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1300 - 1399

1300

13002 = 1690000, 16 = 42 and 90000 = 3002.

13002 = (1 + 2 + 3 + 4)(5)(6 + 7)(8 + 9 + 10 + ... + 72).

13002 = (12 + 1)(32 + 1)(52 + 1)(572 + 1)
= (12 + 1)(32 + 1)(52 + 1)(72 + 1)(82 + 1)
= (12 + 4)(162 + 4)(362 + 4) = (12 + 4)(32 + 4)(42 + 4)(362 + 4)
= (14 + 4)(44 + 4)(64 + 4).

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1301

13012 = 1692601, 169 = 132 and 2601 = 512.

13012 = 1692601, a reversible square (1062961 = 10312).

13012 = 423 + 563 + 1133.

Page of Squares : First Upload January 22, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1302

13022 = 1695204, a square with different digits.

13022 = (1)(2)(3 + 4 + 5 + ... + 33)(34 + 35 + 36 + ... + 64).

13022 = (52 + 6)(62 + 6)(362 + 6).

The 4-by-4 magic squares consisting of different squares with constant 1302:

02 12252262
102332 72 82
192 42222212
292142122112
     
12 82 92342
122192262112
142292162 32
312 62172 42
     
12 92142322
152312 42102
202 22272132
262162192 32
     
42102152312
112212222162
182202232 72
292192 82 62

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1303

13032 = 1697809, 1 + 6 * 97 + 80 * 9 = 1303.

The 5-by-5 magic square consisting of different squares with constant 1303:

02 52112142312
132272102162 72
172182252 12 82
192 92 42292 22
222122212 32152

Page of Squares : First Upload January 22, 2007 ; Last Revised October 5, 2009
by Yoshio Mimura, Kobe, Japan

1304

13042 = 123 + 763 + 1083.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1305

Komachi equation: 13052 = 92 * 872 * 62 / 542 * 32 / 22 * 102.

The 4-by-4 magic square consisting of different squares with constant 1305:

02 92182302
112322 42122
202 22262152
282142172 62

Page of Squares : First Upload January 25, 2010 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1306

The sum of the divisors of 13062 is a square, 17292.

Page of Squares : First Upload November 1, 2011 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan

1307

13072 = 1708249, a square with different digits.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1308

Komachi equation: 13082 = 982 / 72 * 6542 * 32 / 212.

Page of Squares : First Upload July 23, 2010 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1311

13112± 2 are primes.

13112 = 1718721, 1 + 7 * 187 + 2 - 1 = 1 * 7 * 187 + 2 * 1 = 1311.

13112 = 1718721, 12 + 72 + 12 + 82 + 72 + 22 + 12 = 132.

A cubic polynomial :
(X + 6082)(X + 13112)(X + 21242) = X3 + 25692X2 + 31712522X + 16930149122.

Page of Squares : First Upload January 22, 2007 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1312

13122 = 44 + 124 + 124 + 364.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1314

13142 = 1726596, a zigzag square.

13142± 5 are primes.

106k + 266k + 814k + 1314k are squares for k = 1,2,3 (502, 15722, 531802).

The 4-by-4 magic squares consisting of different squares with constant 1314:

02 12232282
32342102 72
242112192162
272 62182152
   
02 32242272
52262172182
82252202152
352 22 72 62
   
02 32242272
92262192142
122252162172
332 22112102
   
02 82172312
122322 52112
212 12262142
272152182 62
   
02 92122332
112302172 22
132182252142
322 32162 52
02 92122332
172302 52102
202 32282112
252182192 22
   
02122212272
132252222 62
192232102182
282 42172152
   
12 62112342
142132302 72
212222172102
262252 22 32
   
22 62 72352
112272202 82
172152282 42
302182 92 32
   
32 92182302
112312 62142
202 42272132
282162152 72

Page of Squares : First Upload January 22, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1316

13162 = 1731856, 1 + 73 * 18 - 5 + 6 = 1316.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1317

13172 = 163 + 663 + 1133 = 413 + 783 + 1063.

The 4-by-4 magic squares consisting of different squares with constant 1317:

02 22232282
42342 82 92
252 62202162
262112182142
     
12 42202302
62262222112
162242172142
322 72122102

Page of Squares : First Upload July 7, 2008 ; Last Revised January 25, 2010
by Yoshio Mimura, Kobe, Japan

1318

82k + 182k + 722k + 1318k are squares for k = 1,2,3 (482, 15162, 516962).

Page of Squares : First Upload April 19, 2011 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1319

1319 is the 9th prime for which the Lendre symbol (a/p) = 1 for a = 1, 2,..., 10
1319 is the third primt for which the Legendre symbol (a/p) = 1 for a = 1, 2,..., 12.

13192 = 1739761, 1 + 7 * 3 * 9 * 7 - 6 + 1 = 1319.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1320

612 + 1320 = 712, 612 - 1320 = 492.

13202 = (1)(2)(3)(4)(5)(6 + 7 + 8 + ... + 170),
13202 = (1)(2)(3)(4)(5 + 6 + 7 + ... + 15)(16 + 17 + 18 + ... + 39),
13202 = (1)(2)(3 + 4 + 5 + ... + 8)(9 + 10 + 11)(12 + 13 + 14 + ... + 43),
13202 = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + ... + 10)(11)(12 + 13 + 14 + ... + 21),
13202 = (1)(2 + 3)(4 + 5 + 6 + 7)(8)(9 + 10 + 11 + ... + 63),
13202 = (1 + 2 + 3 + ... + 11)(12)(13 + 14 + 15 + ... + 67),
13202 = (1 + 2 + 3 + ... + 32)(33 + 34 + 35 + ... + 87),
13202 = (1 + 2 + 3 + 4)(5 + 6 + ... + 10)(11 + 12 + ... + 21)(22),
13202 = (1 + 2 + 3 + ... + 9)(10)(11 + 12 + 13 + ... + 21)(22),
13202 = (1 + 2 + 3)(4)(5)(6 + 7 + 8 + ... + 170),
13202 = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 15)(16 + 17 + 18 + ... + 39).

Page of Squares : First Upload January 22, 2007 ; Last Revised July 26, 2011
by Yoshio Mimura, Kobe, Japan

1321

13212 = 1745041, a zigzag square.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1322

13222 = 1747684, a zigzag square.

13222 = 13 + 273 + 1203.

Page of Squares : First Upload January 22, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1323

13232 = 1750329, a square with different digits.

13232 = 1750329, 12 + 72 + 52 + 02 + 32 + 22 + 92 = 132.

Komachi equation: 13232 = 983 / 73 / 63 * 543 - 33 * 213.

13232 = 843 + 1053 = 513 + 633 + 1113.

13232 = (1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10 + ... + 154),
13232 = (1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10)(11 + 12 +13 + ... + 31).

The 4-by-4 magic squares consisting of different squares with constant 1323:

02 12192312
32272212122
152232202132
332 82112 72
   
02 32152332
132322 92 72
232172212 82
252 12242112
   
12 32172322
112332 82 72
242 92212152
252122232 52
   
12 32232282
92212242152
202272132 52
292122 72172
   
32122212272
132232242 72
192252 92162
282 52152172

Page of Squares : First Upload January 22, 2007 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1324

13242 = 413 + 793 + 1063.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1326

13262 = 1758276, a zigzag square.

13262 = 1758276, 175 * 8 + 2 - 76 = 1326.

13262 = 18 + 18 + 38 + 38 + 48 + 68.

13262 = (1)(2 + 3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 25)(26).

13262 = 13 + 23 + 33 + 43 + ... + 513.

13262 = (22 + 9)(52 + 9)(632 + 9).

The 4-by-4 magic squares consisting of different squares with constant 1326:

02 12102352
112302172 42
232202192 62
262 52242 72
     
02 12102352
172302112 42
192202232 62
262 52242 72
     
02 12132342
22272232 82
192202222 92
312142122 52
     
02 22192312
72132282182
112332102 42
342 82 92 52
02 72112342
132302162 12
142192252122
312 42182 52
     
02112232262
142252212 82
172242102192
292 22162152
     
12 62 82352
132202262 92
162292152 22
302 72192 42
     
12 82192302
102332 42112
212 22252162
282132182 72
22 32232282
82252212142
132262162152
332 42102112
     
22 82132332
112152282142
242262 72 52
252192182 42
     
32 72222282
102242232112
162262132152
312 52122142

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1328

13282 = 1763584, a square with different digits.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1329

13292 = 43 + 893 + 1023.

The 4-by-4 magic square consisting of different squares with constant 1329:

02 22222292
42342112 62
232122202162
282 52182142

Page of Squares : First Upload July 7, 2008 ; Last Revised January 25, 2010
by Yoshio Mimura, Kobe, Japan

1330

13302 = (1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10 + 11 + 12 + ... + 85).

13302 = (1)(2)(3 + 4 + ... + 16)(17 + 18 + ... + 116).

1330k + 3610k + 5690k + 6270k are squares for k = 1,2,3 (1302, 93002, 6929002).
1330k + 8417k + 8626k + 10868k are squares for k = 1,2,3 (1712, 162832, 15887612).
510k + 690k + 1330k + 9570k are squares for k = 1,2,3 (1102, 97002, 9377002).

Komachi equation: 13302 = 982 * 762 * 52 / 42 * 32 / 212.

The 4-by-4 magic square consisting of different squares with constant 1330:

12 22222292
42172252202
232262102 52
282192112 82

Page of Squares : First Upload January 22, 2007 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1331

1331 = 113 = 192 + 212 + 232.

13312 = 1213 = 493 + 843 + 1023.

1 / 1331 = 0.00075131480090157776108...,
and 7512 + 312 + 4802 + 092 + 0152 + 7772 + 6102 + 82 = 13312.

13312 = 30 + 32 + 34 + 311 + 313.

Page of Squares : First Upload January 22, 2007 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

1332

13322 = 263 + 733 + 1113.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1333

Komachi equation: 13332 = 1233 + 453 - 63 - 73 * 83 + 93.

13332 = 463 + 653 + 1123.

74648k + 493210k + 498542k + 710489k are squares for k = 1,2,3 (13332, 10010832, 7765004932).

Page of Squares : First Upload July 7, 2008 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1334

13342 = 1779556, a square with odd digits except the last digit 6.

1334 = (12 + 22 + 32 + ... + 1152) / (12 + 22 + 32 + ... + 102).

Page of Squares : First Upload November 25, 2008 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

1335

The 4-by-4 magic square consisting of different squares with constant 1335:

12 32222292
72262212132
142252172152
332 52112102

Page of Squares : First Upload January 25, 2010 ; Last Revised January 25, 2010
by Yoshio Mimura, Kobe, Japan

1336

13362 = 203 + 603 + 1163.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1338

(32 + 6)(52 + 6)(72 + 6)(82 + 6) = (13382 + 6),
(12 + 6)(52 + 6)(72 + 6)(122 + 6) = (13382 + 6),
(12 + 6)(22 + 6)(32 + 6)(52 + 6)(72 + 6) = (13382 + 6).

210k + 606k + 762k + 1338k are squares for k = 1,2,3 (542, 16682, 554042).

The 4-by-4 magic squares consisting of different squares with constant 1338:

02 42192312
52272222102
172232182142
322 82132 92
   
22 32132342
152322 82 52
222172232 62
252 42242112
   
22 32222292
72282192122
142232182172
332 42132 82
   
22 72142332
152322 52 82
222 32262132
252162212 42
   
22112222272
132242232 82
182252102172
292 42152162

Page of Squares : First Upload January 22, 2007 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1340

13402 = (13 + 9)(53 + 9)(113 + 9).

13402 = 1795600 appears in the decimal expressions of e:
  e = 2.71828•••1795600••• (from the 140676th digit)

Page of Squares : First Upload November 4, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1341

102 + 212 + 322 + 432 + 542 + 652 + ... + 13412 = 86012.

The 4-by-4 magic squares consisting of different squares with constant 1341:

02 32 62362
182222232 22
212282102 42
242 82262 52
     
02 82112342
182302 62 92
212 42282102
242192202 22

Page of Squares : First Upload January 22, 2007 ; Last Revised January 25, 2010
by Yoshio Mimura, Kobe, Japan

1342

1342 = (12 + 22 + 32 + ... + 602) / (12 + 22 + 32 + 42 + 52).

13422 = 34 + 174 + 274 + 334.

13422 = 30 + 32 + 34 + 35 + 36 + 37 + 38 + 39 + 311 + 313.

The 4-by-4 magic square consisting of different squares with constant 1342:

12 42102352
142112312 82
192262162 72
282232 52 22

Page of Squares : First Upload July 7, 2008 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

1343

13432 = 1803649, a square with different digits.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1344

13442± 5 are primes.

13442 = 883 + 1043.

13442 = (32 - 1)(52 - 1)(972 - 1) = (72 - 1)(132 - 1)(152 - 1).

502 + 1344 = 622, 502 - 1344 = 342.

Komachi equations:
13442 = 12 * 22 / 32 * 42 * 5672 * 82 / 92 = 92 * 82 * 72 / 62 / 52 / 42 * 322 * 102
 = 982 / 72 * 62 / 52 / 42 * 322 * 102.

A quartic polynomial: (See 420)
(X + 420)(X + 525)(X + 1344)(X + 1680) = X4 + 632X3 + 23102X2 + 529202X + 7056002.

13442 = (1)(2)(3)(4)(5 + 6 + ... + 11)(12)(13 + 14 + ... + 19),
13442 = (1 + 2 + 3)(4)(5 + 6 + ... + 11)(12)(13 + 14 + ... + 19).

Page of Squares : First Upload January 22, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1347

13472 = 1! + 2! + 3! + 5 x 9!.

Page of Squares : First Upload March 7, 2007 ; Last Revised March 7, 2007
by Yoshio Mimura, Kobe, Japan

1348

13482 = 184 + 244 + 244 + 324.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1350

12 + 22 + 32 + 42 + ... + 13502 = 821036475, which consists of distinct digits
(the second 9-digit sum, there are such 2 sums in all).

13502 = 153 + 453 + 1203 = 303 + 653 + 1153.

13502 = (32 + 9)(42 + 9)(62 + 9)(92 + 9) = (62 + 9)(92 + 9)(212 + 9).

Komachi equations:
13502 = 92 * 82 / 72 / 62 * 52 / 42 * 32 * 2102 = 92 / 82 / 72 * 62 * 52 * 42 / 32 * 2102.

The 4-by-4 magic squares consisting of different squares with constant 1350:

02 22112352
32272242 62
212192222 82
302162132 52
   
02 32212302
72342 82 92
252 42222152
262132192122
   
02 32212302
102112272202
172322 62 12
312142122 72
   
02 32212302
142272 52202
232 62282 12
252242102 72
   
02 52132342
62272212122
152202262 72
332142 82 12
02 52222292
62132282192
152302 92122
332162 12 22
   
02 52222292
62192282132
152302 92122
332 82 12142
   
12 62232282
72322 92142
122132262192
342112 82 32
   
12 72122342
92232262 82
222242132112
282142192 32
   
12 72122342
172312 62 82
222 42272112
242182212 32

13502 = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12 + 13)(14 + 15 + 16),
13502 = (1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 64),
13502 = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + 9)(10)(11 + 12 + 13 + ... + 19),
13502 = (1)(2 + 3)(4 + 5 + 6 + 7 + 8)(9)(10)(11 + 12 + 13 + ... + 19),
13502 = (1)(2 + 3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15),
13502 = (1)(2 + 3 + 4 + ... + 10)(11 + 12 + 13 + 14)(15 + 16 + ... + 39),
13502 = (1)(2 + 3 + 4 + ... + 13)(14 + 15 + 16 + ... + 22)(23 + 24 + 25 + 26 + 27),
13502 = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10 + 11 + 12)(13 + 14 + 15 + 16 + 17)(18),
13502 = (1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 367),
13502 = (1)(2 + 3 + 4)(5)(6 + 7 + 8 + 9)(10)(11 + 12 + ... + 19),
13502 = (1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 116),
13502 = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 142),
13502 = (1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 14)(15)(16 + 17 + 18 + 19 + 20),
13502 = (1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + 9)(10 + 11 + ... + 90),
13502 = (1 + 2 + 3 + ... + 9)(10 + 11 + 12 + 13 + 14)(15 + 16 + 17 + ... + 39),
13502 = (1 + 2 + 3 + ... + 9)(10 + 11 + 12 + ... + 17)(18 + 19 + 20 + ... + 32).

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1351

13512 = 343 + 703 + 1133.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1352

1 / 1352 = 0.0007396..., and 7396 = 862.

13522 = 1827904, a square with different digits.

13522 = 264 + 264 + 264 + 264.

13522 = (22 + 4)(4782 + 4).

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1353

13532 = 483 + 733 + 1103 = 164 + 214 + 224 + 344.

13532 = (192 + 2)(712 + 2).

330k + 1353k + 4026k + 4092k are squares for k = 1,2,3 (992, 59072, 3691712).
1122k + 1353k + 2508k + 4818k are squares for k = 1,2,3 (992, 57092, 3626372).

The 4-by-4 magic squares consisting of different squares with constant 1353:

02 12142342
132322122 42
202182232102
282 22222 92
   
12 22182322
102302172 82
242202162112
262 72222122
   
12 22182322
142222232122
162242202112
302172102 82
   
22 62232282
172262182 82
222252102122
242 42202192
   
22102152322
172182262 82
222232142122
242202162112

Page of Squares : First Upload July 7, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1355

13552 = 1836025, a square with different digits.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1356

Komachi equations: 13562 = - 95 + 85 - 75 - 65 + 545 / 35 + 25 */ 15 = - 95 + 85 - 75 - 65 + 545 / 35 + 25 / 15.

Page of Squares : First Upload July 23, 2010 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1357

278185k + 298540k + 502090k + 762634k are squares for k = 1,2,3 (13572, 10001092, 7862987232).

Page of Squares : First Upload April 19, 2011 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1358

The 4-by-4 magic squares consisting of different squares with constant 1358:

02 22252272
132332 62 82
172 32242222
302162112 92
     
02 22252272
132332 62 82
172122212222
302112162 92
     
12 52 62362
122182292 72
222282 92 32
272152202 22
     
22112122332
152242192142
202252182 32
272 62232 82

Page of Squares : First Upload February 9, 2010 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1359

The 4-by-4 magic squares consisting of different squares with constant 1359:

12 62192312
92152272182
112332102 72
342 32132 52
     
12 62192312
102332 72112
232 32252142
272152182 92

Page of Squares : First Upload February 9, 2010 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1360

13602 = 63 + 693 + 1153.

13602 = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 74).

13602 = (12 + 4)(22 + 4)(82 + 4)(262 + 4) = (22 + 4)(62 + 4)(762 + 4)
= (22 + 4)(62 + 4)(82 + 4)(92 + 4) = (62 + 4)(82 + 4)(262 + 4).

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1361

13612 = 1852321, 1 + 85 / 2 * 32 * 1 = 1 * 85 / 2 * 32 + 1 = 1361.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1362

290106k + 328242k + 409962k + 826734k are squares for k = 1,2,3 (13622, 10215002, 8329147562).

The 4-by-4 magic squares consisting of different squares with constant 1362:

02 42112352
72272222102
172192262 62
322162 92 12
   
12 32142342
42282212112
162202252 92
332132102 22
   
12 62132342
162312 82 92
232 22272102
242192202 52
   
42112212282
122232252 82
192262102152
292 62142172
   
52 92102342
122162292112
132312142 62
322 82152 72

Page of Squares : First Upload February 9, 2010 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1363

13632 = 35 + 75 + 85 + 155 + 165.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1364

A cubic polynomial :
(X + 3632)(X + 13642)(X + 20162) = X3 + 24612X2 + 28882922X + 9981861122.

13642 = 1860496, 1860 - 496 = 1364.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1365

13652 = 53 + 243 +433 + 623 + 813 + 1003.

13652 = 263 + 423 + 1213 = 263 + 493 + 1203.

Komachi equations:
13652 = 92 * 82 * 72 * 652 / 42 / 32 / 22 */ 12 = 92 / 82 * 72 * 652 * 42 / 32 * 22 */ 12
 = 982 / 72 * 652 / 42 * 32 * 22 */ 12.

13652 = (1 + 2)(3 + 4 + 5 + ... + 32)(33 + 34 + 35 + ... + 58),
13652 = (1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 32)(33 + 34 + 35 + 36 +37).

13652 = 1682 + 1692 + 1702 + 1712 + 1722 + 1732 + 1742 + ... + 2172.

The 4-by-4 magic squares consisting of different squares with constant 1365:

12 82202302
122312162 22
142182222192
322 42152102
     
22102192302
122252142202
162242222 72
312 82182 42
     
42 62232282
72222242162
202262 82152
302132142102
     
42 72202302
92322 82142
222 62262132
282162152102

Page of Squares : First Upload January 22, 2007 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1367

13672 = 1868689, a square pegged by 8.

13672 = 105 + 105 + 125 + 175.

Page of Squares : First Upload January 22, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1368

(12 + 22 + 32 + ... + 1612)(12 + 22 + 32 + ... + 13682) = 346339982.

13682 = (12 + 8)(42 + 8)(72 + 8)(122 + 8).

Komachi equations:
13682 = 92 * 82 * 762 / 52 * 42 / 322 * 102 = 92 / 82 * 762 / 52 / 42 * 322 * 102.

13682 = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 12)(13 + 14 + 15 + ... + 44),
13682 = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 25)(26 + 27 + 28 + ... + 31),
13682 = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9)(10 + 11 + ... + 66),
13682 = (1 + 2)(3 + 4 + 5)(6)(7 + 8 + 9)(10 + 11 + ... + 28).

13682 = 138 * 139 + 139 * 140 + 140 * 141 + ... + 201 * 202.

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1369

the square of 37.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1369 - 4930 - 3301 - 1090 - ... - 9442 - 10600 - 37 - 1369
(Note f(1369) = 132 + 692 = 4930,   f(4930) = 492 + 302 = 3301, etc. See 37)

(12 + 22 + 32 + ... + 7422) + (12 + 22 + 32 + ... + 12922) = (12 + 22 + 32 + ... + 13692).

Page of Squares : First Upload January 22, 2007 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

1370

The square root of 1370 is 37.013511....., 37 = 02 + 12 + 32 + 52 + 12 + 12.

Komachi equation: 13702 = 96 + 86 + 76 - 66 + 56 - 46 + 36 + 26 + 106.

Page of Squares : First Upload January 22, 2007 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1371

1371 = 12 + 372 + 12.

1 / 1371 = 0.000729..., 729 = 272.

42k + 129k + 660k + 1194k are squares for k = 1,2,3 (452, 13712, 446312).

The 4-by-4 magic squares consisting of different squares with constant 1371:

02 12232292
52272192162
112252202152
352 42 92 72
     
12 32202312
52282212112
162232192152
332 72132 82

Page of Squares : First Upload January 22, 2007 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1372

13722 = 983 + 983 = 144 + 284 + 284 + 284.

Komachi equations:
13722 = 982 * 72 * 62 * 52 * 42 / 32 / 22 / 102 = 982 * 72 * 62 / 52 / 42 / 32 * 22 * 102
 = 982 * 72 / 62 * 52 * 42 * 32 * 22 / 102 = 982 * 72 / 62 / 52 * 42 * 32 / 22 * 102.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 23, 2010
by Yoshio Mimura, Kobe, Japan

1374

13742 = 1023 + 55 + 77.

The 4-by-4 magic squares consisting of different squares with constant 1374:

02 22232292
72272202142
132252182162
342 42112 92
   
12 22122352
162312112 62
212202222 72
262 32252 82
   
12 52182322
112332 82102
242 22252132
262162192 92
   
22 32202312
52262232122
162252182132
332 82112102
   
42 92112342
102292172122
132162302 72
332142 82 52

Page of Squares : First Upload February 9, 2010 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

1375

13752 = 1890625, a square with different digits.

13752 = 15 + 25 + 45 + 185.

The 5-by-5 magic squars consisting of different squares with constant 1375:

12 22 32202312
112232242 72102
122152272142 92
222162 52212132
252192 62172 82
     
12 22 32202312
112232242 72102
122152272142 92
222192 52212 82
252162 62172132

Page of Squares : First Upload January 22, 2007 ; Last Revised October 5, 2009
by Yoshio Mimura, Kobe, Japan

1376

13762 = (52 + 7)(62 + 7)(372 + 7).

Page of Squares : First Upload December 14, 2013 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1377

The first square which is the sum of 4 4-th powers:
13772 = 64 + 12 4 + 214 + 364 = 94 + 184 + 184 + 364 = 144 + 214 + 284 + 324.

13772 = (52 + 2)(2652 + 2).

Page of Squares : First Upload January 22, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1378

13782± 3 are primes.

13782 = 1898884, a square pegged by 8.

13782 = 13 + 23 + 33 + 43 + 53 + ... + 523.

13782 + 13792 + 13802 + ... + 14042 = 14052 + 14062 + 14072 + ... + 14302.

1378k + 3354k + 4446k + 7722k are squares for k = 1,2,3 (1302, 96202, 7672602).

Page of Squares : First Upload January 22, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1380

13802 = 93 + 433 + 773 + 1113.

13802 = 643 + 653 + 1113.

13802 = (1)(2)(3)(4 + 5 + 6 + ... + 11)(12 + 13 + 14 + ... + 103),
13802 = (1 + 2 + 3)(4 + 5 + 6 + ... + 11)(12 + 13 + 14 + ... + 103).

18k + 282k + 921k + 1380k are squares for k = 1,2,3 (512, 16832, 585812).

Page of Squares : First Upload January 22, 2007 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1381

13812 = 1907161, a zigzag square.

The square root of 1381 is 37.16...., 37 = 12 + 62.

13812 = 1907161, 12 + 92 + 02 + 72 + 12 + 62 + 12 = 132.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1383

The 4-by-4 magic square consisting of different squares with constant 1383:

22 32232292
52342 92112
252 72222152
272132172142

Page of Squares : First Upload February 9, 2010 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1384

1 / 1384 = 0.0007225, 7225 = 852.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1386

1386 = (12 + 22 + 32 + ... + 272) / (12 + 22).

The integral triangle of sides 442, 10305, 10549 (or 291, 17545, 17738) has square area 13862.

The 4-by-4 magic squares consisting of different squares with constant 1386:

02 12192322
92342102 72
242152212122
272 22222132
   
02 12192322
132122282172
162292152 82
312202 42 32
   
02 12192322
162332 52 42
172142262152
292102182112
   
02 32 92362
162212252 82
172302142 12
292 62222 52
   
02 32 92362
192302102 52
202212232 42
252 62262 72
02 92242272
132262212102
162252122192
312 22152142
   
02 92242272
162252122192
172262152142
292 22212102
   
02132162312
192302 52102
202112242172
252142232 62
   
12 42122352
142312152 22
172202242112
302 32212 62
   
12102142332
122192252162
202212232 42
292222 62 52
12122202292
152322 42112
222 72232182
262132212102
   
12142172302
162252122192
202 92282112
272222132 22
   
22 92252262
132342 62 52
222 72232182
272102142192
   
32 62212302
82252242112
172262152142
322 72122132
   
52 62222292
162252212 82
232262102 92
242 72192202

13862 = (1)(2)(3 + 4)(5 + 6)(7 + 8 + 9 + ... + 20)(21 + 22 + 23),
13862 = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 16)(17 + 18 + 19 + ... + 49),
13862 = (1)(2)(3 + 4 + 5 + ... + 11)(12 + 13 + 14 + ... + 32)(33),
13862 = (1)(2)(3 + 4 + 5 + ... + 30)(31 + 32)(33),
13862 = (1)(2 + 3 + 4 + ... + 12)(13 + 14 + 15)(16 + 17)(18),
13862 = (1)(2 + 3 + 4 + ... + 5)(6)(7 + 8 + 9 + ... + 15)(16 + 17 + 18 + ... + 26),
13862 = (1)(2 + 3 + 4 + ... + 9)(10 + 11)(12 + 13 + 14 + ... + 65),
13862 = (1)(2 + 3 + 4 + ... + 9)(10 + 11 + 12 + ... + 23)(24 + 25 + 26 + ... + 30),
13862 = (1)(2 + 3 + 4)(5 + 6)(7 + 8 + 9 + ... + 17)(18 + 19 + 20 + ... + 24),
13862 = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 16)(17 + 18 + 19 + ... + 60),
13862 = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 37)(38 + 39),
13862 = (1 + 2)(3)(4 + 5 + 6 + ... + 59)(60 + 61),
13862 = (1 + 2)(3)(4 + 5 + 6 + ... + 7)(8 + 9 + 10 + ... + 139),
13862 = (1 + 2)(3)(4 + 5 + 6 + ... + 7)(8 + 9 + 10 + ... + 14)(15 + 16 + 17 + ... + 21),
13862 = (1 + 2)(3 + 4)(5 + 6 + 7)(8 + 9 + 10 + 11 + ... + 14)(15 + 16 + 17 + ... + 18),
13862 = (1 + 2)(3 + 4 + 5 + 6)(7)(8 + 9 + 10 + 11 + ... + 14)(15 + 16 + 17 + ... + 18),
13862 = (1 + 2)(3 + 4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 15)(16 + 17 + 18 + ... + 26),
13862 = (1 + 2 + 3 + ... + 27)(28 + 29 + 30 + ... + 104),
13862 = (1 + 2 + 3 + ... + 7)(8 + 9 + 10 + ... + 370).

Page of Squares : First Upload January 22, 2007 ; Last Revised October 4, 2011
by Yoshio Mimura, Kobe, Japan

1387

13872 = 383 + 463 + 1213.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1388

13882 = 1926544, 1 * 92 + 6 * 54 * 4 = 1388.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1389

The 4-by-4 magic squares consisting of different squares with constant 1389:

02 82102352
132202262122
142302172 22
322 52182 42
     
12 22222302
62282202132
142242192162
342 52122 82

Page of Squares : First Upload February 9, 2010 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1390

13902 = 563 + 813 + 1073.

The 5-by-5 magic squares consisting of different squares with constant 1390:

02 32 92122342
42232102242132
72202282112 62
222142192182 52
292162 82152 22
     
02 32 92122342
72202282112 62
182142192222 52
212162 82252 22
242232102 42132

Page of Squares : First Upload July 7, 2008 ; Last Revised October 5, 2009
by Yoshio Mimura, Kobe, Japan

1392

13922 = 1937664, 1 + 9 + 3 * 7 * 66 - 4 = 1392.

13922 = 123 + 793 + 1133 = 483 + 863 + 1063.

Page of Squares : First Upload January 22, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1393

A cubic polynomial :
(X + 5762)(X + 7442)(X +13932) = X3 + 16812X2 + 13789682X + 5969617922.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1394

118490k + 165886k + 748578k + 910282k are squares for k = 1,2,3 (13942, 11960522, 10862689242).

The 4-by-4 magic square consisting of different squares with constant 1394:

02 32192322
92362 42 12
232 82242152
282 52212122

Page of Squares : First Upload February 9, 2010 ; Last Revised April 19, 2011
by Yoshio Mimura, Kobe, Japan

1395

13952 = 1946025, a square with different digits.

The 4-by-4 magic squares consisting of different squares with constant 1395:

02 12132352
152322112 52
212192232 82
272 32242 92
     
02 52232292
92272212122
152252162172
332 42132112

Page of Squares : First Upload January 22, 2007 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1396

1 / 1396 = 0.00071633237, 72 + 162 + 32 + 322 + 32 + 72 = 1396.

13962 = 1948816, 1 - 9 - 4 + 88 * 16 = 1396.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1397

13972 = 1951609, 12 + 92 + 52 + 12 + 62 + 02 + 92 = 152.

13972 = 2272 + 2282 + 2292 + 2302 + 2312 + 2322 + 2332 + ...+2592.

Page of Squares : First Upload January 22, 2007 ; Last Revised January 22, 2007
by Yoshio Mimura, Kobe, Japan

1398

The 4-by-4 magic squares consisting of different squares with constant 1398:

22 32192322
42292212102
172222202152
332 82142 72
     
32 82102352
132182282112
142312152 42
322 72172 62

The 5-by-5 magic square consisting of different squares with constant 1398:

02 32 82132342
72272182142102
122 22282212 52
232162 12242 62
262202152 42 92

Page of Squares : First Upload October 5, 2009 ; Last Revised February 9, 2010
by Yoshio Mimura, Kobe, Japan

1399

The 5-by-5 magic square consisting of different squares with constant 1399:

02 22 52232292
32172222192162
62332 72122 92
252 42212142112
272 12202132102
     
02 32182212252
52352122 12 22
132 72272142162
232102 92202172
262 42112192152

Page of Squares : First Upload October 5, 2009 ; Last Revised October 5, 2009
by Yoshio Mimura, Kobe, Japan