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270

The smallest squares containing k 270's :
2704 = 52²,
22706270596 = 150686²,
285527027027025 = 16897545².

270² = (1² + 5)(5² + 5)(20² + 5) = (3² + 9)(6² + 9)(9² + 9).

Komachi equation: 270² = 9³ - 8³ + 7³ * 6³ - 5³ - 4³ - 3³ * 2³ - 10³.

(1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9)(10) = 270²,
(1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12) = 270²,
(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13) = 270²,
(1 + 2 + 3)(4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14)(15) = 270²,
(1)(2)(3)(4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14)(15) = 270²,
(1 + 2)(3 + 4 + ... + 18)(19 + 20 + ... + 270) = 4284²,
(1 + 2 + ... + 5)(6 + 7 + ... + 170)(171 + 172 + ... + 270) = 69300²,
(1 + 2 + ... + 6)(7 + 8 + ... + 18)(19 + 20 + ... + 270) = 10710²,
(1 + 2 + ... + 9)(10 + 11 + ... + 90)(91 + 92 + ... + 270) = 76950²,
(1 + 2 + ... + 15)(16 + 17 + ... + 170)(171 + 172 + ... + 270) = 195300²,
(1 + 2 + ... + 32)(33 + 34 + ... + 72)(73 + 74 + ... + 270) = 194040²,
(1 + 2 + ... + 49)(50 + 51 + ... + 145)(146 + 147 + ... + 270) = 546000²,
(1 + 2 + ... + 80)(81 + 82 + ... + 269)(270) = 170100²,
(1 + 2 + ... + 100)(101 + 102 + ... + 234)(235 + 236 + ... + 270) = 1015050²,
(1 + 2 + ... + 114)(115 + 116 + ... + 189)(190 + 191 + ... + 270) = 1179900²,
(1 + 2 + ... + 147)(148 + 149 + ... + 221)(222 + 223 + ... + 270) = 1337994²,
(1 + 2 + ... + 169)(170)(171 + 172 + ... + 270) = 232050².

270² = 72900 appears in the decimal expression of e:
  e = 2.71828•••72900••• (from the 137th digit),
  (72900 is the first 5-digit square in the expression of e.)

Page of Squares : First Upload November 1, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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271

The smallest squares containing k 271's :
271441 = 521²,
127127189401 = 356549²,
2713271942711929 = 52089077².

271² is the tenth square which is the sum of 8 fifth powers :(1,2,2,2,2,6,8,8).

Komachi equations:
271² = 9 * 8 * 765 * 4 / 3 + 2 - 1 = 9 + 8 * 765 * 4 * 3 + 2 - 10.

3-by-3 magic squares consisting of different squares with constant 271²:

(1 + 2 + ... + 13)(14 + 15 + ... + 103)(104 + 105 + ... + 271) = 122850²,
(1 + 2 + ... + 19)(20 + 21 + ... + 43)(44 + 45 + ... + 271) = 71820²,
(1 + 2 + ... + 40)(41 + 42 + ... + 148)(149 + 150 + ... + 271) = 464940²,
(1 + 2 + ... + 64)(65 + 66 + ... + 208)(209 + 210 + ... + 271) = 786240²,
(1 + 2 + ... + 127)(128 + 129 + ... + 253)(254 + 255 + ... + 271) = 960120².

271² = 73441 appears in the decimal expressions of π:
  π = 3.14159•••73441••• (from the 111611st digit),
  e = 2.71828•••73441••• (from the 38677th digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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272

The smallest squares containing k 272's :
27225 = 165²,
27252727056 = 165084²,
27212727762724 = 5216582².

(195 / 272)² = 0.513962478... (Komachic).

272² = 73984, a square with different digits.

272² = 4⁴ + 8⁴ + 8⁴ + 16⁴.

272 is the second integer which is the sum of a squre and a prime in 7 ways :
1² + 271, 3² + 263, 7² + 223, 9² + 191, 11² + 151, 13² + 103, 15² + 47.

217^k + 218^k + 272^k + 518^k are squares for k = 1,2,3 (35², 661², 13405²).

Komachi equations:
272² = 9² * 8² * 7² / 6² * 5² - 4² - 32² * 10² = 98² / 7² * 6² * 5² - 4² - 32² * 10².

272² + 273² + 274² + ... + 8552² = 456638².

(1 + 2 + ... + 63)(64 + 65 + ... + 112)(113 + 114 + ... + 272) = 517440².

272² = 73984 appears in the decimal expressions of π:
  π = 3.14159•••73984••• (from the 74860th digit),
  e = 2.71828•••73984••• (from the 29441st digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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273

The smallest squares containing k 273's :
273529 = 523²,
273273961 = 16531²,
70327327327321 = 8386139².

The square root of 273 is 16.522711641..., 16² = 5² + 2² + 2² + 7² + 11² + 6² + 4² + 1².

273² = 74529, a zigzag square with different digits.

273^k + 8853^k + 18213^k + 27417^k are squares for k = 1,2,3 (234², 34086², 5229198²).
3185^k + 21658^k + 22022^k + 27664^k are squares for k = 1,2,3 (273², 41587², 6484023²).

Komachi equations:
273² = 1³ * 2³ + 3³ + 4³ + 5³ + 6³ * 7³ - 8³ + 9³ = 9³ - 8³ + 7³ * 6³ + 5³ + 4³ + 3³ + 2³ * 1³
  = 9³ - 8³ + 7³ * 6³ + 5³ + 4³ + 3³ + 2³ / 1³ = 98³ - 76³ * 5³ / 4³ - 3³ - 21³.

273² = 74529, 7 + 45 + 29 = 9².

(1² + 2² + 3² + 4² + 5² + 6²)(1² + 2² + 3² + 4² + ... + 13²) = 273².

3-by-3 magic squares consisting of different squares with constant 273²:

(1 + 2 + ... + 6)(7 + 8 + ... + 84) = 273²,
(1)(2 + 3 + ... + 86)(87 + 88 + ... + 273) = 11220²,
(1 + 2 + ... + 18)(19 + 20 + ... + 30)(31 + 32 + ... + 273) = 43092²,
(1 + 2 + ... + 27)(28 + 29 + ... + 126)(127 + 128 + ... + 273) = 291060²,
(1 + 2 + ... + 168)(169 + 170 + ... + 272)(273) = 298116².

273² = 74529 appears in the decimal expressions of π:
  π = 3.14159•••74529••• (from the 28032nd digit),
  e = 2.71828•••74529••• (from the 70177th digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised February 25, 2011
by Yoshio Mimura, Kobe, Japan

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274

The smallest squares containing k 274's :
127449 = 357²,
127483274304 = 357048²,
1274069274527449 = 35694107².

274² = 44² + 45² + 46² + ... + 67².

274² = 75076, 7 * 50 - 76 = 274.

274² = 75076, 7 + 5 + 0 + 7 + 6 = 5².

Komachi equations:
274² = - 9² + 87² + 65² * 4² - 3² - 2² + 1²,
274² = - 12³ / 3³ - 4³ - 5³ + 6³ * 7³ + 8³ + 9³.

(1 + 2 + ... + 49)(50 + 51 + ... + 130)(131 + 132 + ... + 274) = 510300²,
(1 + 2 + ... + 50)(51 + 52 + ... + 85)(86 + 87 + ... + 274) = 321300²,
(1 + 2 + ... + 156)(157)(158 + 159 + ... + 274) = 220428².

274² = 75076 appears in the decimal expression of e:
  e = 2.71828•••75076••• (from the 56442nd digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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275

The smallest squares containing k 275's :
27556 = 166²,
1275275521 = 35711²,
242752759275625 = 15580525².

275² = 75625, a zigzag square.

275² = (7² + 6)(37² + 6).

275² is the 6th square which is the sum of 4 fifth powers : (4, 6, 6, 9).

(1² + 2² + 3² + ... + 212²) + (1² + 2² + 3² + ... + 224²) = (1² + 2² + 3² + 4² + ... + 275²).

275² = 4⁵ + 6⁵ + 6⁵ + 9⁵.

67^k + 209^k + 233^k + 275^k are squares for k = 1,2,3 (28², 422², 6548²).

Komachi Square Sum : 275² = 16² + 38² + 97² + 254².

275² = 75625, 7 + 5 + 6 + 2 + 5 = 5².

3-by-3 magic squares consisting of different squares with constant 275²:

(1 + 2)(3 + 4 + ... + 123)(124 + 125 + ... + 275) = 26334²,
(1 + 2 + ... + 8)(9 + 10 + ... + 62)(63 + 64 + ... + 275) = 49842²,
(1 + 2 + ... + 34)(35 + 36 + ... + 149)(150 + 151 + ... + 275) = 410550²,
(1 + 2 + ... + 67)(68 + 69 + ... + 268)(269 + 270 + ... + 275) = 382704²,
(1 + 2 + ... + 119)(120 + 121 + ... + 200)(201 + 202 + ... + 275) = 1285200².

(1³ + 2³ + ... + 77³)(78³ + 79³ + ... + 230³)(231³ + 232³ + ... + 275³) = 2148172986960².

275² = 75625 appears in the decimal expressions of π:
  π = 3.14159•••75625••• (from the 18330th digit),
  e = 2.71828•••75625••• (from the 4646th digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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276

The smallest squares containing k 276's :
12769 = 113²,
2762763844 = 52562²,
2276872762761 = 1508931².

The squares which begin with 276 and end in 276 are
27613798276 = 166174²,   27664338276 = 166326²,   276333154276 = 525674²,
276492982276 = 525826²,   276859078276 = 526174²,...

276² = 76176, a square with 3 kinds of digits.

83² + 84² + 85² + ... + 276² = 2619².

276² = 76176, 7 + 6 + 17 + 6 = 6²,
276² = 76176, 7 + 61 + 7 + 6 = 9²,
276² = 76176, 7 + 61 + 76 = 12².

(1 + 2 + ... + 11)(12 + 13 + ... + 20)(21 + 22 + ... + 276) = 19008²,
(1 + 2 + ... + 95)(96 + 97 + ... + 266)(267 + 268 + ... + 276) = 619020²,
(1 + 2 + ... + 153)(154 + 155 + ... + 174)(175 + 176 + ... + 276) = 966042².

276² = 76176 appears in the decimal expressions of π:
  π = 3.14159•••76176••• (from the 93280th digit),
  e = 2.71828•••76176••• (from the 3278th digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised June 19, 2006
by Yoshio Mimura, Kobe, Japan

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277

The smallest squares containing k 277's :
277729 = 527²,
277122227776 = 526424²,
10127791277277184 = 100636928².

1 / 277 = 0.00361..., 361 = 19².

277² = 76729, a zigzag square.

277² = 76729, 7 + 6 + 7 + 29 = 7².

3-by-3 magic squares consisting of different squares with constant 277²:

277² = 76729 appears in the decimal expressions of π:
  π = 3.14159•••76729••• (from the 47791st digit),
  e = 2.71828•••76729••• (from the 88368th digit).

(1 + 2)(3 + 4 + ... + 52)(53 + 54 + ... + 277) = 12375²,
(1 + 2)(3 + 4 + ... + 272)(273 + 274 + ... + 277) = 12375²,
(1 + 2 + ... + 4)(5 + 6 + ... + 85)(86 + 87 + ... + 277) = 35640²,
(1 + 2 + ... + 9)(10 + 11 + ... + 72)(73 + 74 + ... + 277) = 64575²,
(1 + 2 + ... + 9)(10 + 11 + ... + 85)(86 + 87 + ... + 277) = 75240²,
(1 + 2 + ... + 10)(11 + 12 + ... + 52)(53 + 54 + ... + 277) = 51975²,
(1 + 2 + ... + 10)(11 + 12 + ... + 55)(56 + 57 + ... + 277) = 54945²,
(1 + 2 + ... + 13)(14 + 15 + ... + 112)(113 + 114 + ... + 277) = 135135²,
(1 + 2 + ... + 27)(28 + 29 + ... + 97)(98 + 99 + ... + 277) = 236250²,
(1 + 2 + ... + 54)(55)(56 + 57 + ... + 277) = 54945²,
(1 + 2 + ... + 54)(55 + 56 + ... + 272)(273 + 274 + ... + 277) = 269775²,
(1 + 2 + ... + 60)(61 + 62 + ... + 122)(123 + 124 + ... + 277) = 567300²,
(1 + 2 + ... + 65)(66 + 67 + ... + 129)(130 + 131 + ... + 277) = 634920²,
(1 + 2 + ... + 74)(75 + 76 + ... + 222)(223 + 224 + ... + 277) = 915750²,
(1 + 2 + ... + 77)(78 + 79 + ... + 272)(273 + 274 + ... + 277) = 375375²,
(1 + 2 + ... + 99)(100 + 101 + ... + 261)(262 + 263 + ... + 277) = 790020²,
(1 + 2 + ... + 164)(165 + 166 + ... + 255)(256 + 257 + ... + 277) = 1231230²,
(1 + 2 + ... + 186)(187)(188 + 189 + ... + 277) = 260865²,
(1 + 2 + ... + 195)(196 + 197 + ... + 255)(256 + 257 + ... + 277) = 1231230²,
(1 + 2 + ... + 255)(256 + 257 + ... + 272)(273 + 274 + ... + 277) = 448800².

Page of Squares : First Upload November 1, 2004 ; Last Revised January 20, 2009
by Yoshio Mimura, Kobe, Japan

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278

The smallest squares containing k 278's :
27889 = 167²,
427827856 = 20684²,
278527827857409 = 16689153².

278² is the 9th square which is the sum of 6 fifth powers : (1,5,6,7,7,8).

Komachi equation: 278² = 98² - 7² + 65² + 4² * 3² * 21².

278 is the third integer which is the sum of a square and a prime in 7 ways :
1² + 277, 3² + 269, 7² + 229, 9² + 197, 11² + 157, 13² + 109, 15² + 53.

278² = 77284, 7 + 7 + 2 + 84 = 10²,
278² = 77284, 77 + 284 = 19²,
278² = 77284, 772 + 8 + 4 = 28².

(1 + 2 + ... + 6)(7 + 8 + ... + 126)(127 + 128 + ... + 278) = 71820²,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 278) = 307230²,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 278) = 98154²,
(1 + 2 + ... + 95)(96 + 97 + ... + 113)(114 + 115 + ... + 278) = 526680²,
(1 + 2 + ... + 159)(160 + 161 + ... + 225)(226 + 227 + ... + 278) = 1469160².

(1³ + 2³ + ... + 98³)(99³ + 100³ + ... + 153³)(154³ + 155³ + ... + 278³) = 1924261693200².

278² = 77284 appears in the decimal expressions of π:
  π = 3.14159•••77284••• (from the 98075th digit),
  e = 2.71828•••77284••• (from the 96683rd digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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279

The smallest squares containing k 279's :
279841 = 529²,
27942799921 = 167161²,
279279279145801 = 16711651².

Komachi equations: 279² = - 9³ - 8³ - 7³ - 6³ + 5³ + 43³ + 2³ + 1³.

279² = 21³ + 21³ + 39³.

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

3-by-3 magic squares consisting of different squares with constant 279²:

(1 + 2 + ... + 15)(16 + 17 + ... + 240)(241 + 242 + ... + 279) = 187200²,
(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + 138 + ... + 279) = 204204²,
(1 + 2 + ... + 152)(153 + 154 + ... + 228)(229 + 230 + ... + 279) = 1476756².

(1³ + 2³ + ... + 216³)(217³ + 218³ + ... + 279³) = 732328128².

279² = 77841 appears in the decimal expressions of π:
  π = 3.14159•••77841••• (from the 89157th digit),
  e = 2.71828•••77841••• (from the 68523th digit).

Page of Squares : First Upload November 1, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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