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2200 - 2299

2200

2200 = (12 + 22 + 32 + ... + 1752) / (12 + 22 + 32 + ... + 132).

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

Page of Squares : First Upload November 25, 2008 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2201

22012 = 4844401, a reversible square (1044484 = 10222).

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2202

22022 = 4848804, a square with just 3 kinds of even digits.

22022 = 4848804, a reversible square (4088484 = 20222).

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2203

22032 = 4853209, a squaare with different digits.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2205

22052 = 73 + 1193 + 1473 = 143 + 493 + 1683 = 493 + 1263 + 1403.

A cubic polynomial :
(X + 322)(X + 22052)(X + 42242) = X3 + 47652X2 + 93151682X + 2980454402.

22052 = (20 + 21 + 22)2 + (23 + 24 + 25)2 + (26 + 27 + 28)2 + (29 + 30 + 31)2 + ... + (167 + 168 + 169)2.

22052 = (53 + 54 + 55 + ... + 73)2 + (74 + 75 + 76 + ... + 94)2 + (95 + 96 + 97 + ... + 115)2 + ... + (74 + 75 + 76 + ... + 94)2.

22052 = 2872 + 2882 + 2892 + 2902 + 2912 + 2922 + 2932 + ... + 3362.

Page of Squares : First Upload March 26, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2206

22062 = 193 + 323 + 1693.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2209

The square of 47.

22092 = 4879681, a zigzag square.

Komachi equation: 22092 = 9874 / 64 * 54 * 44 * 34 / 2104.

22092 = 4879681 appears in the decimal expressions of e:
  e = 2.71828•••4879681••• (from the 10075th digit)
  (4879681 is the fifth 7-digit square in the expression of e.)

Page of Squares : First Upload March 26, 2007 ; Last Revised September 21, 2010
by Yoshio Mimura, Kobe, Japan

2210

2210 = (12 + 22 + 32 + ... + 842) / (12 + 22 + 32 + 42 + 52 + 62).

22102 = (12 + 1)(22 + 1)(42 + 1)(82 + 1)(212 + 1)
= (12 + 1)(42 + 1)(182 + 1)(212 + 1) = (12 + 1)(42 + 1)(82 + 1)(472 + 1)
= (12 + 1)(52 + 1)(82 + 1)(382 + 1) = (22 + 1)(212 + 1)(472 + 1)
= (22 + 1)(42 + 1)(52 + 1)(472 + 1) = (32 + 1)(42 + 1)(82 + 1)(212 + 1)
= (42 + 1)(52 + 1)(82 + 1)(132 + 1) = (82 + 1)(132 + 1)(212 + 1)
= (82 + 4)(92 + 4)(292 + 4) = (292 + 4)(762 + 4) = (52 + 9)(3792 + 9).

1666k + 2210k + 5270k + 9350k are squares for k = 1,2,3 (1362, 110842, 9895362).

Page of Squares : First Upload November 25, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2211

22112 = 4888521, a reversible square (1258884 = 11222).

22112± 2 are primes.

22112 = 923 + 1023 + 1453.

22112 + 22122 + 22132 + ... + 22442 = 22452 + 22462 + 22472 + ... + 22772.

Page of Squares : First Upload March 26, 2007 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

2214

22145 = 53197115312713824 : 52 + 32 + 12 + 92 + 72 + 12 + 152 + 312 + 22 + 72 + 132 + 82 + 242 = 52 + 312 + 92 + 72 + 12 + 152 + 32 + 12 + 22 + 72 + 132 + 82 + 242 = 2214.

22145 = (12 + 5)(22 + 5)(62 + 5)(472 + 5) = (62 + 5)(72 + 5)(472 + 5).

Page of Squares : First Upload December 8, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2216

22162 = 4910656, and 4 * 9 * 10 * 6 + 56 = 2216.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2218

22182 = 173 + 993 + 1583.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2219

22194 = 24245391929521, and 22 + 42 + 22 + 42 + 52 + 392 + 12 + 92 + 22 + 92 + 52 + 212 = 2219.

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

2222

22222 = 4937284, a zigzag square.

22222± 3 are primes.

22222 = 1922 + 1932 + 1942 + 1952 + 1962 + 1972 + ... + 2792.

22222 = 583 + 633 + 1653.

Page of Squares : First Upload March 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2223

22232 = 4941729, and 494 + 1729 = 2223.

22232 = 93 + 1303 + 1403.

Page of Squares : First Upload March 26, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2224

22242 = 4946176, a zigzag square.

22242 = 243 + 973 + 1593 = 873 + 973 + 1503.

Page of Squares : First Upload March 26, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2225

22252 = (112 + 4)(1992 + 4).

22252 = 4950625, and 4 = 22, 950625 = 9752,
22252 = 4950625, and 49 = 72, 50625 = 2252.

Page of Squares : First Upload March 26, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2227

22272 = S2(43) + S2(245), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2228

22282 = 753 + 1013 + 1523.

1 / 2228 = 0.00044883303411, 442 + 82 + 82 + 32 + 32 + 032 + 42 + 112 = 2228.

Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2228 - 1268 - 4768 - 6833 - ... - 6596 - 13441 - 2838 - 2228
(Note f(2228) = 222 + 282 = 1268,   f(1268) = 122 + 682 = 4768, etc. See 1268)

Page of Squares : First Upload March 26, 2007 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

2230

22302 = 4972900, and 49 = 72, 72900 = 2702.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2232

22322 = 843 + 843 + 1563.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2233

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

Page of Squares : First Upload May 13, 2011 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2234

22342± 3 are primes.

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

2236

22362 = 4999696, a square with just 3 kinds of digits.

22362 = 4999696, a square pegged by 9.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2237

22372 = 1133 + 1163 + 1263.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2239

22393 = 11224377919, and 112 + 22 + 22 + 432 + 72 + 72 + 92 + 12 + 92 = 2239.

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

2240

22402 = 243 + 643 + 1683 = 27 + 27 + 57 + 57 + 57 + 97.

22402 = (32 - 1)(62 - 1)(92 - 1)(152 - 1).

Page of Squares : First Upload July 17, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2241

22412 = 1063 + 1083 + 1373.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2242

22422 = 30 + 33 + 34 + 36 + 38 + 310 + 311 + 314.

2242k + 4978k + 13262k + 15618k are squares for k = 1,2,3 (1902, 212042, 25053402).

Page of Squares : First Upload May 13, 2011 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

2243

22435 = 56773591412619443 : 52 + 62 + 72 + 72 + 352 + 92 + 142 + 122 + 62 + 192 + 42 + 42 + 32 = 2243.

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

2244

22442 = (32 + 8)(142 + 8)(382 + 8) = (32 + 8)(52 + 8)(62 + 8)(142 + 8) = (32 + 8)(62 + 8)(822 + 8).

22442 = S2(50) + S2(246), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload March 26, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2245

22455 = 57027157741403125 : 52 + 72 + 02 + 22 + 72 + 152 + 72 + 72 + 42 + 12 + 402 + 32 + 122 + 52 = 2245.

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

2247

22472 = 23 + 1263 + 1453.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2248

22482 = 5053504, a zigzag square.

22482± 3 are primes.

22482 = 5053504, 52 + 02 + 52 + 32 + 52 + 02 + 42 = 102.

Page of Squares : First Upload March 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2250

22502 = 573 + 1073 + 1543.

22502 = (12 + 9)(42 + 9)(62 + 9)(212 + 9) = (32 + 9)(62 + 9)(792 + 9).

Page of Squares : First Upload July 17, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2251

22512 = 15 + 95 + 155 + 165 + 205.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2252

22522 = 5071504, a zigzag square.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2254

22542 = 5080516, a zigzag square.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2256

22562 = 46 x 47 x 48 + 48 x 49 x 50 + 50 x 51 x 52 + 52 x 53 x 54 + ... + 80 x 81 x 82.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2257

22572 = 44 + 314 + 384 + 384.

22572 = 2252 + 2262 + 2272 + 2282 + 2292 + 2302 + ... + 2982.

1517k + 2257k + 12173k + 33337k are squares for k = 1,2,3 (2222, 355942, 62344262).

1 / 2257 = 0.0004430660168, 42 + 432 + 02 + 62 + 62 + 02 + 162 + 82 = 2257,
1 / 2257 = 0.0004430660168, 42 + 432 + 02 + 62 + 62 + 0162 + 82 = 2257,
1 / 2257 = 0.0004430660168, 42 + 432 + 062 + 62 + 02 + 162 + 82 = 2257,
1 / 2257 = 0.0004430660168, 42 + 432 + 062 + 62 + 0162 + 82 = 2257.

Page of Squares : First Upload March 26, 2007 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2259

22592 = 5103081, 52 + 12 + 02 + 32 + 02 + 82 + 12 = 102.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2260

22602 = 673 + 1093 + 1523.

Komachi equations:
22602 = 12 / 22 * 32 * 42 * 52 * 6782 / 92 = 12 * 22 / 32 * 452 * 6782 / 92.

Page of Squares : First Upload July 17, 2008 ; Last Revised September 21, 2010
by Yoshio Mimura, Kobe, Japan

2261

22612 = 5112121, a square with just 3 kinds of digits.

22612 = 1663 + 145 + 17.

22k + 73k + 130k + 136k are squares for k = 1,2,3 (192, 2032, 22612).

Page of Squares : First Upload March 26, 2007 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2262

22622 = (72 + 9)(2972 + 9).

390k + 2262k + 4329k + 6708k are squares for k = 1,2,3 (1172, 83072, 6281732).

Page of Squares : First Upload May 13, 2011 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2263

1 / 2263 = 0.000441...., and 441 = 212.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2264

22642± 3 are primes.

22642 = 843 + 1073 + 1493.

1 / 2264 = 0.000441...., and 441 = 212.

Page of Squares : First Upload March 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2265

1 / 2265 = 0.000441...., and 441 = 212.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2267

1 / 2267 = 0.000441...., and 441 = 212.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2268

22682 = 5143824, a zigzag square.

22682 = 543 + 1263 + 1443 = 813 + 843 + 1593 = 124 + 244 + 364 + 424 = 184 + 364 + 364 + 364.

22682 = (12 + 5)(22 + 5)(32 + 5)(72 + 5)(112 + 5).

A cubic polynomial :
(X + 8962)(X + 12692)(X + 22682) = X3 + 27492X2 + 37021322X + 25787704322.

Komachi equations:
22682 = 92 / 82 * 72 / 62 * 542 * 322 */ 12 = 92 * 82 * 72 * 62 * 52 / 42 * 32 * 22 / 102
 = 92 * 82 * 72 * 62 / 52 / 42 * 32 / 22 * 102 = 92 / 82 * 72 * 62 / 52 * 42 * 32 * 22 * 102
 = - 92 * 82 * 72 * 62 + 542 / 32 * 2102.

Page of Squares : First Upload March 26, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2274

22742 = 53 + 1203 + 1513 = 75 + 115 + 135 + 145 + 215.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2275

22752 = 5175625, a zigzag square.

22752 = 5175625, and 517 * 5 - 62 * 5 = 5 * 1 * 75 * 6 + 25 = 5 / 1 * 75 * 6 + 25 = 2275.

22752 = (8 + 9 + 10 + ... + 42)2 + (43 + 44 + 45 + ... + 77)2 + (78 + 79 + 80 + ... + 112)2 + ... + (43 + 44 + 45 + ... + 77)2.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2277

22772 = 5184729, a zigzag square with different digits.

22772 = 5184729, and 5184 = 722, 729 = 272.

1 / 2277 = 0.0004391743522, 42 + 32 + 92 + 172 + 432 + 52 + 22 + 22 = 2277,
1 / 2277 = 0.0004391743522, 432 + 92 + 172 + 42 + 32 + 52 + 22 + 22 = 2277.

22772 = (80 + 81 + 82)2 + (83 + 84 + 85)2 + (86 + 87 + 88)2 + (89 + 90 + 91)2 + ... + (176 + 177 + 178)2.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2280

22802 = 70 x 71 x 72 + 72 x 73 x 74 + 74 x 75 x 76 + 76 x 77 x 78 + ... + 88 x 89 x 90.

22802 = 103 + 273 + 1733 = 193 + 243 + 1733 = 383 + 1363 + 1383.

Page of Squares : First Upload March 26, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2282

22824 = 27118306210576, and 272 + 12 + 12 + 82 + 302 + 62 + 212 + 02 + 52 + 72 + 62 = 2282.

2282k + 3010k + 3626k + 6958k are squares for k = 1,2,3 (1262, 87082, 6509162).

Page of Squares : First Upload December 1, 2008 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2283

22832± 2 are primes.

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

2285

22852 = 5221225, a palindromic square which consists of just 3 kinds of digits.

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2286

(22862 + 4) = (12 + 4)(22 + 4)(52 + 4)(72 + 4)(92 + 4).

Page of Squares : First Upload March 26, 2007 ; Last Revised March 26, 2007
by Yoshio Mimura, Kobe, Japan

2288

2288 = (12 + 22 + 32 + ... + 322) / (12 + 22).

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

2291

2291k + 2465k + 2755k + 5945k are squares for k = 1,2,3 (1162, 73662, 5079642).

Page of Squares : First Upload May 13, 2011 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2293

22934 = 27644976106801, and 22 + 72 + 62 + 442 + 92 + 72 + 62 + 12 + 02 + 62 + 82 + 02 + 12 = 2293.

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

2294

22942 = 5262436, a zigzag square.

22942 = (112 + 3)(2062 + 3).

Page of Squares : First Upload March 26, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2295

Komachi equations:
22952 = 92 * 82 * 7652 / 42 / 32 / 22 */ 12 = 92 / 82 * 7652 * 42 / 32 * 22 */ 12.

Page of Squares : First Upload September 21, 2010 ; Last Revised September 21, 2010
by Yoshio Mimura, Kobe, Japan

2296

22962 = 5271616, a zigzag square.

22962± 3 are primes.

22962 = 213 + 643 + 1713 = 563 + 1003 + 1603.

22962 + 22972 + 22982 + ... + 57292 = 57302 + 57312 + 57322 + ... + 71402.

Page of Squares : First Upload March 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2297

22972 = 164 + 324 + 344 + 414.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2298

22982 = 5280804, a zigzag square.

22982± 5 are primes.

Page of Squares : First Upload March 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan