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3300 - 3399

3300

The square root of 3300 is 57.445...., and 57 = 42 + 42 + 52.

33002 = 6! + 6! + 6! + 6! + 6! + 10! + 10! + 10!.

33002 = (22 + 6)(42 + 6)(122 + 6)(182 + 6) = (92 + 9)(182 + 9)(192 + 9) = (92 + 9)(3512 + 9).

Page of Squares : First Upload June 18, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3301

The square root of 3301 is 57.454...., and 57 = 42 + 52 + 42.

33012 = 1! + 5! + 7! + 7! + 10! + 10! + 10!.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
3301 - 1090 - 8200 - 6724 - 9442 - ... - 37 - 1369 - 4930 - 3301
(Note f(3301) = 332 + 012 = 1090,   f(1090) = 102 + 902 = 8200, etc. See 37)

Page of Squares : First Upload June 18, 2007 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

3303

33032 = 11012 + 22022 + 22022 : 20222 + 20222 + 10112 = 30332.

33032 = 873 + 953 + 2113 = 1023 + 1203 + 2013 = 64 + 124 + 244 + 574.

Page of Squares : First Upload August 10, 2008 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3304

33042± 3 are primes.

1 / 3304 = 0.000302663438256, 32 + 0262 + 62 + 32 + 432 + 82 + 252 + 62 = 3304.

Page of Squares : First Upload June 18, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

3305

1 / 3305 = 0.0003025...., and 3025 = 552.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3306

33062 = 11022 + 22042 + 22042 : 40222 + 40222 + 20112 = 60332.

Page of Squares : First Upload September 18, 2013 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3307

33072 = 10936249, 1 * 0 + 9 * 362 + 49 = 3307.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3309

33092 = 10949481, a zigzag square.

33092 = 11032 + 22062 + 22062 : 60222 + 60222 + 30112 = 90332.

33092 = 27 + 67 + 67 + 77 + 97 + 97.

Page of Squares : First Upload June 18, 2007 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3311

385k + 1232k + 2156k + 2156k are squares for k = 1,2,3 (772, 33112, 1482252).

Page of Squares : First Upload May 31, 2011 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3312

33122 = 124 + 164 + 324 + 564 = 124 + 244 + 484 + 484.

33122 = 10969344, 1 * 0 + 9 * 69 / 3 * 4 * 4 = 1 * 0 * 9 + 69 * 3 * 4 * 4 = 3312.

Komachi equation: 33122 = 12 * 232 / 42 * 562 / 72 * 82 * 92.

Page of Squares : First Upload June 18, 2007 ; Last Revised October 1, 2010
by Yoshio Mimura, Kobe, Japan

3315

33152 = (60 + 61 + 62 + 63 + 64)2 + (65 + 66 + 67 + 68 + 69)2 + (70 + 71 + 72 + 73 + 74)2 + ... + (185 + 186 + 187 + 188 + 189)2.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3317

33173 = 36495256013, and 32 + 62 + 42 + 92 + 52 + 22 + 562 + 02 + 12 + 32 = 3317.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

3318

33182 = 253 + 1383 + 2033.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3320

33202 = 753 + 1173 + 2083 = 1363 + 1603 + 1643.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3321

33212 = 304 + 304 + 454 + 484.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3324

33242 = 343 + 1143 + 2123.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3325

33252 = 11055625, 1 * 10 * 55 * 6 + 25 = 3325.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3328

33285 = 408240970811834368 : 42 + 02 + 82 + 22 + 402 + 92 + 72 + 02 + 82 + 112 + 82 + 342 + 32 + 62 + 82 = 402 + 82 + 22 + 42 + 02 + 92 + 72 + 02 + 82 + 112 + 82 + 342 + 32 + 62 + 82 = 3328.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3329

33295 = 408854682322747649 : 402 + 82 + 82 + 52 + 42 + 62 + 82 + 232 + 22 + 272 + 42 + 72 + 62 + 42 + 92 = 3329.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3330

3330 = (12 + 22 + 32 + ... + 249752) / (12 + 22 + 32 + ... + 16722).

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

3332

33322 = 84 + 144 + 444 + 524 = 144 + 284 + 284 + 564.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3333

33332 = 503 + 1423 + 2013.

462k + 1034k + 2200k + 2233k are squares for k = 1,2,3 (772, 33332, 1516132).

Page of Squares : First Upload August 10, 2008 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3334

33342 = 11115556, a square with just 3 kinds of digits.

33342 = 11115556, a square with odd digits except the last digit 6.

33342 = 11115556, 1 + 1 + 1 + 1 + 555 * 6 = 3334.

Page of Squares : First Upload June 18, 2007 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

3335

33352 = 11122225, a square with just 3 kinds of digits.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3336

33362 = 11122 + 22242 + 22242 : 42222 + 42222 + 21112 = 63332.

Page of Squares : First Upload September 18, 2013 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3337

33372 = 11135569, a square with nondecreasing digits.

A cubic polynomial :
(X + 14162)(X + 15682)(X + 25832) =X3 + 33372X2 + 58915922X + 57350039042.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3338

33382 = 11142244, a square with just 3 kinds of digits.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3339

33392 = 11132 + 22262 + 22262 : 62222 + 62222 + 31112 = 93332.

33392 = 11148921, 12 + 12 + 12 + 42 + 82 + 92 + 22 + 12 = 132.

Page of Squares : First Upload June 18, 2007 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3344

152k + 209k + 3344k + 5320k are squares for k = 1,2,3 (952, 62892, 4335612).

Page of Squares : First Upload May 31, 2011 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3346

33462 = 11195716, a square with odd digits except the last digit 6.

Page of Squares : First Upload May 31, 2011 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

3348

33482 = 483 + 543 + 2223.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3354

33542 = 11249316, 1 + 124 * 9 * 3 - 1 + 6 = 1 * 124 * 9 * 3 + 1 * 6
  = 1 * 124 * 9 * 3 * 1 + 6 = 3354.

33542 = 3! + 3! + 4! + 9! + 10! + 10! + 10!.

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

Page of Squares : First Upload June 18, 2007 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3355

33552 = 11256025, 1 * 12 * 560 / 2 - 5 = 112 * 5 * 6 + 0 * 2 - 5 = 3355 .

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3358

33582 = 11276164, 12 + 12 + 22 + 72 + 62 + 12 + 62 + 42 = 122.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3359

33592 = 11282881, a square with just 3 kinds of digits.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3360

33602 = 8! + 9! + 10! + 10! + 10!.

33602 = (22 - 1)(32 - 1)(62 - 1)(92 - 1)(132 - 1) = (22 - 1)(32 - 1)(72 - 1)(992 - 1)
= (32 - 1)(292 - 1)(412 - 1) = (32 - 1)(52 - 1)(62 - 1)(412 - 1) = (32 - 1)(62 - 1)(72 - 1)(292 - 1)
= (42 - 1)(92 - 1)(972 - 1) = (52 - 1)(62 - 1)(92 - 1)(132 - 1) = (52 - 1)(72 - 1)(992 - 1)
= (92 - 1)(132 - 1)(292 - 1).

1132 + 3360 = 1272, 1132 - 3360 = 972,
742 + 3360 = 942, 742 - 3360 = 462,
582 + 3360 = 822, 582 - 3360 = 22.

Komachi equations:
33602 = 92 * 82 * 72 / 62 * 52 / 42 * 322 */ 12 = 982 / 72 * 62 * 52 / 42 * 322 */ 12
 = 92 * 82 * 72 * 62 / 542 * 32 * 22 * 102.

1065k + 3360k + 6180k + 7620k are squares for k = 1,2,3 (1352, 104252, 8471252).

Page of Squares : First Upload June 18, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3361

1 / 3361 = 0.000297530496, 292 + 72 + 52 + 32 + 02 + 492 + 62 = 3361,
1 / 3361 = 0.000297530496, 292 + 72 + 52 + 32 + 0492 + 62 = 3361.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3362

33622 = 414 + 414 + 414 + 414.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3363

33632 = 11212 + 22422 + 22422 : 24222 + 24222 + 12112 = 36332.

Page of Squares : First Upload September 18, 2013 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3364

The square of 58.

Komachi equations: 33642 = 94 * 874 * 64 / 544 / 34 * 24 */ 14.

Page of Squares : First Upload June 18, 2007 ; Last Revised October 1, 2010
by Yoshio Mimura, Kobe, Japan

3366

33662 = 293 + 603 + 2233.

33662 = 11329956, 11 + 3299 + 56 = 113 * 29 + 95 - 6 = 3366.

33662 = 11222 + 22442 + 22442 : 44222 + 44222 + 22112 = 66332.

33662 = 2442 + 2452 + 2462 + 2472 + 2482 + 2492 + ... + 3642.

33662 = (12 + 2)(32 + 2)(102 + 2)(582 + 2) = (12 + 2)(32 + 2)(72 + 2)(82 + 2)(102 + 2)
= (12 + 2)(42 + 2)(192 + 2)(242 + 2) = (22 + 2)(32 + 2)(102 + 2)(412 + 2)
= (22 + 2)(32 + 2)(72 + 2)(582 + 2) = (22 + 2)(72 + 2)(102 + 2)(192 + 2) = (242 + 2)(1402 + 2)
= (32 + 2)(52 + 2)(82 + 2)(242 + 2) = (32 + 2)(72 + 2)(102 + 2)(142 + 2)
= (72 + 2)(82 + 2)(582 + 2) = (82 + 2)(102 + 2)(412 + 2).

3366k + 7446k + 15708k + 38505k are squares for k = 1,2,3 (2552, 423812, 78368132).

Page of Squares : First Upload June 18, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3367

33672 = 11336689, a square with non-decreasing digits.

33674 = 128520517482721, and 12 + 22 + 82 + 52 + 202 + 52 + 12 + 72 + 482 + 22 + 72 + 212 = 3367.

33672 = 11336689, 1 + 1 * 33 / 6 * 68 * 9 = 1 * 1 + 33 / 6 * 68 * 9 = 1 + 1 + 3366 + 8 - 9
    = 1 - 1 + 3366 - 8 + 9 = 1 * 1 * 3366 - 8 + 9 = 3367.

Page of Squares : First Upload June 18, 2007 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

3368

33682 = 11343424, a square every digit of which is non-zero and smaller than 5.

33682 = 95 + 155 + 155 + 255.

Page of Squares : First Upload August 10, 2008 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

3369

33692 = 11232 + 22462 + 22462 : 64222 + 64222 + 32112 = 96332.

Page of Squares : First Upload September 18, 2013 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3372

33725 = 435951152140741632 : 42 + 32 + 52 + 92 + 52 + 12 + 152 + 22 + 142 + 02 + 72 + 412 + 62 + 322 = 3372.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3375

33752 = 253 + 1503 + 2003 = 313 + 953 + 2193 = 363 + 1533 + 1983 = 1143 + 1293 + 1983.

Komachi equations:
33752 = 96 * 86 * 76 / 66 * 56 / 46 * 36 / 216 = 96 * 86 / 76 / 66 * 56 / 46 / 36 * 216
 = 96 / 86 * 76 * 66 * 56 * 46 / 36 / 216 = 986 / 76 * 66 * 56 / 46 * 36 / 216.

Page of Squares : First Upload August 10, 2008 ; Last Revised October 1, 2010
by Yoshio Mimura, Kobe, Japan

3376

33762 = 11397376, a square with odd digits except the last digit 6.

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

3377

3377 = (12 + 22 + 32 + ... + 3072) / (12 + 22 + 32 + ... + 202).

33772 = 24 + 84 + 174 + 584.

Page of Squares : First Upload August 10, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

3379

3379 = (12 + 22 + 32 + ... + 2172) / (12 + 22 + 32 + ... + 142).

33792 = 11417641, 12 + 12 + 42 + 12 + 72 + 62 + 42 + 12 = 112.

33792 = (45 + 46 + 47 + ... + 75)2 + (76 + 77 + 78 + ... + 106)2.

Page of Squares : First Upload June 18, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

3381

33812 = 423 + 493 + 2243 = 1053 + 1403 + 1963.

3045k + 3381k + 4305k + 5145k are squares for k = 1,2,3 (1262, 81062, 5318462).

Page of Squares : First Upload August 10, 2008 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3383

33832 = 11444689, a square with non-decreasing digits.

513k + 1292k + 3382k + 3838k are squares for k = 1,2,3 (952, 53012, 3122652).

Page of Squares : First Upload June 18, 2007 ; Last Revised May 31, 2011
by Yoshio Mimura, Kobe, Japan

3384

33842 = 11451456, 12 + 12 + 42 + 52 + 12 + 42 + 52 + 62 = 112.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3387

33872 = 11471769, 1 + 1 + 4 + 7 * 1 * 7 * 69 = 1 - 1 - 4 * 7 + 854 * 4 = 3388.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3388

33882 = 224 + 444 + 444 + 444.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3389

33892 = 11485321, 12 + 12 + 42 + 82 + 52 + 32 + 22 + 12 = 112.

Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007
by Yoshio Mimura, Kobe, Japan

3390

3390 = (12 + 22 + 32 + ... + 1122) / (12 + 22 + 32 + ... + 72).

Komachi equations:
33902 = 122 * 32 / 42 + 52 * 6782 - 92 = 122 * 32 / 42 * 52 * 6782 / 92
 = - 122 * 32 / 42 + 52 * 6782 + 92.

Page of Squares : First Upload November 25, 2008 ; Last Revised October 1, 2010
by Yoshio Mimura, Kobe, Japan

3392

33922 = 163 + 643 + 2243.

Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008
by Yoshio Mimura, Kobe, Japan

3393

33932 = 11312 + 22622 + 22622 : 26222 + 26222 + 13112 = 39332.

Komachi equation: 33932 = 92 * 872 * 652 * 42 / 32 / 22 / 102.

Page of Squares : First Upload October 1, 2010 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3394

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
3394 - 9925 - 10426 - 693 - ... - 11565 - 4451 - 4537 - 3394
(Note f(3394) = 332 + 942 = 9925,   f(9925) = 992 + 252 = 10426, etc. See 37)

33942 = 573 + 943 + 2193 = 883 + 1573 + 1913.

Page of Squares : First Upload August 10, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

3395

3395 = (12 + 22 + 32 + ... + 972) / (12 + 22 + 32 + ... + 62).

33955 = 451023211560621875 : 452 + 102 + 22 + 322 + 12 + 12 + 52 + 62 + 02 + 62 + 22 + 12 + 82 + 72 + 52 = 3395.

Page of Squares : First Upload November 25, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3396

33962 = 11322 + 22642 + 22642 : 46222 + 46222 + 23112 = 69332.

Page of Squares : First Upload September 18, 2013 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan

3399

33992 = 11332 + 22662 + 22662 : 66222 + 66222 + 33112 = 99332,
33992 = 11772 + 20242 + 24642 : 46422 + 42022 + 77112 = 99332.

33992 = 1812 + 1832 + 1852 + 1872 + 1882 + 1892 + ... + 4212.

Page of Squares : First Upload June 18, 2007 ; Last Revised September 18, 2013
by Yoshio Mimura, Kobe, Japan