## 5200

5200^{2} = (1^{2} + 4)(10^{2} + 4)(14^{2} + 4)(16^{2} + 4) = (1^{2} + 4)(2^{2} + 4)(3^{2} + 4)(14^{2} + 4)(16^{2} + 4)

= (1^{2} + 4)(2^{2} + 4)(3^{2} + 4)(6^{2} + 4)(36^{2} + 4) = (1^{2} + 4)(2^{2} + 4)(4^{2} + 4)(6^{2} + 4)(29^{2} + 4)

= (1^{2} + 4)(3^{2} + 4)(4^{2} + 4)(10^{2} + 4)(14^{2} + 4) = (1^{2} + 4)(6^{2} + 4)(10^{2} + 4)(36^{2} + 4)

= (10^{2} + 4)(14^{2} + 4)(36^{2} + 4) = (2^{2} + 4)(10^{2} + 4)(11^{2} + 4)(16^{2} + 4) = (2^{2} + 4)(16^{2} + 4)(114^{2} + 4)

= (2^{2} + 4)(3^{2} + 4)(14^{2} + 4)(36^{2} + 4) = (2^{2} + 4)(3^{2} + 4)(4^{2} + 4)(10^{2} + 4)(11^{2} + 4)

= (2^{2} + 4)(3^{2} + 4)(4^{2} + 4)(114^{2} + 4) = (2^{2} + 4)(4^{2} + 4)(14^{2} + 4)(29^{2} + 4)

= (3^{2} + 4)(6^{2} + 4)(14^{2} + 4)(16^{2} + 4) = (4^{2} + 4)(10^{2} + 4)(114^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 5201

5201^{2} = 38^{3} + 217^{3} + 256^{3}.

5201^{2} = 27050401, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5202

5202^{2} = 51^{4} + 51^{4} + 51^{4} + 51^{4}.

5202^{2} = 27060804, a zigzag square.

5202^{2} = 27060804, 2^{2} + 7^{2} + 0^{2} + 6^{2} + 0^{2} + 8^{2} + 0^{2} + 4^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 5203

5203^{2} = 27071209, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5204

5204^{2} = 27081616, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5205

5205^{2} = 4^{3} + 105^{3} + 296^{3}.

by Yoshio Mimura, Kobe, Japan

## 5207

5207 = (1^{2} + 2^{2} + 3^{2} + ... + 11049^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 637^{2}).

5207^{2} = 53^{3} + 237^{3} + 239^{3} = 7^{4} + 46^{4} + 58^{4} + 58^{4}.

by Yoshio Mimura, Kobe, Japan

## 5208

5208^{2} = (1^{2} + 3)(3^{2} + 3)(9^{2} + 3)(82^{2} + 3) = (2^{2} + 3)(3^{2} + 3)(11^{2} + 3)(51^{2} + 3)

= (3^{2} + 3)(11^{2} + 3)(135^{2} + 3) = (9^{2} + 3)(11^{2} + 3)(51^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 5209

A cubic polynomial :

(X + 1044^{2})(X + 2639^{2})(X + 4368^{2}) = X^{3} + 5209^{2}X^{2} + 12698868^{2}X + 12034346688^{2}.

by Yoshio Mimura, Kobe, Japan

## 5211

5211^{2} = 72^{3} + 186^{3} + 273^{3}.

by Yoshio Mimura, Kobe, Japan

## 5216

5216^{2} = 144^{3} + 188^{3} + 260^{3}.

by Yoshio Mimura, Kobe, Japan

## 5217

5217^{2} = 24^{4} + 38^{4} + 43^{4} + 68^{4}.

The square root of 5217 is 72.228..., and 72 = 2^{2} + 2^{2} + 8^{2}.

by Yoshio Mimura, Kobe, Japan

## 5220

5220^{2} = (1^{2} + 9)(3^{2} + 9)(7^{2} + 9)(51^{2} + 9).

Komachi equations:

5220^{2} = 9^{2} * 87^{2} / 6^{2} * 5^{2} / 4^{2} * 32^{2} */ 1^{2} = 9^{2} * 87^{2} * 6^{2} / 54^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 5221

5221^{2} = 27258841 appears in the decimal expressions of π:

π = 3.14159•••27258841••• (from the 32175th digit)

(27258841 is the sixth 8-digit square in the expression of π.)

by Yoshio Mimura, Kobe, Japan

## 5223

5223^{2} = 27279729, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5225

5225^{5} = 3894318555166015625 : 3^{2} + 8^{2} + 9^{2} + 4^{2} + 3^{2} + 18^{2} + 5^{2} + 5^{2} + 5^{2} + 1^{2} + 66^{2} + 0^{2} + 15^{2} + 6^{2} + 2^{2} + 5^{2} = 5225.

by Yoshio Mimura, Kobe, Japan

## 5226

1326^{k} + 2730^{k} + 5226^{k} + 7618^{k} are squares for k = 1,2,3 (130^{2}, 9724^{2}, 779428^{2}).

by Yoshio Mimura, Kobe, Japan

## 5229

5229^{2} = 120^{3} + 174^{3} + 273^{3}.

5229^{2} = 27342441, 2 * 7 * 342 + 441 = 5229.

by Yoshio Mimura, Kobe, Japan

## 5232

5232^{2} = 3^{5} + 9^{5} + 20^{5} + 25^{5} + 27^{5}.

5232^{2} = 27373824, a zigzag square.

Komachi equation: 5232^{2} = 9^{2} * 8^{2} * 7^{2} * 654^{2} / 3^{2} / 21^{2}.

5232^{2} = 27373824, 2 * 7 * 373 + 8 - 2 + 4 = 27 / 3 * 73 * 8 - 24 = 5232.

by Yoshio Mimura, Kobe, Japan

## 5234

5234^{2} = 27394756, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5235

5235^{2} = 1^{3} + 74^{3} + 300^{3}.

5235^{2} = 27405225, 2 * 7 - 4 + 0 + 5225 = 5235.

by Yoshio Mimura, Kobe, Japan

## 5237

5237^{2} = 198^{3} + 201^{3} + 226^{3}.

by Yoshio Mimura, Kobe, Japan

## 5238

5238^{2} = 129^{3} + 222^{3} + 243^{3}.

Komachi equations: 5238^{2} = 9^{2} / 8^{2} / 7^{2} * 6^{2} * 5432^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 5240

5240^{2} = 27457600, 2 * 74 * 5 * 7 + 60 + 0 = 5240.

by Yoshio Mimura, Kobe, Japan

## 5242

5242^{2} = 8^{3} + 219^{3} + 257^{3}.

by Yoshio Mimura, Kobe, Japan

## 5243

5243^{2} = 77^{3} + 231^{3} + 245^{3}.

by Yoshio Mimura, Kobe, Japan

## 5244

5244^{2} = 122^{3} + 186^{3} + 268^{3}.

5244^{2} = 27499536, 2 - 7 - 4 + 99 * 53 + 6 = 5244.

by Yoshio Mimura, Kobe, Japan

## 5246

5246^{2} = 27520516, 2^{2} + 7^{2} + 5^{2} + 2^{2} + 0^{2} + 5^{2} + 1^{2} + 6^{2} = 12^{2}.

5246^{2} = (13^{2} + 3)(400^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 5247

5247^{2} = 12^{4} + 18^{4} + 27^{4} + 72^{4}.

5247^{2} = 27531009, 2^{2} + 7^{2} + 5^{2} + 3^{2} + 1^{2} + 0^{2} + 0^{2} + 9^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 5248

5248^{2} = 8^{4} + 24^{4} + 24^{4} + 72^{4}.

by Yoshio Mimura, Kobe, Japan

## 5250

5250^{2} = (1^{2} + 6)(12^{2} + 6)(162^{2} + 6) = (1^{2} + 6)(2^{2} + 6)(3^{2} + 6)(12^{2} + 6)(13^{2} + 6)

= (1^{2} + 6)(2^{2} + 6)(3^{2} + 6)(162^{2} + 6) = (3^{2} + 6)(8^{2} + 6)(12^{2} + 6)(13^{2} + 6)

= (3^{2} + 6)(8^{2} + 6)(162^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 5251

5251^{k} + 7257^{k} + 15635^{k} + 27553^{k} are squares for k = 1,2,3 (236^{2}, 32922^{2}, 5026564^{2}).

by Yoshio Mimura, Kobe, Japan

## 5252

5252^{2} = 27583504, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5253

5253^{2} = 27594009, 2^{2} + 7^{2} + 5^{2} + 9^{2} + 4^{2} + 0^{2} + 0^{2} + 9^{2} = 16^{2}.

5253^{2} + 5254^{2} + 5255^{2} + ... + 5304^{2} = 5305^{2} + 5306^{2} + 5307^{2} + ... + 5355^{2}.

by Yoshio Mimura, Kobe, Japan

## 5255

5255^{2} = 27615025, 2^{2} + 7^{2} + 6^{2} + 1^{2} + 5^{2} + 0^{2} + 2^{2} + 5^{2} = 12^{2}.

by Yoshio Mimura, Kobe, Japan

## 5256

5256^{2}± 5 are primes.

5256^{2} = 72^{3} + 142^{3} + 290^{3}.

5256^{2} = 27625536, 2 * 7 * 625 / 5 * 3 + 6 = 5256.

by Yoshio Mimura, Kobe, Japan

## 5257

5257^{2} = 193^{3} + 201^{3} + 231^{3}.

by Yoshio Mimura, Kobe, Japan

## 5258

1276^{k} + 1529^{k} + 1738^{k} + 5258^{k} are squares for k = 1,2,3 (99^{2}, 5885^{2}, 395307^{2}).

by Yoshio Mimura, Kobe, Japan

## 5259

Komachi equation: 5259^{2} = 9^{2} * 8765^{2} * 4^{2} / 3^{2} / 2^{2} / 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 5265

5265^{2} = 33^{3} + 156^{3} + 288^{3}.

Komachi equation: 5265^{2} = 12^{2} * 3^{2} / 4^{2} * 5^{2} / 6^{2} * 78^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 5270

1666^{k} + 2210^{k} + 5270^{k} + 9350^{k} are squares for k = 1,2,3 (136^{2}, 11084^{2}, 989536^{2}).

496^{k} + 5270^{k} + 7161^{k} + 11098^{k} are squares for k = 1,2,3 (155^{2}, 14229^{2}, 1371347^{2}).

by Yoshio Mimura, Kobe, Japan

## 5273

1 / 5273 = 0.0001896453631, 1^{2} + 8^{2} + 9^{2} + 64^{2} + 5^{2} + 3^{2} + 6^{2} + 31^{2} = 5273.

by Yoshio Mimura, Kobe, Japan

## 5277

5277^{2} = 25^{3} + 66^{3} + 302^{3}.

by Yoshio Mimura, Kobe, Japan

## 5278

5278^{2} = 57^{3} + 98^{3} + 299^{3}.

by Yoshio Mimura, Kobe, Japan

## 5280

5280^{2} = (11^{2} - 1)(21^{2} - 1)(23^{2} - 1) = (3^{2} - 1)(21^{2} - 1)(89^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(21^{2} - 1)(23^{2} - 1)

= (3^{2} - 1)(9^{2} - 1)(10^{2} - 1)(21^{2} - 1) = (4^{2} - 1)(21^{2} - 1)(65^{2} - 1).

5280^{2} = 145^{3} + 146^{3} + 279^{3}.

The integral triangle of sides 697, 131144, 131697 has square area 5280^{2}.

122^{2} + 5280 = 142^{2}, 122^{2} - 5280 = 98^{2},

73^{2} + 5280 = 103^{2}, 73^{2} - 5280 = 7^{2}.

The square root of 5280 is 72.66..., 72 = 6^{2} + 6^{2}.

by Yoshio Mimura, Kobe, Japan

## 5281

5281^{2} = 27888961, 2 - 7 * 8 + 889 * 6 + 1 = 5281.

by Yoshio Mimura, Kobe, Japan

## 5282

5282^{2} = 27899524, 2^{2} + 7^{2} + 8^{2} + 9^{2} + 9^{2} + 5^{2} + 2^{2} + 4^{2} = 18^{2}.

by Yoshio Mimura, Kobe, Japan

## 5283

5283^{2} = 66^{3} + 220^{3} + 257^{3} = 129^{3} + 178^{3} + 272^{3}.

by Yoshio Mimura, Kobe, Japan

## 5286

5286^{2}± 5 are primes.

5286^{2} = (2^{2} + 2)(2158^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 5287

5287^{2} = 27952369, 2^{2} + 7^{2} + 9^{2} + 5^{2} + 2^{2} + 3^{2} + 6^{2} + 9^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 5288

5288^{2} = 27962944, 27 / 9 * 6 * 294 - 4 = 5288.

by Yoshio Mimura, Kobe, Japan

## 5289

5289^{2} = (6^{2} + 5)(826^{2} + 5).

5289^{k} + 9202^{k} + 12986^{k} + 18748^{k} are squares for k = 1,2,3 (215^{2}, 25155^{2}, 3115565^{2}).

by Yoshio Mimura, Kobe, Japan

## 5292

5292^{2} = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(11^{2} + 5)(17^{2} + 5) = (1^{2} + 5)(3^{2} + 5)(4^{2} + 5)(7^{2} + 5)(17^{2} + 5)

= (3^{2} + 5)(7^{2} + 5)(11^{2} + 5)(17^{2} + 5).

5292^{2} = 84^{3} + 126^{3} + 294^{3} = 102^{3} + 222^{3} + 252^{3}.

5292^{2} = 28005264, 28 + 0 + 0 + 5264 = 28 + 0 * 0 + 5264 = 5292.

by Yoshio Mimura, Kobe, Japan

## 5294

5294^{2} = 28026436, 2^{2} + 8^{2} + 0^{2} + 2^{2} + 6^{2} + 4^{2} + 3^{2} + 6^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 5298

5298^{2} = 28068804, a square with even digits.

by Yoshio Mimura, Kobe, Japan