## 8301

8301^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 8303

8303^{2} = 19^{4} + 38^{4} + 76^{4} + 76^{4}.

by Yoshio Mimura, Kobe, Japan

## 8304

8304^{2}± 5 are primes.

8304^{5} = 39485414602020225024 : 39^{2} + 48^{2} + 5^{2} + 41^{2} + 46^{2} + 0^{2} + 2^{2} + 0^{2} + 2^{2} + 0^{2} + 2^{2} + 25^{2} + 0^{2} + 2^{2} + 4^{2} = 8304.

by Yoshio Mimura, Kobe, Japan

## 8305

8305^{2} = 68973025, a square with different digits.

1 / 8305 = 0.000120409, and 120409 = 347^{2}

by Yoshio Mimura, Kobe, Japan

## 8306

8306^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 8307

390^{k} + 2262^{k} + 4329^{k} + 6708^{k} are squares for k = 1,2,3 (117^{2}, 8307^{2}, 628173^{2}).

by Yoshio Mimura, Kobe, Japan

## 8308

8308^{2} = 158^{3} + 310^{3} + 328^{3}.

by Yoshio Mimura, Kobe, Japan

## 8313

8313^{2} = 161^{3} + 274^{3} + 354^{3}.

by Yoshio Mimura, Kobe, Japan

## 8315

8315^{2} = 81^{3} + 298^{3} + 348^{3} = 121^{3} + 278^{3} + 358^{3}.

by Yoshio Mimura, Kobe, Japan

## 8316

8316^{2} = 42^{3} + 282^{3} + 360^{3}.

by Yoshio Mimura, Kobe, Japan

## 8317

8317^{2} = 177^{3} + 255^{3} + 361^{3}.

by Yoshio Mimura, Kobe, Japan

## 8319

8319^{2} = 57^{3} + 128^{3} + 406^{3} = 10^{4} + 35^{4} + 38^{4} + 90^{4}.

by Yoshio Mimura, Kobe, Japan

## 8320

8320^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 8322

8322^{2} = 69255684, and 6925 / 5 * 6 + 8 + 4 = 8322.

by Yoshio Mimura, Kobe, Japan

## 8323

8323^{2} = 69272329, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 8327

8327^{5} = 40035275672417850407 : 4^{2} + 0^{2} + 0^{2} + 3^{2} + 52^{2} + 7^{2} + 56^{2} + 7^{2} + 24^{2} + 1^{2} + 7^{2} + 8^{2} + 5^{2} + 0^{2} + 40^{2} + 7^{2} = 4^{2} + 0^{2} + 0^{2} + 3^{2} + 5^{2} + 27^{2} + 56^{2} + 7^{2} + 2^{2} + 41^{2} + 7^{2} + 8^{2} + 50^{2} + 4^{2} + 0^{2} + 7^{2} =
40^{2} + 0^{2} + 3^{2} + 52^{2} + 7^{2} + 56^{2} + 7^{2} + 24^{2} + 1^{2} + 7^{2} + 8^{2} + 5^{2} + 0^{2} + 4^{2} + 0^{2} + 7^{2} = 8327.

by Yoshio Mimura, Kobe, Japan

## 8329

8329^{2} = 17^{3} + 219^{3} + 389^{3}.

by Yoshio Mimura, Kobe, Japan

## 8330

8330^{2} = 203^{3} + 294^{3} + 329^{3}.

8330^{2} = (366 + 367 + 368 + 369)^{2} + (370 + 371 + 372 + 373)^{2} + (374 + 375 + 376 + 377)^{2} + ... + (462 + 463 + 464 + 465)^{2}.

by Yoshio Mimura, Kobe, Japan

## 8334

8334^{2} = 69455556, and

6945 * 5 / 5 / 5 * 6 = 6945 / 5 * 5 / 5 * 6 = 6945 / 5 / 5 * 5 * 6 = 8334.

by Yoshio Mimura, Kobe, Japan

## 8335

8335^{5} = 40227961037165959375 : 4^{2} + 0^{2} + 22^{2} + 7^{2} + 9^{2} + 61^{2} + 0^{2} + 3^{2} + 7^{2} + 16^{2} + 5^{2} + 9^{2} + 59^{2} + 3^{2} + 7^{2} + 5^{2} = 4^{2} + 0^{2} + 22^{2} + 7^{2} + 9^{2} + 61^{2} + 0^{2} + 3^{2} + 7^{2} + 16^{2} + 59^{2} + 5^{2} + 9^{2} + 3^{2} + 7^{2} + 5^{2} = 8335.

by Yoshio Mimura, Kobe, Japan

## 8338

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

## 8339

8339^{2} = 69538921, and 695 / 3 * 8 * 9 / 2 - 1 = 8339.

by Yoshio Mimura, Kobe, Japan

## 8342

8342^{2} = 69588964, 6 + 95 * 88 - 96 / 4 = 8342.

by Yoshio Mimura, Kobe, Japan

## 8343

8343^{2} = 18^{4} + 63^{4} + 72^{4} + 72^{4}.

by Yoshio Mimura, Kobe, Japan

## 8346

8346^{2} = 69655716, and 6^{2} + 9^{2} + 6^{2} + 5^{2} + 5^{2} + 7^{2} + 1^{2} + 6^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 8347

8347^{2} = 69672409, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 8349

8349^{2}± 2 are primes.

8349^{2} = 69705801, and 6^{2} + 9^{2} + 7^{2} + 0^{2} + 5^{2} + 8^{2} + 0^{2} + 1^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 8350

8350^{4} = 4861227006250000, and 4^{2} + 86^{2} + 12^{2} + 27^{2} + 0^{2} + 0^{2} + 6^{2} + 2^{2} + 5^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 8350.

by Yoshio Mimura, Kobe, Japan

## 8352

8352^{2} = (62 + 63 + 64 + ... + 93)^{2} + (94 + 95 + 96 + ... + 125)^{2} + (126 + 127 + 128 + ... + 157)^{2} + ... + (158 + 159 + 160 + ... + 189)^{2}.

8352^{2} = 48^{3} + 120^{3} + 408^{3}.

Komachi equations: 8352^{2} = 9^{2} * 87^{2} / 6^{2} * 5^{2} * 4^{2} * 32^{2} / 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 8353

8353^{5} = 40664216322568456993 : 4^{2} + 0^{2} + 6^{2} + 6^{2} + 4^{2} + 2^{2} + 16^{2} + 3^{2} + 2^{2} + 2^{2} + 5^{2} + 68^{2} + 4^{2} + 56^{2} + 9^{2} + 9^{2} + 3^{2} = 8353.

by Yoshio Mimura, Kobe, Japan

## 8354

1 / 8354 = 0.000119703136222169, 1^{2} + 19^{2} + 7^{2} + 031^{2} + 36^{2} + 22^{2} + 21^{2} + 69^{2} = 8354.

by Yoshio Mimura, Kobe, Japan

## 8356

8356^{5} = 40737292048864715776 : 40^{2} + 7^{2} + 3^{2} + 7^{2} + 29^{2} + 2^{2} + 0^{2} + 48^{2} + 8^{2} + 6^{2} + 4^{2} + 7^{2} + 1^{2} + 57^{2} + 7^{2} + 6^{2} = 8356.

by Yoshio Mimura, Kobe, Japan

## 8358

8358^{2} = 74^{3} + 181^{3} + 399^{3}.

8358^{5} = 40786067536627740768 : 4^{2} + 0^{2} + 7^{2} + 8^{2} + 6^{2} + 0^{2} + 6^{2} + 7^{2} + 53^{2} + 66^{2} + 27^{2} + 7^{2} + 4^{2} + 0^{2} + 7^{2} + 6^{2} + 8^{2} = 8358.

by Yoshio Mimura, Kobe, Japan

## 8360

8360^{5} = 40834889733017600000 : 4^{2} + 0^{2} + 8^{2} + 34^{2} + 8^{2} + 8^{2} + 9^{2} + 7^{2} + 33^{2} + 0^{2} + 1^{2} + 76^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 8360.

by Yoshio Mimura, Kobe, Japan

## 8362

8362^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 8364

8364^{2} = (3^{2} + 8)(14^{2} + 8)(142^{2} + 8).

8364^{2} = 49^{3} + 231^{3} + 386^{3}.

by Yoshio Mimura, Kobe, Japan

## 8365

8365^{2} = 69973225, and 6^{2} + 9^{2} + 9^{2} + 7^{2} + 3^{2} + 2^{2} + 2^{2} + 5^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 8369

8369^{5} = 41055168535669652849 : 4^{2} + 10^{2} + 5^{2} + 5^{2} + 1^{2} + 6^{2} + 8^{2} + 5^{2} + 3^{2} + 5^{2} + 6^{2} + 69^{2} + 6^{2} + 5^{2} + 28^{2} + 49^{2} = 8369.

by Yoshio Mimura, Kobe, Japan

## 8374

8374^{k} + 11289^{k} + 22366^{k} + 28196^{k} are squares for k = 1,2,3 (265^{2}, 38637^{2}, 5969125^{2}).

by Yoshio Mimura, Kobe, Japan

## 8375

The square root of 8375 is 91.515026...., and 91 = 5^{2} + 1^{2} + 5^{2} + 0^{2} + 2^{2} + 6^{2}.

8375^{5} = 41202548431396484375 : 4^{2} + 12^{2} + 0^{2} + 2^{2} + 5^{2} + 4^{2} + 8^{2} + 4^{2} + 3^{2} + 1^{2} + 3^{2} + 9^{2} + 6^{2} + 48^{2} + 4^{2} + 3^{2} + 75^{2} = 4^{2} + 12^{2} + 0^{2} + 2^{2} + 5^{2} + 48^{2} + 4^{2} + 3^{2} + 1^{2} + 3^{2} + 9^{2} + 6^{2} + 4^{2} + 8^{2} + 4^{2} + 3^{2} + 75^{2} =
4^{2} + 12^{2} + 0^{2} + 25^{2} + 4^{2} + 8^{2} + 43^{2} + 1^{2} + 3^{2} + 9^{2} + 64^{2} + 8^{2} + 4^{2} + 37^{2} + 5^{2} = 8375.

by Yoshio Mimura, Kobe, Japan

## 8376

8376^{2} = (70 + 71 + 72 + ... + 81)^{2} + (82 + 83 + 84 + ... + 93)^{2} + (94 + 95 + 96 + ... + 105)^{2} + ... + (250 + 251 + 252 + ... + 261)^{2}.

8376^{5} = 41227152842752229376 : 4^{2} + 1^{2} + 2^{2} + 2^{2} + 71^{2} + 5^{2} + 28^{2} + 42^{2} + 7^{2} + 5^{2} + 2^{2} + 22^{2} + 9^{2} + 3^{2} + 7^{2} + 6^{2} = 4^{2} + 1^{2} + 2^{2} + 2^{2} + 71^{2} + 5^{2} + 28^{2} + 42^{2} + 7^{2} + 5^{2} + 22^{2} + 2^{2} + 9^{2} + 3^{2} + 7^{2} + 6^{2} =
4^{2} + 1^{2} + 22^{2} + 71^{2} + 5^{2} + 28^{2} + 42^{2} + 7^{2} + 5^{2} + 2^{2} + 2^{2} + 2^{2} + 9^{2} + 3^{2} + 7^{2} + 6^{2} = 8376.

by Yoshio Mimura, Kobe, Japan

## 8378

8378^{2} = 29^{4} + 37^{4} + 67^{4} + 83^{4}.

by Yoshio Mimura, Kobe, Japan

## 8381

8381^{2} = (332 + 333 + 334 + ... + 348)^{2} + (349 + 350 + 351 + ... + 365)^{2}.

by Yoshio Mimura, Kobe, Japan

## 8383

8383^{5} = 41399713035453662143 : 4^{2} + 1^{2} + 39^{2} + 9^{2} + 7^{2} + 1^{2} + 30^{2} + 3^{2} + 5^{2} + 4^{2} + 5^{2} + 3^{2} + 6^{2} + 62^{2} + 1^{2} + 43^{2} = 4^{2} + 1^{2} + 3^{2} + 9^{2} + 9^{2} + 7^{2} + 1^{2} + 30^{2} + 3^{2} + 5^{2} + 4^{2} + 53^{2} + 66^{2} + 2^{2} + 1^{2} + 4^{2} + 3^{2} = 8383.

by Yoshio Mimura, Kobe, Japan

## 8384

8384^{2} = 70291456, a square with different digits.

8384^{2} = 27^{3} + 293^{3} + 356^{3} = 152^{3} + 272^{3} + 360^{3}.

by Yoshio Mimura, Kobe, Japan

## 8385

8385^{2} = 78^{3} + 97^{3} + 410^{3}.

8385^{4} = 4943246502650625, and 4^{2} + 9^{2} + 43^{2} + 2^{2} + 46^{2} + 5^{2} + 0^{2} + 2^{2} + 65^{2} + 0^{2} + 6^{2} + 2^{2} + 5^{2} = 8385.

by Yoshio Mimura, Kobe, Japan

## 8390

8390^{2} = S_{2}(4) + S_{2}(595), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 8393

(8393^{2} + 1) = (2^{2} + 1)(5^{2} + 1)(6^{2} + 1)(10^{2} + 1)(12^{2} + 1) = (1^{2} + 1)(6^{2} + 1)(8^{2} + 1)(10^{2} + 1)(12^{2} + 1).

8393^{4} = 4962138621117601, and 4^{2} + 9^{2} + 6^{2} + 21^{2} + 38^{2} + 6^{2} + 21^{2} + 11^{2} + 76^{2} + 0^{2} + 1^{2} = 8393.

8393^{5} = 41647229447040025193 : 4^{2} + 16^{2} + 4^{2} + 72^{2} + 29^{2} + 4^{2} + 4^{2} + 7^{2} + 0^{2} + 40^{2} + 0^{2} + 2^{2} + 5^{2} + 19^{2} + 3^{2} = 4^{2} + 16^{2} + 47^{2} + 22^{2} + 9^{2} + 4^{2} + 4^{2} + 70^{2} + 4^{2} + 0^{2} + 0^{2} + 2^{2} + 5^{2} + 19^{2} + 3^{2} = 8393.

by Yoshio Mimura, Kobe, Japan

## 8394

8394^{2} = 70459236, a square with different digits.

8394^{2} = 70459236, and 70 * 4 * 5 * 9 * 2 / 3 - 6 = 8394.

by Yoshio Mimura, Kobe, Japan

## 8396

8396^{2} = 70492816, a square with different digits.

8396^{5} = 41721714755492350976 : 4^{2} + 1^{2} + 7^{2} + 21^{2} + 7^{2} + 1^{2} + 47^{2} + 55^{2} + 49^{2} + 2^{2} + 3^{2} + 5^{2} + 0^{2} + 9^{2} + 7^{2} + 6^{2} = 8396.

by Yoshio Mimura, Kobe, Japan

## 8397

8397^{2} = 102^{3} + 249^{3} + 378^{3}.

by Yoshio Mimura, Kobe, Japan

## 8399

8399^{2} = 70543201, and 70 * 5 * 4 * 3 * 2 + 0 - 1 = 8399.

by Yoshio Mimura, Kobe, Japan