## 7400

7400^{2} = (1^{2} + 4)(14^{2} + 4)(234^{2} + 4) = (2^{2} + 4)(11^{2} + 4)(234^{2} + 4) = (4^{2} + 4)(12^{2} + 4)(136^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 7402

7402^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 7403

7403^{2} = 119^{3} + 165^{3} + 365^{3}.

by Yoshio Mimura, Kobe, Japan

## 7405

7405^{5} = 22265134054473128125 :

2^{2} + 2^{2} + 2^{2} + 6^{2} + 5^{2} + 1^{2} + 3^{2} + 4^{2} + 0^{2} + 5^{2} + 4^{2} + 4^{2} + 7^{2} + 3^{2} + 1^{2} + 2^{2} + 81^{2} + 25^{2} = 7405.

by Yoshio Mimura, Kobe, Japan

## 7406

7406^{4} = 3008394810554896,

and 3^{2} + 0^{2} + 0^{2} + 83^{2} + 9^{2} + 4^{2} + 8^{2} + 10^{2} + 5^{2} + 5^{2} + 4^{2} + 8^{2} + 9^{2} + 6^{2} = 7406.

by Yoshio Mimura, Kobe, Japan

## 7407

7407^{2} = 135^{3} + 295^{3} + 299^{3},

7407^{2} = 23^{4} + 32^{4} + 44^{4} + 84^{4}.

Komachi equations:

7407^{2} = 9876^{2} * 5^{2} / 4^{2} * 3^{2} * 2^{2} / 10^{2} = 9876^{2} / 5^{2} / 4^{2} * 3^{2} / 2^{2} * 10^{2}.

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 0^{2} + 0^{2} + 74^{2} + 25^{2} + 4^{2} + 08^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 0^{2} + 074^{2} + 25^{2} + 4^{2} + 0^{2} + 8^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 0^{2} + 074^{2} + 25^{2} + 4^{2} + 08^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 00^{2} + 74^{2} + 25^{2} + 4^{2} + 0^{2} + 8^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 00^{2} + 74^{2} + 25^{2} + 4^{2} + 08^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 0074^{2} + 25^{2} + 4^{2} + 0^{2} + 8^{2} = 7407,

1 / 7407 = 0.000135007425408, 1^{2} + 35^{2} + 0074^{2} + 25^{2} + 4^{2} + 08^{2} = 7407.

by Yoshio Mimura, Kobe, Japan

## 7408

7408^{2} = 54878464, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 7409

7409^{5} = 22325334462883216049 :

2^{2} + 2^{2} + 3^{2} + 2^{2} + 5^{2} + 33^{2} + 44^{2} + 62^{2} + 8^{2} + 8^{2} + 3^{2} + 2^{2} + 16^{2} + 0^{2} + 4^{2} + 9^{2} = 7409,

2^{2} + 23^{2} + 25^{2} + 33^{2} + 4^{2} + 4^{2} + 62^{2} + 8^{2} + 8^{2} + 32^{2} + 1^{2} + 6^{2} + 0^{2} + 4^{2} + 9^{2} = 7409.

by Yoshio Mimura, Kobe, Japan

## 7412

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

7412 - 5620 - 3536 - 2521 - ... - 2874 - 6260 - 7444 - 7412

(Note f(7412) = 74^{2} + 12^{2} = 5620, f(5620) = 56^{2} + 20^{2} = 3536, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 7414

7414^{2} = 54967396, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 7415

7415^{2} = 54982225, and 5 * 4 + 9 * 822 + 2 - 5 = 7415.

by Yoshio Mimura, Kobe, Japan

## 7419

7419^{5} = 22476405331452753099 :

2^{2} + 2^{2} + 47^{2} + 6^{2} + 4^{2} + 0^{2} + 5^{2} + 33^{2} + 14^{2} + 52^{2} + 7^{2} + 5^{2} + 30^{2} + 9^{2} + 9^{2} = 7419.

by Yoshio Mimura, Kobe, Japan

## 7421

The square root of 7421 is 86.145226...., and 86 = 1^{2} + 4^{2} + 5^{2} + 2^{2} + 2^{2} + 6^{2}.

7421^{2} = 55071241, and 5^{2} + 5^{2} + 0^{2} + 7^{2} + 1^{2} + 2^{2} + 4^{2} + 1^{2} = 11^{2}.

Loop of length 5 : 7421 --- 74^{2} + 21^{2} = 5917 --- 59^{2} + 17^{2} = 3770 -- 37^{2} + 70^{2} = 6269 --- 62^{2} + 69^{2} = 8605 --- 86^{2} + 05^{2} = 7421

by Yoshio Mimura, Kobe, Japan

## 7422

The square root of 7422 is 86.15103017...., and 86 = 1^{2} + 5^{2} + 1^{2} + 0^{2} + 3^{2} + 0^{2} + 1^{2} + 7^{2}.

by Yoshio Mimura, Kobe, Japan

## 7424

7424^{2} = 55115776, a square with odd digits except the last digit 6.

7424^{2} = (1^{2} + 7)(3^{2} + 7)(15^{2} + 7)(43^{2} + 7) = (11^{2} + 7)(15^{2} + 7)(43^{2} + 7).

The square root of 7424 is 86.16263...., and 86 = 1^{2} + 6^{2} + 2^{2} + 6^{2} + 3^{2}.

7424^{2} = 64^{3} + 224^{3} + 352^{3}.

by Yoshio Mimura, Kobe, Japan

## 7426

The square root of 7426 is 86.17424...., and 86 = 1^{2} + 7^{2} + 4^{2} + 2^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 7432

7432^{2} = 14^{4} + 14^{4} + 26^{4} + 86^{4}.

by Yoshio Mimura, Kobe, Japan

## 7435

7435^{2} = 156^{3} + 169^{3} + 360^{3}.

by Yoshio Mimura, Kobe, Japan

## 7438

1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + 7438^{2} = 137193913719 is the first 12-digit sum consisting of odd digits (there are such two 12-digit sums, see 12149).

by Yoshio Mimura, Kobe, Japan

## 7440

7440^{2} = (4^{2} - 1)(1921^{2} - 1).

7440^{2} = 44^{3} + 292^{3} + 312^{3}.

by Yoshio Mimura, Kobe, Japan

## 7441

7441^{5} = 22811640709248613201 :

2^{2} + 2^{2} + 8^{2} + 1^{2} + 16^{2} + 40^{2} + 70^{2} + 9^{2} + 2^{2} + 4^{2} + 8^{2} + 6^{2} + 1^{2} + 3^{2} + 20^{2} + 1^{2} = 7441.

by Yoshio Mimura, Kobe, Japan

## 7442

7442^{2} = 61^{3} + 183^{3} + 366^{3},

7442^{2} = 61^{4} + 61^{4} + 61^{4} + 61^{4}.

7442^{5} = 22826973173212243232 :

2^{2} + 2^{2} + 8^{2} + 2^{2} + 6^{2} + 9^{2} + 73^{2} + 17^{2} + 3^{2} + 2^{2} + 1^{2} + 2^{2} + 24^{2} + 3^{2} + 2^{2} + 32^{2}

= 2^{2} + 2^{2} + 8^{2} + 2^{2} + 6^{2} + 9^{2} + 73^{2} + 17^{2} + 3^{2} + 2^{2} + 1^{2} + 2^{2} + 24^{2} + 32^{2} + 3^{2} + 2^{2}

= 2^{2} + 2^{2} + 8^{2} + 2^{2} + 6^{2} + 9^{2} + 73^{2} + 17^{2} + 32^{2} + 1^{2} + 2^{2} + 24^{2} + 3^{2} + 2^{2} + 3^{2} + 2^{2}

= 2^{2} + 28^{2} + 2^{2} + 6^{2} + 9^{2} + 7^{2} + 31^{2} + 73^{2} + 2^{2} + 12^{2} + 2^{2} + 4^{2} + 3^{2} + 2^{2} + 3^{2} + 2^{2} = 7442.

by Yoshio Mimura, Kobe, Japan

## 7443

7443^{2} = 55398249, and 5 - 5 + 3 * 9 + 824 * 9 = 5 / 5 * 3 * 9 + 824 * 9 = 7443.

7443^{2} = 118^{3} + 152^{3} + 369^{3}.

by Yoshio Mimura, Kobe, Japan

## 7444

7444^{2} = 55413136, and 5 + 5 + 413 * 1 * 3 * 6 = 7444.

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

7444 - 7412 - 5620 - 3536 - ... - 10853 - 2874 - 6260 - 7444

(Note f(7444) = 74^{2} + 44^{2} = 7412, f(7412) = 74^{2} + 12^{2} = 5620, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 7445

7445^{5} = 22873020037957653125 :

2^{2} + 2^{2} + 8^{2} + 7^{2} + 30^{2} + 20^{2} + 0^{2} + 3^{2} + 7^{2} + 9^{2} + 5^{2} + 7^{2} + 65^{2} + 31^{2} + 25^{2} = 7445.

by Yoshio Mimura, Kobe, Japan

## 7446

7446^{2} = (10^{2} + 2)(12^{2} + 2)(61^{2} + 2).

3366^{k} + 7446^{k} + 15708^{k} + 38505^{k} are squares for k = 1,2,3 (255^{2}, 42381^{2}, 7836813^{2}).

by Yoshio Mimura, Kobe, Japan

## 7447

7447^{2} = 44^{3} + 300^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 7448

7448^{2} = (1^{2} + 3)(2^{2} + 3)(23^{2} + 3)(61^{2} + 3) = (1^{2} + 3)(2^{2} + 3)(4^{2} + 3)(5^{2} + 3)(61^{2} + 3)

= (1^{2} + 3)(4^{2} + 3)(23^{2} + 3)(37^{2} + 3) = (5^{2} + 3)(23^{2} + 3)(61^{2} + 3).

7448^{2} = 20^{3} + 84^{3} + 380^{3} = 224^{3} + 266^{3} + 294^{3}.

by Yoshio Mimura, Kobe, Japan

## 7449

7449^{2} = 201^{3} + 260^{3} + 310^{3}.

7449^{5} = 22934531418412512249 :

2^{2} + 29^{2} + 3^{2} + 45^{2} + 31^{2} + 4^{2} + 1^{2} + 8^{2} + 4^{2} + 1^{2} + 25^{2} + 1^{2} + 22^{2} + 49^{2} = 7449.

by Yoshio Mimura, Kobe, Japan

## 7450

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

7450 - 7976 - 12017 - 690 - ... 4441 - 3617 - 1585 - 7450

(Note f(7450) = 74^{2} + 50^{2} = 7976, f(7976) = 79^{2} + 76^{2} = 12017, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 7451

7451^{2} = 55517401, 55 - 5 + 1 * 7401 = 55 - 5 * 1 + 7401 = 7451.

by Yoshio Mimura, Kobe, Japan

## 7452

7452^{2} = (59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67)^{2} + (68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76)^{2} + (77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85)^{2} + ... + (257 + 258 + 259 + 260 + 261 + 262 + 263 + 264 + 265)^{2}.

7452^{2} = 18^{4} + 24^{4} + 48^{4} + 84^{4} = 18^{4} + 36^{4} + 72^{4} + 72^{4} = 24^{4} + 26^{4} + 68^{4} + 76^{4}.

7452^{5} = 22980751740810796032 :

2^{2} + 29^{2} + 8^{2} + 0^{2} + 7^{2} + 5^{2} + 1^{2} + 7^{2} + 40^{2} + 8^{2} + 1^{2} + 0^{2} + 7^{2} + 9^{2} + 60^{2} + 32^{2} = 7452.

by Yoshio Mimura, Kobe, Japan

## 7453

7453^{2} = 55547209, and 5^{2} + 5^{2} + 5^{2} + 4^{2} + 7^{2} + 2^{2} + 0^{2} + 9^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 7455

7455^{5} = 23027046552026409375 :

23^{2} + 0^{2} + 27^{2} + 0^{2} + 4^{2} + 65^{2} + 5^{2} + 20^{2} + 2^{2} + 6^{2} + 4^{2} + 0^{2} + 9^{2} + 37^{2} + 5^{2} = 7455.

by Yoshio Mimura, Kobe, Japan

## 7456

7456^{2} = 55591936, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 7458

7458^{2} = 147^{3} + 242^{3} + 337^{3},

7458^{2} = 3^{4} + 41^{4} + 61^{4} + 79^{4}.

by Yoshio Mimura, Kobe, Japan

## 7462

7462^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 7463

7463^{2} = 287^{2} + 288^{2} + 289^{2} + 290^{2} + 291^{2} + ... + 575^{2}.

by Yoshio Mimura, Kobe, Japan

## 7466

7466^{2} = 55741156, 5 * 5 + 7411 + 5 * 6 = 7466.

by Yoshio Mimura, Kobe, Japan

## 7468

7468^{2} = 55771024, and 5^{2} + 5^{2} + 7^{2} + 7^{2} + 1^{2} + 0^{2} + 2^{2} + 4^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 7469

517^{k} + 869^{k} + 3245^{k} + 7469^{k} are squares for k = 1,2,3 (110^{2}, 8206^{2}, 672034^{2}).

by Yoshio Mimura, Kobe, Japan

## 7471

7471^{2} = 1^{3} - 2^{2} + 3^{3} - 4^{3} + 5^{3} - 6^{3} + 7^{3} - 8^{3} + ... + 481^{3}.

by Yoshio Mimura, Kobe, Japan

## 7472

7472^{2} = (94 + 95 + 96 + 97)^{2} + (98 + 99 + 100 + 101)^{2} + (102 + 103 + 104 + 105)^{2} + ... + (346 + 347 + 348 + 349)^{2}.

by Yoshio Mimura, Kobe, Japan

## 7473

7473^{2} = 55845729, and 5^{2} + 5^{2} + 8^{2} + 4^{2} + 5^{2} + 7^{2} + 2^{2} + 9^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 7475

7475^{2} = 65^{3} + 90^{3} + 380^{3}.

by Yoshio Mimura, Kobe, Japan

## 7476

7476^{2} = 55890576, 5 - 5 + 890 / 5 * 7 * 6 = 5 / 5 * 890 / 5 * 7 * 6 = 7476.

by Yoshio Mimura, Kobe, Japan

## 7479

1 / 7479 = 0.000133707714935, 13^{2} + 3^{2} + 7^{2} + 077^{2} + 1^{2} + 4^{2} + 9^{2} + 35^{2} = 7479.

The square root of 7479 is 86.48121...., and 86 = 4^{2} + 8^{2} + 1^{2} + 2^{2} + 1^{2}.

7479^{5} = 23400097454226453399 :

23^{2} + 4^{2} + 0^{2} + 0^{2} + 0^{2} + 9^{2} + 74^{2} + 5^{2} + 4^{2} + 2^{2} + 2^{2} + 6^{2} + 4^{2} + 5^{2} + 33^{2} + 9^{2} + 9^{2} = 7479.

by Yoshio Mimura, Kobe, Japan

## 7480

7480^{2} = 109^{3} + 147^{3} + 372^{3}.

7480^{5} = 23415745505996800000 :

2^{2} + 3^{2} + 41^{2} + 5^{2} + 7^{2} + 45^{2} + 5^{2} + 0^{2} + 59^{2} + 9^{2} + 6^{2} + 8^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 7480.

by Yoshio Mimura, Kobe, Japan

## 7481

7481^{2} = 15^{5} + 23^{5} + 27^{5} + 28^{5} + 28^{5}.

by Yoshio Mimura, Kobe, Japan

## 7482

7482^{2} = 149^{3} + 295^{3} + 300^{3}.

7482^{4} = 3133796675144976,

and 3^{2} + 1^{2} + 3^{2} + 3^{2} + 7^{2} + 9^{2} + 6^{2} + 67^{2} + 51^{2} + 4^{2} + 4^{2} + 9^{2} + 7^{2} + 6^{2} = 7482.

7482^{5} = 23447066723434710432 :

2^{2} + 3^{2} + 44^{2} + 7^{2} + 0^{2} + 6^{2} + 6^{2} + 72^{2} + 3^{2} + 4^{2} + 3^{2} + 4^{2} + 7^{2} + 10^{2} + 4^{2} + 3^{2} + 2^{2} = 7482.

by Yoshio Mimura, Kobe, Japan

## 7483

7483^{5} = 23462739895818117643 :

2^{2} + 34^{2} + 62^{2} + 7^{2} + 39^{2} + 8^{2} + 9^{2} + 5^{2} + 8^{2} + 18^{2} + 1^{2} + 17^{2} + 6^{2} + 4^{2} + 3^{2} = 7483.

by Yoshio Mimura, Kobe, Japan

## 7484

7484^{2} = 162^{3} + 236^{3} + 338^{3}.

7484^{5} = 23478421448456551424 :

2^{2} + 3^{2} + 4^{2} + 7^{2} + 8^{2} + 42^{2} + 14^{2} + 4^{2} + 8^{2} + 45^{2} + 6^{2} + 55^{2} + 14^{2} + 2^{2} + 4^{2} = 7484.

by Yoshio Mimura, Kobe, Japan

## 7487

7487^{5} = 23525516421508166207 :

2^{2} + 35^{2} + 25^{2} + 5^{2} + 16^{2} + 4^{2} + 21^{2} + 5^{2} + 0^{2} + 8^{2} + 1^{2} + 66^{2} + 20^{2} + 7^{2} = 7487.

by Yoshio Mimura, Kobe, Japan

## 7490

7490^{2}= 434 x 435 + 435 x 436 + 436 x 437 + 437 x 438 + ... + 629 x 630.

by Yoshio Mimura, Kobe, Japan

## 7492

The square root of 7492 is 86.556...., and 86 = 5^{2} + 5^{2} + 6^{2}.

by Yoshio Mimura, Kobe, Japan

## 7493

7493^{4} = 3152266527212401,

and 31^{2} + 5^{2} + 2^{2} + 26^{2} + 6^{2} + 5^{2} + 2^{2} + 72^{2} + 1^{2} + 24^{2} + 0^{2} + 1^{2} = 7493.

by Yoshio Mimura, Kobe, Japan

## 7494

7494^{2} = 15^{3} + 196^{3} + 365^{3}.

by Yoshio Mimura, Kobe, Japan

## 7495

7495^{2} = 56175025, and 5 - 6 - 1 + 7502 - 5 = 7495.

by Yoshio Mimura, Kobe, Japan

## 7496

7496^{2} = 36^{3} + 144^{3} + 376^{3}.

by Yoshio Mimura, Kobe, Japan

## 7497

Komachi equation: 7497^{2} = 1^{2} / 2^{2} * 34^{2} * 56^{2} * 7^{2} / 8^{2} * 9^{2}.

7497^{2} is written as the sum of four fourths in four ways :

7497^{2} = 12^{4} + 21^{4} + 66^{4} + 78^{4} = 12^{4} + 26^{4} + 64^{4} + 79^{4} = 14^{4} + 28^{4} + 49^{4} + 84^{4} = 21^{4} + 42^{4} + 42^{4} + 84^{4}.

7497^{2} = 25^{3} + 26^{3} + 27^{3} + 28^{3} + 29^{3} + 30^{3} + ... + 122^{3}.

by Yoshio Mimura, Kobe, Japan

## 7498

7498^{k} + 13662^{k} + 15778^{k} + 15962^{k} are squares for k = 1,2,3 (230^{2}, 27324^{2}, 3311540^{2}).

by Yoshio Mimura, Kobe, Japan

## 7499

7499^{2} = 56235001, and 5^{2} + 6^{2} + 2^{2} + 3^{2} + 5^{2} + 0^{2} + 0^{2} + 1^{2} = 10^{2}.

7499^{4} = 3162375337470001,

and 3^{2} + 1^{2} + 6^{2} + 2^{2} + 37^{2} + 5^{2} + 33^{2} + 7^{2} + 4^{2} + 70^{2} + 0^{2} + 0^{2} + 1^{2} = 7499.

by Yoshio Mimura, Kobe, Japan