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, 2013by 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)
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, 2013by 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, 2014by Yoshio Mimura, Kobe, Japan
3305
1 / 3305 = 0.0003025...., and 3025 = 552.
Page of Squares : First Upload June 18, 2007 ; Last Revised June 18, 2007by 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, 2013by 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, 2007by 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, 2013by 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, 2011by 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, 2010by 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, 2007by 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, 2008by Yoshio Mimura, Kobe, Japan
3318
33182 = 253 + 1383 + 2033.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by Yoshio Mimura, Kobe, Japan
3320
33202 = 753 + 1173 + 2083 = 1363 + 1603 + 1643.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by Yoshio Mimura, Kobe, Japan
3321
33212 = 304 + 304 + 454 + 484.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by Yoshio Mimura, Kobe, Japan
3324
33242 = 343 + 1143 + 2123.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by 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, 2007by 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, 2008by 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, 2008by 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, 2008by 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, 2008by 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, 2011by 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, 2013by 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, 2007by 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, 2013by 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.
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, 2007by 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, 2013by 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, 2011by 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, 2013by Yoshio Mimura, Kobe, Japan
3348
33482 = 483 + 543 + 2223.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by 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, 2011by 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, 2007by 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, 2007by 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, 2007by 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, 2013by 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.
by Yoshio Mimura, Kobe, Japan
3362
33622 = 414 + 414 + 414 + 414.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by 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, 2013by 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, 2010by 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, 2013by 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.
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, 2013by 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, 2013by 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, 2008by 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.
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, 2013by 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, 2008by 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, 2008by 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, 2011by 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, 2011by 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, 2007by 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, 2007by Yoshio Mimura, Kobe, Japan
3388
33882 = 224 + 444 + 444 + 444.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by 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, 2007by 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.
by Yoshio Mimura, Kobe, Japan
3392
33922 = 163 + 643 + 2243.
Page of Squares : First Upload August 10, 2008 ; Last Revised August 10, 2008by 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, 2013by 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, 2008by 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, 2008by 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, 2013by 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, 2013by Yoshio Mimura, Kobe, Japan