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280 - 289

280

The smallest squares containing k 280's :
2809 = 532,
280328049 = 167432,
128028025612809 = 113149472.

2802 = 63 + 163 + 423.

2802 = (32 - 1)(992 - 1).

22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).

Komachi equations:
2802 = 12 * 22 * 32 / 42 * 52 * 62 * 72 * 82 / 92 = 12 / 22 / 32 * 42 * 52 / 62 * 72 * 82 * 92
  = 12 * 22 * 32 * 452 / 62 * 72 * 82 / 92 = 12 / 22 / 32 * 452 * 62 * 72 * 82 / 92
  = 92 * 82 * 72 / 62 * 52 * 42 / 32 / 22 * 12 = 92 * 82 * 72 / 62 * 52 * 42 / 32 / 22 / 12
  = 982 / 72 * 62 * 52 * 42 / 32 / 22 * 12 = 982 / 72 * 62 * 52 * 42 / 32 / 22 / 12
  = 982 / 72 / 62 * 52 * 42 * 32 * 22 * 12 = 982 / 72 / 62 * 52 * 42 * 32 * 22 / 12
  = 92 / 82 * 72 * 62 / 542 * 322 * 102.

2802 + 2812 + 2822 + 2832 + 2842 + ... + 6312 = 87562.

(1 + 2 + ... + 8)(9 + 10 + ... + 16)(17 + 18 + ... + 280) = 118802,
(1 + 2 + ... + 8)(9 + 10 + ... + 144)(145 + 146 + ... + 280) = 1040402,
(1 + 2 + ... + 42)(43 + 44 + ... + 258)(259 + 260 + ... + 280) = 4171862,
(1 + 2 + ... + 242)(243 + 244 + ... + 269)(270 + 271 + ... + 280) = 7840802,
(1 + 2 + ... + 243)(244 + 245 + ... + 268)(269 + 270 + ... + 280) = 7905602.

2802 = 78400 appears in the decimal expression of π:
  π = 3.14159•••78400••• (from the 14634th digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

281

The smallest squares containing k 281's :
8281 = 912,
1528106281 = 390912,
1192811281281 = 10921592.

The squares which begin with 281 and end in 281 are
2818654281 = 530912,   28109540281 = 1676592,   28170601281 = 1678412,
28193432281 = 1679092,   281068565281 = 5301592,...

2812 = 78961, a square with different digits.

Komachi equations:
2812 = 987 * 6 * 5 * 4 / 3 * 2 + 1 = 9 * 8765 + 43 * 2 - 10.

3-by-3 magic squares consisting of different squares with constant 2812:

925222762
8422642472
2682812242
      
9212422522
16422072962
22821442792
      
12214422412
19621772962
201216421082

(1 + 2 + ... + 14)(15 + 16 + ... + 273)(274 + 275 + ... + 281) = 932402,
(1 + 2 + ... + 17)(18 + 19 + ... + 177)(178 + 179 + ... + 281) = 2386802,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + ... + 281) = 615602,
(1 + 2 + ... + 56)(57 + 58 + ... + 231)(232 + 233 + ... + 281) = 7182002,
(1 + 2 + ... + 91)(92 + 93 + ... + 155)(156 + 157 + ... + 281) = 9544082,
(1 + 2 + ... + 108)(109 + 110 + ... + 263)(264 + 265 + ... + 281) = 9123302,
(1 + 2 + ... + 153)(154 + 155 + ... + 230)(231 + 232 + ... + 281) = 15079682,
(1 + 2 + ... + 224)(225 + 226 + ... + 231)(232 + 233 + ... + 281) = 7182002.

2812 = 78961 appears in the decimal expressions of π:
  π = 3.14159•••78961••• (from the 47930th digit),
  e = 2.71828•••78961••• (from the 127766th digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised May 25, 2010
by Yoshio Mimura, Kobe, Japan

282

The smallest squares containing k 282's :
28224 = 1682,
2823328225 = 531352,
1282810282628281 = 358163412.

2822 = 79524, a square with different digits.

18k + 282k + 921k + 1380k are squares for k = 1,2,3 (512, 16832, 585812).
60k + 241k + 282k + 378k are squares for k = 1,2,3 (312, 5332,95212).

Komachi Square Sum : 2822 = 152 + 472 + 892 + 2632.

27252 = 652 + 662 + 672 + 682 + 692 + 702 + ... + 2822.

(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 282) = 78752,
(1 + 2)(3 + 4 + ... + 92)(93 + 94 + ... + 282) = 213752,
(1 + 2)(3 + 4 + ... + 122)(123 + 124 + ... + 282) = 270002,
(1 + 2 + ... + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 282) = 204752,
(1 + 2 + ... + 6)(7 + 8 + ... + 56)(57 + 58 + ... + 282) = 355952,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 282) = 346502,
(1 + 2 + ... + 18)(19 + 20 + ... + 56)(57 + 58 + ... + 282) = 966152,
(1 + 2 + ... + 18)(19 + 20 + ... + 92)(93 + 94 + ... + 282) = 1581752,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 282) = 519752,
(1 + 2 + ... + 24)(25 + 26 + ... + 56)(57 + 58 + ... + 282) = 1220402,
(1 + 2 + ... + 24)(25 + 26 + ... + 122)(123 + 124 + ... + 282) = 2646002,
(1 + 2 + ... + 25)(26 + 27 + ... + 217)(218 + 219 + ... + 282) = 3510002,
(1 + 2 + ... + 25)(26 + 27 + ... + 246)(247 + 248 + ... + 282) = 3049802,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 282) = 472502,
(1 + 2 + ... + 35)(36 + 37 + ... + 42)(43 + 44 + ... + 282) = 819002,
(1 + 2 + ... + 35)(36 + 37 + ... + 230)(231 + 232 + ... + 282) = 4668302,
(1 + 2 + ... + 36)(37)(38 + 39 + ... + 282) = 310802,
(1 + 2 + ... + 37)(38 + 39 + ... + 97)(98 + 99 + ... + 282) = 3163502,
(1 + 2 + ... + 44)(45 + 46 + ... + 95)(96 + 97 + ... + 282) = 3534302,
(1 + 2 + ... + 54)(55 + 56 + ... + 135)(136 + 137 + ... + 282) = 5925152,
(1 + 2 + ... + 65)(66 + 67 + ... + 120)(121 + 122 + ... + 282) = 5984552,
(1 + 2 + ... + 74)(75 + 76 + ... + 87)(88 + 89 + ... + 282) = 3246752,
(1 + 2 + ... + 84)(85 + 86 + ... + 95)(96 + 97 + ... + 282) = 3534302.

2822 = 79524 appears in the decimal expressions of π:
  π = 3.14159•••79524••• (from the 6011st digit),
  e = 2.71828•••79524••• (from the 69920th digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

283

The smallest squares containing k 283's :
283024 = 5322,
1552832836 = 394062,
228328328428369 = 151105372.

2832 = 80089, a square with 3 kinds of digits.

2832 = 80089, 8 + 0 + 0 + 8 + 9 = 52,
2832 = 80089, 80 + 0 + 89 = 132.

3-by-3 magic squares consisting of different squares with constant 2832:

23211422582
174219821032
22221672542
      
57216622222
194213821532
19821832862

(1 + 2 + ... + 8)(9 + 10 + ... + 13)(14 + 15 + ... + 283) = 89102,
(1 + 2 + ... + 8)(9 + 10 + ... + 178)(179 + 180 + ... + 283) = 1178102,
(1 + 2 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 283) = 89102,
(1 + 2 + ... + 10)(11 + 12 + ... + 178)(179 + 180 + ... + 283) = 1455302,
(1 + 2 + ... + 10)(11 + 12 + ... + 220)(221 + 222 + ... + 283) = 1455302,
(1 + 2 + ... + 49)(50 + 51 + ... + 67)(68 + 69 + ... + 283) = 2211302,
(1 + 2 + ... + 143)(144 + 145 + ... + 184)(185 + 186 + ... + 283) = 12664082,
(1 + 2 + ... + 143)(144 + 145 + ... + 220)(221 + 222 + ... + 283) = 15135122.

2832 = 80089 appears in the decimal expression of π:
  π = 3.14159•••78400••• (from the 33641st digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised January, 2009
by Yoshio Mimura, Kobe, Japan

284

The smallest squares containing k 284's :
42849 = 2072,
1662845284 = 407782,
284284924284001 = 168607512.

The squares which begin with 284 and end in 284 are
28467113284 = 1687222,   28486013284 = 1687782,   284325701284 = 5332222,
284385425284 = 5332782,   284859173284 = 5337222,...

2842 = 2(2! + 3! + 8!)

1 / 284 = 0.00352112676, 32 + 52 + 22 + 112 + 22 + 62 + 72 + 62 = 284.

2842 = 80656, a zigzag square.

2842 = 80656, 8 + 0 + 6 + 5 + 6 = 52.

3621k + 16543k + 16685k + 43807k are squares for k = 1,2,3 (2842, 498422, 96585562).

(1 + 2 + ... + 24)(25 + 26 + ... + 39)(40 + 41 + ... + 284) = 756002.

(12 + 22 + 32 + 42 + ... + 392) + (12 + 22 + 32 + 42 + ... + 562) = 2842.

(13 + 23 + ... + 153)(163 + 173 + ... + 393)(403 + 413 + ... + 2843) = 37422000002.

2842 = 80656 appears in the decimal expressions of π:
  π = 3.14159•••80656••• (from the 63610th digit),
  e = 2.71828•••80656••• (from the 130562nd digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

285

The smallest squares containing k 285's :
28561 = 1692,
28528561216 = 1689042,
128522852851264 = 113367922.

2852 = 81225, 81 = 92 and 225 = 152.

285 = 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92.

Komachi Fractions : 8041275 / 396 = (285 / 2)2, 8649 / 731025 = (31 / 285)2.

Komachi equations:
2852 = 92 / 82 * 762 * 52 * 42 / 32 / 22 * 12 = 92 / 82 * 762 * 52 * 42 / 32 / 22 / 12.

3-by-3 magic squares consisting of different squares with constant 2852:

8212522562
18122002922
22021602852
      
1125222802
14022452402
24821362352
      
1529022702
16222252662
23421502632
      
20211522602
16022202852
23521402802
20213922482
160220021252
23521482642
      
40211222592
14022352802
24521162882

2852 = 81225, 8 + 1 + 2 + 25 = 62,
2852 = 81225, 8 + 1 + 22 + 5 = 62.

2852 = 81225 appears in the decimal expression of e:
  e = 3.14159•••81225••• (from the 30784th digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised May 25, 2010
by Yoshio Mimura, Kobe, Japan

286

The smallest squares containing k 286's :
286225 = 5352,
2862864 = 16922,
590286286372864 = 242958082.

2862 = 81796, a square with different digits.

260k + 286k + 1742k + 5993k are squares for k = 1,2,3 (912, 62532, 4696512).
858k + 16302k + 21450k + 43186k are squares for k = 1,2,3 (2862, 509082, 97337242).
5005k + 17017k + 23881k + 35893k are squares for k = 1,2,3 (2862, 466182, 80569062).
8437k + 10153k + 30745k + 32461k are squares for k = 1,2,3 (2862, 466182, 80569062).

Komachi Square Sum : 2862 = 42 + 72 + 532 + 812 + 2692.

2862 = 81796, 8 + 17 + 96 = 112.

2862 + 2872 + 2882 + 2892 + 2902 + ... + 135102 = 9066602.

(1 + 2 + ... + 255)(256 + 257 + ... + 271)(272 + 273 + ... + 286) = 7588802.

2862 = 81796 appears in the decimal expression of π:
  π = 3.14159•••81796••• (from the 101291st digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

287

The smallest squares containing k 287's :
287296 = 5362,
128735287209 = 3587972,
328742879287824 = 181312682.

2872 = 82369, a square with different digits.

2872 = 143 + 253 + 403.

The sum of the squares of divisors of 287 is a square, 2902.

Komachi Fraction : 2872 = 5189247 / 63.

Komachi equation: 2872 = 123 / 33 + 43 * 53 + 63 * 73 - 83 + 93.

3-by-3 magic squares consisting of different squares with constant 2872:

625322822
11722582462
26221142272
    
18211822612
181219821022
22221712622
    
26210222672
12322462822
25821072662
    
30216222352
190216521382
21321702902
    
53213822462
17422132822
222213421232

2872 = 82369, 8 + 23 + 69 = 102,
2872 = 82369, 82 + 3 + 6 + 9 = 102.

(22 - 1)(32 - 1)(52 - 1)(122 - 1) = 2872 - 1.

2872 + 2882 + 2892 + 2902 + 2912 + ... + 20472 = 534172,
2872 + 2882 + 2892 + 2902 + 2912 + ... + 3362 = 22052,
2872 + 2882 + 2892 + 2902 + 2912 + ... + 5752 = 74632.

(1 + 2 + ... + 224)(225 + 226 + ... + 287) = 201602.

2872 = 82369 appears in the decimal expression of e:
  e = 3.14159•••82369••• (from the 18421st digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised May 25, 2010
by Yoshio Mimura, Kobe, Japan

288

The smallest squares containing k 288's :
128881 = 3592,
124522882884 = 3528782,
242880442882884 = 155846222.

1 + 2 + 3 + ... + 288 = 2042.

(151 / 288)2 = 0.274896315... (Komachic).

Cubic Polynomials : (X + 1052)(X + 1402)(X + 2882) = X3 + 3372X2 + 525002X + 42336002,
(X + 1832)(X + 2882)(X + 34162) = X3 + 34332X2 + 11668082X + 1800368642,
(X + 2882)(X + 3162)(X + 3572) = X3 + 5572X2 + 1777082X + 324898562,
(X + 2882)(X + 6042)(X + 6272) = X3 + 9172X2 + 4541882X + 1090679042.

2882 - 2872 + 2862 - 2852 + .. + 32 - 22 + 12 = 2042.

2882 = 153 + 223 + 413 = 203 + 253 + 393 = 124 + 124 + 124 + 124.

Komachi equations:
2882 = 12 * 3 * 4 * 56 / 7 * 8 * 9,
2882 = - 92 * 82 * 72 / 62 + 52 * 42 * 32 / 22 * 102 = - 982 / 72 * 62 + 52 * 42 * 32 / 22 * 102,
2882 = 93 - 83 * 73 - 63 + 53 * 43 + 33 * 213 = - 93 + 83 - 73 * 63 + 543 - 33 * 23 + 13.

2882 = 82944, 8 + 29 + 44 = 92.

23722 = 1932 + 1942 + 1952 + 1962 + 1972 + ... + 2882.

(1 + 2 + ... + 63)(64 + 65 + ... + 96)(97 + 98 + ... + 288) = 4435202.

(13 + 23 + ... + 493)(503 + 513 + ... + 1193)(1203 + 1213 + ... + 2883) = 3532798815002.

2882 = 82944 appears in the decimal expressions of π:
  π = 3.14159•••82944••• (from the 24556th digit),
  e = 2.71828•••82944••• (from the 9553rd digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised May 25, 2010
by Yoshio Mimura, Kobe, Japan

289

The squre of 17.

The smallest squares containing k 289's :
289 = 172,
289578289 = 170172,
528917289289 = 7272672.

The squares which begin with 289 and end in 289 are
289578289 = 170172,   2890890289 = 537672,   28905780289 = 1700172,
28979274289 = 1702332,   28990851289 = 1702672,...

2892 = 83521, a square with different digits.

2892 = 173 + 343 + 343.

289 is the sum of m squares for m = 1, 2, 3, ... , 275. (the third square)

Komachi Fractions : 324 / 751689 = (6 / 289)2, 2304 / 751689 = (16 / 289)2.

Komachi equations:
2892 = 9 * 87 / 6 * 5 * 4 * 32 + 1,
2892 = 93 - 83 + 73 * 63 + 53 * 43 + 33 * 23 + 103.

2892 = 83521, 8 + 3 + 52 + 1 = 82,
2892 = 83521, 8 + 35 + 21 = 82,
2892 = 83521, 8 + 352 + 1 = 192.

3-by-3 magic squares consisting of different squares with constant 2892:

24215322442
172220421112
231213621082
      
2828122762
14422442572
24921322642
      
39213622522
18822042812
216215321162

(1 + 2 + ... + 24)(25 + 26 + ... + 240)(241 + 2 + ... + 289) = 3339002,
(1 + 2 + ... + 51)(52 + 53 + ... + 237)(238 + 239 + ... + 289) = 6988022,
(1 + 2 + ... + 72)(73)(74 + 75 + ... + 289) = 867242,
(1 + 2 + ... + 102)(103 + 104 + ... + 186)(187 + 188 + ... + 289) = 12502142,
(1 + 2 + ... + 128)(129 + 130 + ... + 160)(161 + 162 + ... + 289) = 10526402,
(1 + 2 + ... + 128)(129 + 130 + ... + 214)(215 + 216 + ... + 289) = 15170402,
(1 + 2 + ... + 169)(170 + 171 + ... + 289) = 198902,
(1 + 2 + ... + 175)(176 + 177 + ... + 223)(224 + 225 + ... + 289) = 15800402,
(1 + 2 + ... + 216)(217)(218 + 219 + ... + 289) = 3046682,
(1 + 2 + ... + 288)(289) = 34682.

2892 = 83521 appears in the decimal expressions of π:
  π = 3.14159•••83521••• (from the 71600th digit),
  e = 2.71828•••83521••• (from the 14279th digit).

Page of Squares : First Upload November 8, 2004 ; Last Revised May 25, 2010
by Yoshio Mimura, Kobe, Japan