (14 + 15 + 16 + ... + 22)^{2} + (23 + 24 + 25 + ... + 31)^{2} + ... + (212 + 213 + 214 + ... + 220)^{2}.

Komachi equations:

5670^{2} = 1^{2} * 2^{2} * 3^{2} * 4^{2} * 5^{2} * 6^{2} * 7^{2} / 8^{2} * 9^{2} = 1^{2} / 2^{2} * 3^{2} / 4^{2} * 5^{2} * 6^{2} * 7^{2} * 8^{2} * 9^{2}

= 1^{2} / 2^{2} * 3^{2} * 45^{2} / 6^{2} * 7^{2} * 8^{2} * 9^{2} = 9^{2} * 8^{2} * 7^{2} * 6^{2} * 5^{2} / 4^{2} * 3^{2} / 2^{2} */ 1^{2}

= 9^{2} / 8^{2} * 7^{2} * 6^{2} * 5^{2} * 4^{2} * 3^{2} * 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 5671

5671^{2} + 5672^{2} + 5673^{2} + ... + 5724^{2} = 5725^{2} + 5726^{2} + 5727^{2} + ... + 5777^{2}.

by Yoshio Mimura, Kobe, Japan

## 5672

5672^{2} = 32171584, 3^{2} + 2^{2} + 1^{2} + 7^{2} + 1^{2} + 5^{2} + 8^{2} + 4^{2} = 13^{2}.

5672^{5} = 5870581354415685632 : 5^{2} + 8^{2} + 7^{2} + 0^{2} + 5^{2} + 8^{2} + 1^{2} + 3^{2} + 5^{2} + 44^{2} + 15^{2} + 6^{2} + 8^{2} + 56^{2} + 3^{2} + 2^{2} = 5672.

by Yoshio Mimura, Kobe, Japan

## 5673

5673^{2} = 25^{3} + 210^{3} + 284^{3}.

by Yoshio Mimura, Kobe, Japan

## 5676

5676^{2} = (2^{2} + 8)(6^{2} + 8)(247^{2} + 8).

5676^{2} = 170^{2} + 171^{2} + 172^{2} + 173^{2} + 174^{2} + ... + 466^{2}.

The integral triangle of sides 3180, 20339, 20857 has square area 5676^{2}.

by Yoshio Mimura, Kobe, Japan

## 5678

5678^{2}± 3 are primes.

Komachi equations:

5678^{2} = 12^{2} * 3^{2} / 4^{2} + 5678^{2} - 9^{2} = 12^{2} * 3^{2} / 4^{2} * 5678^{2} / 9^{2}

= - 12^{2} * 3^{2} / 4^{2} + 5678^{2} + 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 5681

5681^{2} = S_{2}(325) + S_{2}(396), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

5681^{2} = 168^{2} + 169^{2} + 170^{2} + 171^{2} + 172^{2} + ... + 466^{2}.

by Yoshio Mimura, Kobe, Japan

## 5683

5683^{2} = 32296489, 3 + 2 - 2 * 9 + 64 * 89 = 5683.

by Yoshio Mimura, Kobe, Japan

## 5684

5684^{2} = 32307856, 3^{2} + 2^{2} + 3^{2} + 0^{2} + 7^{2} + 8^{2} + 5^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 5689

5689^{2} = 33^{3} + 213^{3} + 283^{3}.

by Yoshio Mimura, Kobe, Japan

## 5690

1330^{k} + 3610^{k} + 5690^{k} + 6270^{k} are squares for k = 1,2,3 (130^{2}, 9300^{2}, 692900^{2}).

by Yoshio Mimura, Kobe, Japan

## 5692

5692^{2} = 104^{3} + 246^{3} + 254^{3}.

5692^{2} = 32398864, 3 * 23 - 9 + 88 * 64 = 3 * 239 * 8 - 8 * 6 + 4 = 5692.

by Yoshio Mimura, Kobe, Japan

## 5693

5693^{2} = 131^{3} + 243^{3} + 251^{3}.

by Yoshio Mimura, Kobe, Japan

## 5696

1 / 5696 = 0.000175561...., 175561 = 419^{2}.

5696^{2} = 32444416, 32 * 4 * 44 + 4 * 16 = 32 * 44 * 4 + 4 * 16 = 5696.

by Yoshio Mimura, Kobe, Japan

## 5697

5697^{2} = 153^{3} + 232^{3} + 254^{3}.

5697^{2} = (78 + 79 + 80)^{2} + (81 + 82 + 83)^{2} + (84 + 85 + 86)^{2} + ... + (318 + 319 + 320)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5698

5698^{k} + 6622^{k} + 8338^{k} + 10318^{k} are squares for k = 1,2,3 (176^{2}, 15884^{2}, 1467488^{2}).

by Yoshio Mimura, Kobe, Japan

## 5699

5699^{2} = 32478601, a square with different digits.

by Yoshio Mimura, Kobe, Japan