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290

The smallest squares containing k 290's :
52900 = 230²,
7529032900 = 86770²,
290899290292900 = 17055770².

290² = (1² + 1)(12² + 1)(17² + 1)
= (2² + 1)(3² + 1)(41² + 1) = (7² + 1)(41² + 1).

Komachi Square Sum : 290² = 4² + 7² + 83² + 95² + 261².

1450^k + 12818^k + 24186^k + 45646^k are squares for k = 1,2,3 (290², 53244², 10552868²).

(1)(2 + 3 + ... + 52)(53 + 54 + ... + 290) = 7497²,
(1 + 2 + ... + 4)(5 + 6 + ... + 34)(35 + 36 + ... + 290) = 15600²,
(1 + 2 + ... + 4)(5 + 6 + ... + 70)(71 + 72 + ... + 290) = 31350²,
(1 + 2 + ... + 4)(5 + 6 + ... + 240)(241 + 242 + ... + 290) = 61950²,
(1 + 2 + ... + 13)(14 + 15 + ... + 34)(35 + 36 + ... + 290) = 43680²,
(1 + 2 + ... + 13)(14 + 15 + ... + 238)(239 + 240 + ... + 290) = 188370²,
(1 + 2 + ... + 17)(18 + 19 + ... + 34)(35 + 36 + ... + 290) = 53040²,
(1 + 2 + ... + 17)(18 + 19 + ... + 52)(53 + 54 + ... + 290) = 87465²,
(1 + 2 + ... + 17)(18 + 19 + ... + 171)(172 + 173 + ... + 290) = 247401²,
(1 + 2 + ... + 17)(18 + 19 + ... + 238)(239 + 240 + ... + 290) = 243984²,
(1 + 2 + ... + 32)(33 + 34 + ... + 66)(67 + 68 + ... + 290) = 188496²,
(1 + 2 + ... + 45)(46 + 47 + ... + 129)(130 + 131 + ... + 290) = 507150²,
(1 + 2 + ... + 49)(50 + 51 + 52)(53 + 54 + ... + 290) = 87465²,
(1 + 2 + ... + 80)(81 + 82 + ... + 269)(270 + 271 + ... + 290) = 793800²,
(1 + 2 + ... + 139)(140 + 141 + ... + 265)(266 + 267 + ... + 290) = 1313550²,
(1 + 2 + ... + 147)(148 + 149 + ... + 227)(228 + 229 + ... + 290) = 1631700²,
(1 + 2 + ... + 158)(159 + 160 + ... + 237)(238 + 239 + ... + 290) = 1658052²,
(1 + 2 + ... + 225)(226 + 227 + ... + 274)(275 + 276 + ... + 290) = 1186500².

(1³ + 2³ + ... + 42³)(43³ + 44³ + ... + 174³)(175³ + 176³ + ... + 290³) = 540072308160²,
(1³ + 2³ + ... + 87³)(88³ + 89³ + ... + 290³) = 160856388².

290² = 84100 appears in the decimal expression of e:
  e = 3.14159•••84100••• (from the 16067th digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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291

The smallest squares containing k 291's :
2916 = 54²,
2912329156 = 53966²,
1583291291291649 = 39790593².

(140 / 291)² = 0.231456879... (Komachic).

291² = 84681, 8 + 4 + 68 + 1 = 9².

291² + 292² + 293² + 294² + 295² + ... + 8243² = 432113².

3-by-3 magic squares consisting of different squares with constant 291²:

(1 + 2)(3 + 4 + ... + 291) = 357²,
(1 + 2 + ... + 4)(5 + 6 + ... + 115)(116 + 117 + ... + 291) = 48840²,
(1 + 2 + ... + 4)(5 + 6 + ... + 284)(285 + 286 + ... + 291) = 28560²,
(1 + 2 + ... + 9)(10 + 11 + ... + 225)(226 + 227 + ... + 291) = 139590²,
(1 + 2 + ... + 16)(17 + 18 + ... + 203)(204 + 205 + ... + 291) = 246840²,
(1 + 2 + ... + 20)(21 + 22 + ... + 83)(84 + 85 + ... + 291) = 163800²,
(1 + 2 + ... + 30)(31 + 32 + ... + 185)(186 + 187 + ... + 291) = 443610²,
(1 + 2 + ... + 32)(33 + 34 + ... + 275)(276 + 277 + ... + 291) = 299376²,
(1 + 2 + ... + 34)(35 + 36 + ... + 203)(204 + 205 + ... + 291) = 510510²,
(1 + 2 + ... + 41)(42 + 43 + ... + 86)(87 + 88 + ... + 291) = 309960²,
(1 + 2 + ... + 81)(82 + 83 + ... + 86)(87 + 88 + ... + 291) = 232470²,
(1 + 2 + ... + 95)(96 + 97 + ... + 284)(285 + 286 + ... + 291) = 574560²,
(1 + 2 + ... + 158)(159 + 160 + ... + 212)(213 + 214 + ... + 291) = 1582686²,
(1 + 2 + ... + 160)(161 + 162 + ... + 214)(215 + 216 + ... + 291) = 1593900².

291² = 84681 appears in the decimal expression of π:
  π = 3.14159•••84681••• (from the 2193rd digit),
  (84681 is the fifth 5-digit square in the expression of π.)

Page of Squares : First Upload November 15, 2004 ; Last Revised January 27, 2009
by Yoshio Mimura, Kobe, Japan

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292

The smallest squares containing k 292's :
29241 = 171²,
6292772929 = 79327²,
292122927292816 = 17091604².

292² = 85264, a square with different digits.

The sum of (8x + 3)² (x = 0,1,..,15) is 292².

292² = 85264, 8 + 5 + 2 + 6 + 4 = 5².

(1 + 2 + ... + 15)(16 + 17 + ... + 92)(93 + 94 + ... + 292) = 138600²,
(1 + 2 + ... + 17)(18 + 19 + ... + 67)(68 + 69 + ... + 292) = 114750²,
(1 + 2 + ... + 20)(21 + 22 + ... + 139)(140 + 141 + ... + 292) = 257040²,
(1 + 2 + ... + 159)(160 + 161 + ... + 239)(240 + 241 + ... + 292) = 1691760²,
(1 + 2 + ... + 215)(216 + 217 + ... + 257)(258 + 259 + ... + 292) = 1489950².

292² = 85264 appears in the decimal expressions of π and e:
  π = 3.14159•••85264••• (from the 18732nd digit),
  e = 2.71828•••85264••• (from the 21042nd digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised June 22, 2006
by Yoshio Mimura, Kobe, Japan

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293

The smallest squares containing k 293's :
293764 = 542²,
29302934761 = 171181²,
1829312932939024 = 42770468².

293² = 85849, a zigzag square.

(2² + 1)(3² + 1)(4² + 1)(10² + 1) = 293² + 1.

49^k + 98^k + 170^k + 212^k are squares for k = 1,2,3 (23², 293², 3937²).

Komachi equation: 293² = 1³ * 23³ - 4³ - 5³ + 6³ * 7³ + 8³ - 9³.

3-by-3 magic squares consisting of different squares with constant 293²:

(1 + 2 + ... + 5)(6 + 7 + ... + 291)(292 + 293) = 19305²,
(1 + 2 + ... + 10)(11 + 12 + ... + 124)(125 + 126 + ... + 293) = 122265²,
(1 + 2 + ... + 13)(14 + 15 + ... + 111)(112 + 113 + ... + 293) = 143325²,
(1 + 2 + ... + 17)(18 + 19 + ... + 97)(98 + 99 + ... + 293) = 164220²,
(1 + 2 + ... + 26)(27 + 28 + ... + 98)(99 + 100 + ... + 293) = 245700²,
(1 + 2 + ... + 26)(27 + 28 + ... + 291)(292 + 293) = 93015²,
(1 + 2 + ... + 47)(48 + 49 + ... + 282)(283 + 284 + ... + 293) = 372240²,
(1 + 2 + ... + 58)(59 + 60 + ... + 235)(236 + 237 + ... + 293) = 826413²,
(1 + 2 + ... + 62)(63 + 64 + ... + 231)(232 + 233 + ... + 293) = 888615²,
(1 + 2 + ... + 63)(64 + 65 + ... + 97)(98 + 99 + ... + 293) = 459816²,
(1 + 2 + ... + 74)(75 + 76 + ... + 147)(148 + 149 + ... + 293) = 850815²,
(1 + 2 + ... + 90)(91 + 92 + ... + 98)(99 + 100 + ... + 293) = 343980²,
(1 + 2 + ... + 90)(91 + 92 + ... + 286)(287 + 288 + ... + 293) = 554190²,
(1 + 2 + ... + 114)(115 + 116 + ... + 276)(277 + 278 + ... + 293) = 1002915²,
(1 + 2 + ... + 139)(140 + 141 + ... + 154)(155 + 156 + ... + 293) = 817320²,
(1 + 2 + ... + 146)(147)(148 + 149 + ... + 293) = 225351²,
(1 + 2 + ... + 164)(165 + 166 + ... + 245)(246 + 247 + ... + 293) = 1704780²,
(1 + 2 + ... + 194)(195 + 196 + ... + 291)(292 + 293) = 510705².

293² = 85849 appears in the decimal expression of e:
  e = 3.14159•••85849••• (from the 79397th digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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294

The smallest squares containing k 294's :
82944 = 288²,
294294025 = 17155²,
294129462941209 = 17150203².

The square root of 294 is 17.14..., 17 = 1² + 4².

Magic Square : 294² + 753² + 618² = 492² + 357² + 816², (the same for 456, 978, 231).

294² = 86436, 864 / 3 + 6 = 294.

294² = (1² + 5)(3² + 5)(32² + 5) = (2² + 3)(9² + 3)(12² + 3) = (3² + 5)(4² + 5)(17² + 5).

204^k + 294^k + 561^k + 966^k are squares for k = 1,2,3 (45², 1173², 33345²).
294^k + 1288^k + 1617^k + 2730^k are squares for k = 1,2,3 (77², 3437², 163513²).

Komachi equations:
294² = 1² / 2² / 3² * 4² * 56² * 7² / 8² * 9² = 9² * 8² * 7² / 6² / 5² / 4² / 3² * 210²
294² = 98² / 7² * 6² / 5² / 4² / 3² * 210².

294² = 86436 is exchangeable, 36864 = 192².

294² = 86436, 8 + 64 + 3 + 6 = 9²,
294² = 86436, 864 + 36 = 30²,
294² = 86436, 8643 + 6 = 93².

294⁵ = 2196527536224, 2² + 1² + 9² + 6² + 5² + 2² + 7² + 5² + 3² + 6² + 2² + 2² + 4².

294² + 295² + 296² + 297² + 298² + ... + 367² = 2849²,
294² + 295² + 296² + 297² + 298² + ... + 6149² = 278404².

(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 294) = 1530²,
(1 + 2 + ... + 8)(9 + 10 + ... + 30)(31 + 32 + ... + 294) = 25740²,
(1 + 2 + ... + 22)(23 + 24 + ... + 46)(47 + 48 + ... + 294) = 94116²,
(1 + 2 + ... + 22)(23 + 24 + ... + 142)(143 + 144 + ... + 294) = 288420²,
(1 + 2 + ... + 27)(28 + 29 + ... + 119)(120 + 121 + ... + 294) = 304290²,
(1 + 2 + ... + 37)(38)(39 + 40 + ... + 294) = 33744²,
(1 + 2 + ... + 99)(100 + 101 + ... + 275)(276 + 277 + ... + 294) = 940500².

(1³ + 2³ + ... + 8³)(9³ + 10³ + ... + 41³)(42³ + 43³ + ... + 294³) = 1342701360²,
(1³ + 2³ + ... + 69³)(70³ + 71³ + ... + 293³)(294³) = 523527293520².

294² = 86436 appears in the decimal expression of π:
  π = 3.14159•••86436••• (from the 22665th digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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295

The smallest squares containing k 295's :
29584 = 172²,
1829529529 = 42773²,
229542952955625 = 15150675².

295² = 87025, a square with different digits.

253^k + 295^k + 341^k + 407^k are squares for k = 1,2,3 (36², 658², 12204²).
6490^k + 17110^k + 28320^k + 35105^k are squares for k = 1,2,3 (295², 48675², 8441425²).

Komachi Fraction : 108 / 2349675 = (2 / 295)².

3-by-3 magic squares consisting of different squares with constant 295²:

(1² + 7)(2² + 7)(4² + 7)(6² + 7) = 295² + 7.

(1 + 2 + ... + 71)(72 + 73 + ... + 82)(83 + 84 + ... + 295) = 295218²,
(1 + 2 + ... + 120)(121 + 122 + ... + 204)(205 + 206 + ... + 295) = 1501500²,
(1 + 2 + ... + 231)(232)(233 + 234 + ... + 295) = 321552².

295² = 87025 appears in the decimal expressions of π and e:
  π = 3.14159•••87025••• (from the 21300th digit),
  e = 2.71828•••87025••• (from the 32521st digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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296

The smallest squares containing k 296's :
1296 = 36²,
1329623296 = 36464²,
29647632961296 = 5444964².

The squares which begin with 296 and end in 296 are
2966327296 = 54464²,   296441047296 = 544464²,   296519455296 = 544536²,
296985761296 = 544964²,   2960244127296 = 1720536²,...

12358^k + 15762^k + 20498^k + 38998^k are squares for k = 1,2,3 (296², 48396², 8586368²).

Komachi Square Sum : 296² = 1² + 6² + 49² + 53² + 287².

296² = 87616, 8 + 76 + 16 = 10²,
296² = 87616, 87 + 6 + 1 + 6 = 10².

1² + 2² + 3² + 4² + 5² + ... + 296² = 8688636, which consists of 3 kinds of digits.

(1 + 2 + ... + 9)(10 + 11 + ... + 41)(42 + 43 + ... + 296) = 39780²,
(1 + 2 + ... + 155)(156 + 157 + ... + 168)(169 + 170 + ... + 296) = 870480².

(1³ + 2³ + ... + 28³)(29³ + 30³ + ... + 36³)(37³ + 38³ + ... + 296³) = 9420596640²,
(1³ + 2³ + ... + 28³)(29³ + 30³ + ... + 231³)(232³ + 233³ + ... + 296³) = 379030491840².

296² = 87616 appears in the decimal expression of π:
  π = 3.14159•••87616••• (from the 54377th digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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297

The smallest squares containing k 297's :
297025 = 545²,
2972975625 = 54525²,
15297297702976 = 3911176².

Kaprekar number : 297² = 88209, 297 = 88 + 209
Other examples : 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272,....

297² = 88209, 88 + 209 = 297.

297³ + 792³ = 22869².

297³ = (1² + 8)(5² + 8)(17² + 8).

297² = 88209, 8 + 8 + 209 = 15².

3-by-3 magic squares consisting of different squares with constant 297²:

(1 + 2 + ... + 12)(13 + 14 + ... + 167)(168 + 169 + ... + 297) = 181350²,
(1 + 2 + ... + 24)(25 + 26 + ... + 47)(48 + 49 + ... + 297) = 103500²,
(1 + 2 + ... + 31)(32 + 33 + ... + 62)(63 + 64 + ... + 297) = 174840²,
(1 + 2 + ... + 36)(37 + 38)(39 + 40 + ... + 297) = 46620²,
(1 + 2 + ... + 94)(95 + 96 + ... + 234)(235 + 236 + ... + 297) = 1312710²,
(1 + 2 + ... + 144)(145 + 146 + ... + 195)(196 + 197 + ... + 297) = 1508580²,
(1 + 2 + ... + 152)(153 + 154 + ... + 272)(273 + 274 + ... + 297) = 1453500².

297² = 88209 appears in the decimal expression of π:
  π = 3.14159•••88209••• (from the 3470th digit),
  (88209 is the sixth 5-digit square in the expression of π.)

Page of Squares : First Upload November 15, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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298

The smallest squares containing k 298's :
298116 = 546²,
229832989281 = 479409²,
2983582984929856 = 54622184².

298² = 88804, a square with 3 kinds of even digits.

298²± 3 are primes.

82^k + 298^k + 628^k + 673^k are squares for k = 1,2,3 (41², 971², 24073²).

Komachi Square Sums : 298² = 19² + 43² + 65² + 287² = 4² + 5² + 7² + 8² + 63² + 291².

298² = 88804, 8³ + 8³ + 8³ + 0³ + 4³ = 40²,
298² = 88804, 8⁴ + 8⁴ + 8⁴ + 0⁴ + 4⁴ = 112²,
298² = 88804, 8 + 8 + 80 + 4 = 10²,
298² = 88804, 8 + 88 + 0 + 4 = 10²,
298² = 88804, 88 + 8 + 0 + 4 = 10².

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

(1)(2 + 3 + ... + 133)(134 + 135 + ... + 298) = 17820²,
(1)(2 + 3 + ... + 295)(296 + 297 + 298) = 6237²,
(1 + 2)(3 + 4 + ... + 39)(40 + 41 + ... + 298) = 10101²,
(1 + 2 + ... + 11)(12 + 13 + ... + 42)(43 + 44 + 45 + ... + 298) = 49104²,
(1 + 2 + ... + 14)(15 + 16 + ... + 56)(57 + 58 + ... + 298) = 82005²,
(1 + 2 + ... + 34)(35 + 36 + ... + 196)(197 + 198 + ... + 298) = 530145²,
(1 + 2 + ... + 42)(43 + 44 + ... + 174)(175 + 176 + ... + 298) = 615846²,
(1 + 2 + ... + 44)(45 + 46 + ... + 99)(100 + 101 + ... + 298) = 394020²,
(1 + 2 + ... + 44)(45 + 46 + ... + 133)(134 + 135 + ... + 298) = 528660²,
(1 + 2 + ... + 53)(54 + 55 + ... + 86)(87 + 88 + ... + 298) = 367290²,
(1 + 2 + ... + 53)(54 + 55 + ... + 284)(285 + 286 + ... + 298) = 477477²,
(1 + 2 + ... + 60)(61 + 62 + ... + 128)(129 + 130 + ... + 298) = 653310²,
(1 + 2 + ... + 62)(63 + 64 + ... + 279)(280 + 281 + ... + 298) = 630819²,
(1 + 2 + ... + 76)(77 + 78 + ... + 133)(134 + 135 + ... + 298) = 790020²,
(1 + 2 + ... + 98)(99)(100 + 101 + ... + 298) = 137907²,
(1 + 2 + ... + 98)(99 + 100 + ... + 295)(296 + 297 + 298) = 409563²,
(1 + 2 + ... + 150)(151)(152 + 153 + ... + 298) = 237825².

298² = 88804 appears in the decimal expression of e:
  e = 3.14159•••88804••• (from the 128581st digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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299

The smallest squares containing k 299's :
12996 = 114²,
1232992996 = 35114²,
1299299192998849 = 36045793².

299² = 89401, a square with different digits.

299² = 89401, 8 + 9401 = 97².

17^k + 91^k + 299^k + 377^k are squares for k = 1,2,3 (28², 490², 9004²).

(1³ + 2³ + ... + 275³)(276³ + 277³ + ... + 299³)(300³) = 4713390000000².

3-by-3 magic squares consisting of different squares with constant 299²:

299² = 89401 appears in the decimal expression of e:
  e = 3.14159•••89401••• (from the 89487th digit).

Page of Squares : First Upload November 15, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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