## 5400

5400^{2} = (11^{2} - 1)(19^{2} - 1)(26^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(11^{2} - 1)(26^{2} - 1)

= (2^{2} - 1)(4^{2} - 1)(26^{2} - 1)(31^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(19^{2} - 1)(26^{2} - 1)

= (4^{2} - 1)(5^{2} - 1)(11^{2} - 1)(26^{2} - 1).

5400^{2} = 60^{3} + 210^{3} + 270^{3}.

75^{2} + 5400 = 105^{2}, 75^{2} - 5400 = 15^{2}.

Komachi equation: 5400^{2} = 9^{2} * 8^{2} / 7^{2} * 6^{2} * 5^{2} / 4^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 5401

5401^{2} = 29170801, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5402

5402^{2} = 29181604, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5403

5403^{2} = 29192409, a zigzag square.

5403^{2}± 2 are primes.

5403^{2} = 54^{3} + 137^{3} + 298^{3}.

by Yoshio Mimura, Kobe, Japan

## 5405

5405^{k} + 35485^{k} + 74965^{k} + 105045^{k} are squares for k = 1,2,3 (470^{2}, 133950^{2}, 40314250^{2}).

by Yoshio Mimura, Kobe, Japan

## 5406

1 / N = 0.0001849...., and 1849 = 43^{2} for N = 5406, 5407, 5408.

by Yoshio Mimura, Kobe, Japan

## 5407

5407^{2} = 29235649, 2^{2} + 9^{2} + 2^{2} + 3^{2} + 5^{2} + 6^{2} + 4^{2} + 9^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 5408

5408^{2} = 52^{4} + 52^{4} + 52^{4} + 52^{4}.

by Yoshio Mimura, Kobe, Japan

## 5409

5409^{2} = 48^{3} + 129^{3} + 300^{3}.

by Yoshio Mimura, Kobe, Japan

## 5412

5412^{2} = (38^{2} + 8)(142^{2} + 8) = (5^{2} + 8)(6^{2} + 8)(142^{2} + 8).

5412^{2} = 90^{3} + 128^{3} + 298^{3} = 32^{4} + 42^{4} + 44^{4} + 68^{4}.

by Yoshio Mimura, Kobe, Japan

## 5416

5416^{3} = 158867831296, and 1^{2} + 58^{2} + 8^{2} + 6^{2} + 7^{2} + 8^{2} + 31^{2} + 29^{2} + 6^{2} = 5416.

by Yoshio Mimura, Kobe, Japan

## 5418

5418^{2} = 29273454, a zigzag square.

5418^{2} = (1^{2} + 5)(4^{2} + 5)(9^{2} + 5)(52^{2} + 5) = (2^{2} + 5)(3^{2} + 5)(9^{2} + 5)(52^{2} + 5)

= (3^{2} + 5)(4^{2} + 5)(9^{2} + 5)(34^{2} + 5) = (9^{2} + 5)(11^{2} + 5)(52^{2} + 5) = (9^{2} + 5)(17^{2} + 5)(34^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 5419

5419^{2} = 89^{3} + 130^{3} + 298^{3}.

5419^{2} = 29365561, 2 * 9 + 36 * 5 * 5 * 6 + 1 = 5419.

by Yoshio Mimura, Kobe, Japan

## 5423

5423^{5} = 4690271924019343343 : 46^{2} + 9^{2} + 0^{2} + 27^{2} + 19^{2} + 24^{2} + 0^{2} + 19^{2} + 3^{2} + 4^{2} + 3^{2} + 34^{2} + 3^{2} = 46^{2} + 9^{2} + 0^{2} + 27^{2} + 19^{2} + 24^{2} + 0^{2} + 19^{2} + 34^{2} + 3^{2} + 3^{2} + 4^{2} + 3^{2} = 5423.

by Yoshio Mimura, Kobe, Japan

## 5424

5424^{2} = 88^{3} + 180^{3} + 284^{3} = 169^{3} + 222^{3} + 239^{3}.

by Yoshio Mimura, Kobe, Japan

## 5426

5426^{2}± 3 are primes.

The squaree root of 5426 is 73.661...., and 73 = 6^{2} + 6^{2} + 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 5427

5427^{2} = 29452329, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5432

5432^{2} = 9^{5} + 9^{5} + 15^{5} + 31^{5}.

by Yoshio Mimura, Kobe, Japan

## 5433

5433^{2} = 97^{3} + 102^{3} + 302^{3}.

by Yoshio Mimura, Kobe, Japan

## 5435

The square root of 5435 is 73.7224...., and 73 = 7^{2} + 2^{2} + 2^{2} + 4^{2}.

5435^{2} = 29539225, 2 * 9 + 5392 + 25 = 5435.

by Yoshio Mimura, Kobe, Japan

## 5437

5437^{2} = 29560969, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5438

5438^{2} = 29571844, 2^{2} + 9^{2} + 5^{2} + 7^{2} + 1^{2} + 8^{2} + 4^{2} + 4^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 5441

5441^{2} = 29604481, 2 + 9 * 604 - 4 + 8 - 1 = 5441.

by Yoshio Mimura, Kobe, Japan

## 5443

5443^{2} = 29626249, 29 * 6 / 2 * 62 + 49 = 29 * 62 * 6 / 2 + 49 = 5443.

by Yoshio Mimura, Kobe, Japan

## 5444

5444^{2} = 29637136, 2^{2} + 9^{2} + 6^{2} + 3^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 5445

5445^{2} = 13^{5} + 19^{5} + 22^{5} + 25^{5} + 26^{5}.

5445^{2} = (83 + 84 + 85 + ... + 115)^{2} + (116 + 117 + 118 + ... + 148)^{2} + (149 + 150 + 151 + ... + 181)^{2} + ... + (116 + 117 + 118 + ... + 148)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5446

5446^{2} = 29658916, 29 * 6 + 58 * 91 - 6 = 5446.

by Yoshio Mimura, Kobe, Japan

## 5447

The square root of 5447 is 73.803...., and 73 = 8^{2} + 0^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 5448

5448^{2} = 29680704, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5449

5449^{2} = 29691601, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5451

The square root of 5451 is 73.83...., and 73 = 8^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 5452

5452^{2}± 3 are primes.

The square root of 5452 is 73.83...., and 73 = 8^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 5453

Komachi equation: 5453^{2} = 123^{2} * 456^{2} * 7^{2} / 8^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 5455

5455^{2} = 29757025, 2 / 9 * 7 * 5 * 702 - 5 = 2 + 97 * 7 * 8 + 8 * 4 - 9 = 5457.

by Yoshio Mimura, Kobe, Japan

## 5457

5457^{2} = 4^{4} + 24^{4} + 32^{4} + 73^{4}.

by Yoshio Mimura, Kobe, Japan

## 5458

5458 = (1^{2} + 2^{2} + 3^{2} + ... + 1364^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 77^{2}).

by Yoshio Mimura, Kobe, Japan

## 5460

5460^{2} = (14^{2} - 1)(391^{2} - 1).

5460^{2} + 5461^{2} + 5462^{2} + ... + 5512^{2} = 5513^{2} + 5514^{2} + 5515^{2} + ... + 5564^{2}.

Komachi equations:

5460^{2} = 9^{2} * 8^{2} * 7^{2} * 65^{2} / 4^{2} / 3^{2} * 2^{2} */ 1^{2} = 98^{2} / 7^{2} * 65^{2} * 4^{2} * 3^{2} / 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 5461

1 / 5461 = 0.000183116645303, 18^{2} + 31^{2} + 1^{2} + 6^{2} + 64^{2} + 5^{2} + 3^{2} + 03^{2} = 5461.

by Yoshio Mimura, Kobe, Japan

## 5463

5463^{2} = 18^{4} + 33^{4} + 36^{4} + 72^{4} = 28^{4} + 38^{4} + 46^{4} + 69^{4}.

5463^{4} = 890686361008161, and 8^{2} + 9^{2} + 0^{2} + 6^{2} + 8^{2} + 6^{2} + 36^{2} + 10^{2} + 0^{2} + 8^{2} + 1^{2} + 61^{2} = 5463.

by Yoshio Mimura, Kobe, Japan

## 5464

5464^{2} = 619^{2} + 621^{2} + 623^{2} + 625^{2} + 627^{2} + 629^{2} + ... + 745^{2}.

by Yoshio Mimura, Kobe, Japan

## 5465

Loop of length 6 by the function f(N) = b^{2} + a^{2} for N = 100b + a:

5465 - 7141 - 6722 - 4973 - 7730 - 6829 - 5465 (Note f(5465) = 54^{2} + 65^{2} = 7141.)

by Yoshio Mimura, Kobe, Japan

## 5471

5471^{2} = 99^{3} + 205^{3} + 273^{3}.

by Yoshio Mimura, Kobe, Japan

## 5472

5472^{2} = (2^{2} + 8)(4^{2} + 8)(12^{2} + 8)(26^{2} + 8).

Komachi equation: 5472^{2} = 9^{2} / 8^{2} * 76^{2} * 5^{2} * 4^{2} * 32^{2} / 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 5475

5475^{2} = 29975625, 2 * 9 + 97 * 56 + 25 = 5475.

by Yoshio Mimura, Kobe, Japan

## 5476

The square of 74.

5476^{2} = 75^{3} + 164^{3} + 293^{3}.

5476 is a zigzag square with different digits.

Page of Squares : First Upload November 26, 2007 ; Last Revised September 18, 2008by Yoshio Mimura, Kobe, Japan

## 5477

5477^{2} = 9^{3} + 92^{3} + 308^{3}.

by Yoshio Mimura, Kobe, Japan

## 5478

5478^{2} = (2^{2} + 2)(3^{2} + 2)(9^{2} + 2)(74^{2} + 2) = (8^{2} + 2)(9^{2} + 2)(74^{2} + 2).

5478^{2} = 109^{3} + 231^{3} + 254^{3}.

5478^{2} = 30008484, 3^{2} + 0^{2} + 0^{2} + 0^{2} + 8^{2} + 4^{2} + 8^{2} + 4^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 5480

5480^{2} = 4^{3} + 160^{3} + 296^{3} = 79^{3} + 113^{3} + 304^{3} = 190^{3} + 208^{3} + 242^{3}.

by Yoshio Mimura, Kobe, Japan

## 5481

5481^{2} = (4^{2} + 5)(16^{2} + 5)(74^{2} + 5).

5481^{2} = 99^{3} + 227^{3} + 259^{3} = 12^{4} + 45^{4} + 60^{4} + 60^{4}.

Komachi equation: 5481^{2} = 9^{2} * 87^{2} * 6^{2} / 54^{2} * 3^{2} * 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 5484

5484^{2} = 119^{3} + 162^{3} + 289^{3}.

by Yoshio Mimura, Kobe, Japan

## 5487

5487^{2} = 81^{3} + 146^{3} + 298^{3}.

by Yoshio Mimura, Kobe, Japan

## 5488

5488^{2} = 28^{4} + 56^{4} + 56^{4} + 56^{4} = 14^{6} + 14^{6} + 14^{6} + 14^{6}.

The square root of 5488 ia 74.08103...., and 74 = 0^{2} + 8^{2} + 1^{2} + 0^{2} + 3^{2}.

Komachi equations:

5488^{2} = 98^{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}

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

by Yoshio Mimura, Kobe, Japan

## 5489

5489^{2} = 74^{3} + 169^{3} + 292^{3}.

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

## 5491

The square root of 5491 is 74.101282, and 74 = 1^{2} + 0^{2} + 1^{2} + 2^{2} + 8^{2} + 2^{2}.

by Yoshio Mimura, Kobe, Japan

## 5493

5493^{3} = 165740558157, and 16^{2} + 57^{2} + 40^{2} + 5^{2} + 5^{2} + 8^{2} + 15^{2} + 7^{2} = 5493.

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

5493 - 11565 - 4451 - 4537 - ... - 6736 - 5785 - 10474 - 5493

(Note f(5493) = 54^{2} + 93^{2} = 11565, f(11565) = 1^{2} + 15^{2} + 65^{2} = 4451, etc. See 37)

by Yoshio Mimura, Kobe, Japan

## 5497

5497^{2} = 30217009, 3^{2} + 0^{2} + 2^{2} + 1^{2} + 7^{2} + 0^{2} + 0^{2} + 9^{2} = 12^{2}.

by Yoshio Mimura, Kobe, Japan

## 5499

5499^{2} = 2^{3} + 169^{3} + 294^{3}.

5499^{5} = 5028270100947527499 : 5^{2} + 0^{2} + 2^{2} + 8^{2} + 2^{2} + 70^{2} + 10^{2} + 0^{2} + 9^{2} + 4^{2} + 7^{2} + 5^{2} + 2^{2} + 7^{2} + 4^{2} + 9^{2} + 9^{2} = 5499.

by Yoshio Mimura, Kobe, Japan