## 5301

5301^{2}± 2 are primes.

513^{k} + 1292^{k} + 3382^{k} + 3838^{k} are squares for k = 1,2,3 (95^{2}, 5301^{2}, 312265^{2}).

by Yoshio Mimura, Kobe, Japan

## 5302

1 / 5302 = 0.000188608072425, 18^{2} + 8^{2} + 60^{2} + 8^{2} + 0^{2} + 7^{2} + 24^{2} + 25^{2} = 5302,

1 / 5302 = 0.000188608072425, 18^{2} + 8^{2} + 60^{2} + 8^{2} + 07^{2} + 24^{2} + 25^{2} = 5302.

by Yoshio Mimura, Kobe, Japan

## 5303

5303^{2} = 28121809, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5304

5304^{2} = 28132416, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5306

5306^{2} = 28153636, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5313

5313^{2} = 28^{4} + 31^{4} + 48^{4} + 68^{4}.

by Yoshio Mimura, Kobe, Japan

## 5316

5316^{2} = 70^{3} + 72^{3} + 302^{3}.

by Yoshio Mimura, Kobe, Japan

## 5319

5319^{2} = 169^{3} + 192^{3} + 254^{3}.

by Yoshio Mimura, Kobe, Japan

## 5320

5320 = (1^{2} + 2^{2} + 3^{2} + ... + 3552^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 203^{2}).

5320^{2} = 64^{3} + 138^{3} + 294^{3}.

152^{k} + 209^{k} + 3344^{k} + 5320^{k} are squares for k = 1,2,3 (95^{2}, 6289^{2}, 433561^{2}).

by Yoshio Mimura, Kobe, Japan

## 5321

5321^{2} = 28313041, 2 - 8 - 3 + 130 * 41 = 5321.

by Yoshio Mimura, Kobe, Japan

## 5322

5322^{2} = 129^{3} + 152^{3} + 283^{3}.

by Yoshio Mimura, Kobe, Japan

## 5323

5323^{2} = 28334329, 2^{2} + 8^{2} + 3^{2} + 3^{2} + 4^{2} + 3^{2} + 2^{2} + 9^{2} = 14^{2}.

5323^{2} = 28334329, 2 * 8 * 334 - 3 - 2 * 9 = 5323.

by Yoshio Mimura, Kobe, Japan

## 5324

5324^{2} = 242^{3} + 242^{3}.

by Yoshio Mimura, Kobe, Japan

## 5325

5325^{2}± 2 are primes.

5325^{2} = 28355625, 2 / 8 * 355 * 6 * 2 * 5 = 5325.

by Yoshio Mimura, Kobe, Japan

## 5327

5327^{2} = 4^{4} + 44^{4} + 46^{4} + 67^{4}.

5327^{2} = 28376929, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5329

The square of 73.

5329^{2} = 73^{3} + 146^{3} + 292^{3}.

by Yoshio Mimura, Kobe, Japan

## 5330

5330^{2} = (8^{2} + 1)(9^{2} + 1)(73^{2} + 1) = (3^{2} + 4)(23^{2} + 4)(64^{2} + 4).

5330^{k} + 6682^{k} + 7514^{k} + 13598^{k} are squares for k = 1,2,3 (182^{2}, 17732^{2}, 1840748^{2}).

by Yoshio Mimura, Kobe, Japan

## 5331

5331^{2} = 176^{3} + 196^{3} + 249^{3}.

by Yoshio Mimura, Kobe, Japan

## 5333

5333^{4} = 808884167110321, and 8^{2} + 0^{2} + 8^{2} + 8^{2} + 8^{2} + 4^{2} + 1^{2} + 67^{2} + 11^{2} + 0^{2} + 3^{2} + 21^{2} = 5333.

A cubic polynomial :

(X + 828^{2})(X + 2688^{2})(X + 4531^{2}) = X^{3} + 5333^{2}X^{2} + 12936948^{2}X + 10084483584^{2}.

by Yoshio Mimura, Kobe, Japan

## 5334

5334^{2} = 28451556, 2^{2} + 8^{2} + 4^{2} + 5^{2} + 1^{2} + 5^{2} + 5^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 5336

5336^{2} = (4^{2} + 7)(15^{2} + 7)(73^{2} + 7).

5336^{2} = 40^{3} + 68^{3} + 304^{3} = 40^{3} + 152^{3} + 292^{3}.

5336^{4} = 810705806626816, and 8^{2} + 1^{2} + 0^{2} + 7^{2} + 0^{2} + 5^{2} + 8^{2} + 0^{2} + 66^{2} + 26^{2} + 8^{2} + 1^{2} + 6^{2} = 5336.

by Yoshio Mimura, Kobe, Japan

## 5338

The square root of 5338 is 73.0616...., and 73 = 0^{2} + 6^{2} + 1^{2} + 6^{2}.

by Yoshio Mimura, Kobe, Japan

## 5340

5340^{2} = 39^{3} + 97^{3} + 302^{3}.

by Yoshio Mimura, Kobe, Japan

## 5345

Loop of length 4 :

5345 --- 53^{2} + 45^{2} = 4834 --- 48^{2} + 34^{2} = 3460 --- 34^{2} + 60^{2} = 4756 --- 47^{2} + 56^{2} = 5345

by Yoshio Mimura, Kobe, Japan

## 5346

5346^{2} = (1^{2} + 2)(22^{2} + 2)(140^{2} + 2) = (1^{2} + 2)(3^{2} + 2)(5^{2} + 2)(8^{2} + 2)(22^{2} + 2)

= (1^{2} + 2)(4^{2} + 2)(5^{2} + 2)(140^{2} + 2) = (2^{2} + 2)(5^{2} + 2)(19^{2} + 2)(22^{2} + 2)

= (3^{2} + 2)(5^{2} + 2)(14^{2} + 2)(22^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 5347

5347^{2} = 28590409, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5349

5349^{2} = 305^{3} + 11^{5} + 5^{7}.

by Yoshio Mimura, Kobe, Japan

## 5352

5352^{4} = 820473236361216, and 8^{2} + 2^{2} + 0^{2} + 4^{2} + 7^{2} + 32^{2} + 3^{2} + 63^{2} + 6^{2} + 12^{2} + 1^{2} + 6^{2} = 5352.

by Yoshio Mimura, Kobe, Japan

## 5353

1 / 5353 = 0.000186811133943, 18^{2} + 68^{2} + 11^{2} + 13^{2} + 3^{2} + 9^{2} + 4^{2} + 3^{2} = 5353.

by Yoshio Mimura, Kobe, Japan

## 5354

1 / 5354 = 0.00018677624206, 18^{2} + 67^{2} + 7^{2} + 6^{2} + 2^{2} + 4^{2} + 20^{2} + 6^{2} = 5354.

by Yoshio Mimura, Kobe, Japan

## 5355

5355^{2} = (2^{2} - 1)(4^{2} - 1)(16^{2} - 1)(50^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 5356

5356^{2} = 28686736, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5358

5358^{2} = 2^{5} + 5^{5} + 7^{5} + 9^{5} + 31^{5}.

5358^{2} = 28708164, 2 * 87 + 0 + 81 * 64 = 5358.

141^{k} + 1645^{k} + 2773^{k} + 4277^{k} are squares for k = 1,2,3 (94^{2}, 5358^{2}, 322514^{2}).

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

## 5359

5359^{2} = 98^{3} + 225^{3} + 254^{3}.

5359^{2} = 28718881, 2 * 87 + 1 + 8 * 8 * 81 = 5359.

713^{k} + 5359^{k} + 10511^{k} + 17273^{k} are squares for k = 1,2,3 (184^{2}, 20930^{2}, 2543432^{2}).

by Yoshio Mimura, Kobe, Japan

## 5360

5360^{5} = 4424089950617600000 : 44^{2} + 24^{2} + 0^{2} + 8^{2} + 9^{2} + 9^{2} + 50^{2} + 6^{2} + 1^{2} + 7^{2} + 6^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 5360.

by Yoshio Mimura, Kobe, Japan

## 5361

5361^{2} = 49^{3} + 63^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 5363

403^{k} + 961^{k} + 5363^{k} + 8649^{k} are squares for k = 1,2,3 (124^{2}, 10230^{2}, 895652^{2}).

by Yoshio Mimura, Kobe, Japan

## 5364

5364^{2} = 66^{3} + 160^{3} + 290^{3}.

Komachi equation: 5364^{2} = 9^{2} / 87^{2} * 6^{2} / 5^{2} * 43210^{2}.

by Yoshio Mimura, Kobe, Japan

## 5365

5365^{2} = (6^{2} + 1)(882^{2} + 1).

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

5365 - 7034 - 6056 - 6736 - ... - 1446 - 2312 - 673 - 5365

(Note f(5365) = 53^{2} + 65^{2} = 7034, f(7034) = 70^{2} + 34^{2} = 6056, etc. See 37)

5365^{5} = 4444763232114353125 : 44^{2} + 44^{2} + 7^{2} + 6^{2} + 3^{2} + 23^{2} + 2^{2} + 1^{2} + 14^{2} + 3^{2} + 5^{2} + 3^{2} + 1^{2} + 25^{2} = 5365.

by Yoshio Mimura, Kobe, Japan

## 5366

5366^{2} = 28793956, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5368

5368^{2} = 224^{3} + 260^{3} = 6^{4} + 34^{4} + 54^{4} + 66^{4}.

by Yoshio Mimura, Kobe, Japan

## 5370

5370^{2} = 43^{3} + 185^{3} + 282^{3}.

5370^{2} = 28836900, 2 - 8 - 8 * 3 + 6 * 900 = 5370.

by Yoshio Mimura, Kobe, Japan

## 5371

5371^{2} = 28847641, 2 - 8 + 8 * 4 * 7 * 6 * 4 + 1 = 5371.

by Yoshio Mimura, Kobe, Japan

## 5372

5372^{2} = 28858384, 2 * 8 * 8 * 5 * 8 + 3 * 84 = 5372.

by Yoshio Mimura, Kobe, Japan

## 5373

5373^{2} = 28^{3} + 78^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 5375

5375^{2} = 28890625, 2 + 8 + 890 * 6 + 25 = 28 + 890 * 6 + 2 + 5 = 5375.

by Yoshio Mimura, Kobe, Japan

## 5376

5376^{2} = (3^{2} - 1)(15^{2} - 1)(127^{2} - 1).

5376^{2} = 96^{3} + 224^{3} + 256^{3}.

5376^{2}=28901376, a square with different digits.

10^{4} + 5376 = 124^{2}, 10^{4} - 5376 = 68^{2}.

Komachi equations:

5376^{2} = 9^{2} * 8^{2} * 7^{2} / 6^{2} * 5^{2} * 4^{2} * 32^{2} / 10^{2} = 98^{2} / 7^{2} * 6^{2} * 5^{2} * 4^{2} * 32^{2} / 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 5377

5377^{2} = 44^{3} + 228^{3} + 257^{3}.

by Yoshio Mimura, Kobe, Japan

## 5380

5380^{2} = (22 + 23 + 24 + 25 + 26 + 27 + 28 + 29)^{2} + (30 + 31 + 32 + 33 + 34 + 35 + 36 + 37)^{2} + (38 + 39 + 40 + 41 + 42 + 43 + 44 + 45)^{2} + ... + (214 + 215 + 216 + 217 + 218 + 219 + 220 + 221)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5382

1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + ... + 5382^{2} = 51979351515, which consisting of odd digits,

(The second 11-digit sum. There are 3 such sums in all.)

5382^{5} = 4515631221581262432 : 45^{2} + 1^{2} + 5^{2} + 6^{2} + 31^{2} + 2^{2} + 2^{2} + 15^{2} + 8^{2} + 12^{2} + 6^{2} + 2^{2} + 43^{2} + 2^{2} = 45^{2} + 15^{2} + 6^{2} + 31^{2} + 2^{2} + 2^{2} + 1^{2} + 5^{2} + 8^{2} + 12^{2} + 6^{2} + 2^{2} + 43^{2} + 2^{2} =
4^{2} + 5^{2} + 15^{2} + 63^{2} + 12^{2} + 2^{2} + 15^{2} + 8^{2} + 1^{2} + 26^{2} + 2^{2} + 4^{2} + 3^{2} + 2^{2} = 5382.

by Yoshio Mimura, Kobe, Japan

## 5383

5383^{2} = 49^{3} + 211^{3} + 269^{3} = 52^{3} + 204^{3} + 273^{3} = 105^{3} + 129^{3} + 295^{3} = 23^{4} + 38^{4} + 56^{4} + 64^{4}.

by Yoshio Mimura, Kobe, Japan

## 5384

5384^{5} = 4524027707214823424 : 4^{2} + 52^{2} + 4^{2} + 0^{2} + 27^{2} + 7^{2} + 0^{2} + 7^{2} + 2^{2} + 1^{2} + 4^{2} + 8^{2} + 2^{2} + 34^{2} + 24^{2} = 5384.

by Yoshio Mimura, Kobe, Japan

## 5385

The square root of 5385 is 73.38...., and 73 = 3^{2} + 8^{2}.

by Yoshio Mimura, Kobe, Japan

## 5386

The square root of 5386 is 73.38...., and 73 = 3^{2} + 8^{2}.

by Yoshio Mimura, Kobe, Japan

## 5390

5390^{2} = (1^{2} + 6)(2^{2} + 6)(22^{2} + 6)(29^{2} + 6) = (1^{2} + 6)(22^{2} + 6)(92^{2} + 6)

= (8^{2} + 6)(22^{2} + 6)(29^{2} + 6).

5390^{2} = 2^{3} + 49^{3} + 307^{3} = 66^{3} + 225^{3} + 259^{3}.

5390^{2} = 29052100, 2 * 90 + 5210 + 0 = 5390.

5390^{k} + 9482^{k} + 12034^{k} + 12298^{k} are squares for k = 1,2,3 (198^{2}, 20372^{2}, 2147508^{2}).

by Yoshio Mimura, Kobe, Japan

## 5392

5392^{2} = 36^{4} + 48^{4} + 48^{4} + 64^{4}.

by Yoshio Mimura, Kobe, Japan

## 5394

5394^{2} = 29095236, 2 * 90 * 9 * 5 * 2 / 3 - 6 = 5394.

by Yoshio Mimura, Kobe, Japan

## 5395

5395^{2} = 29106025, 2 + 9 * 10 * 60 - 2 - 5 = 2 * 9 * 10 * 60 / 2 - 5 = 5395.

by Yoshio Mimura, Kobe, Japan

## 5396

5396^{2} = 80^{3} + 102^{3} + 302^{3}.

5396^{3} = 157114339136, and 1^{2} + 57^{2} + 1^{2} + 1^{2} + 43^{2} + 3^{2} + 9^{2} + 13^{2} + 6^{2} = 5396.

5396^{2} = 29116816 appears in the decimal expressions of π:

π = 3.14159•••29116816••• (from the 34309th digit)

(29116816 is the seventh 8-digit square in the expression of π.)

by Yoshio Mimura, Kobe, Japan

## 5397

5397^{2} = 29127609, 2^{2} + 9^{2} + 1^{2} + 2^{2} + 7^{2} + 6^{2} + 0^{2} + 9^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan