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260

The smallest squares containing k 260's :
2601 = 51²,
3260752609 = 57103²,
260260072926084 = 16132578².

260 = (1² + 2² + 3² + ... + 175²) / (1² + 2² + 3² + ... + 27²).

260² = 39 x 40 + 40 x 41 + 41 x 42 + ... + 63 x 64.

260² = (3² + 4)(4² + 4)(16² + 4).

260^k + 286^k + 1742^k + 5993^k are squares for k = 1,2,3 (91², 6253², 469651²).
20^k + 260^k + 265^k + 680^k are squares for k = 1,2,3 (35², 775², 18725²).

Komachi equations:
260² = 12² - 3² * 4² + 5² * 6² * 78² / 9² = 12² / 3² - 4² + 5² * 6² * 78² / 9²
  = 12² / 3² / 4² * 5² * 6² * 78² / 9² = 12² / 3² * 45² / 6² * 78² / 9²
  = - 12² + 3² * 4² + 5² * 6² * 78² / 9² = - 12² / 3² + 4² + 5² * 6² * 78² / 9².

(1 + 2 + ... + 11)(12 + 13 + ... + 164)(165 + 166 + ... + 260) = 134640²,
(1 + 2 + ... + 40)(41 + 42 + ... + 190)(191 + 192 + ... + 260) = 473550²,
(1 + 2 + ... + 90)(91 + 92 + ... + 259)(260) = 177450²,
(1 + 2 + ... + 116)(117 + 118 + ... + 260) = 13572²,
(1 + 2 + ... + 144)(145 + 146 + ... + 260) = 15660².

260² = 67600 appears in the decimal expression of π:
  π = 3.14159•••67600••• (from the 11097th digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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261

The smallest squares containing k 261's :
261121 = 511²,
26126196496 = 161636²,
261659126126161 = 16175881².

261² = 68121, a zigzag square.

18^k + 84^k + 129^k + 210^k are squares for k = 1,2,3 (21², 261², 3465²).
2958^k + 7134^k + 24708^k + 33321^k are squares for k = 1,2,3 (261², 42195², 7243533²).

261² = 68121, 6 + 8 + 1 + 21 = 6²,
261² = 68121, 68 + 12 + 1 = 9².

261² = (59 + 60 + 61)² + (62 + 63 + 64)².

(1³ + 2³ + 3⁵) + (4³ + 5³ + 6⁵) + (7³ + 8³ + 9⁵) = 261².

3-by-3 magic squares consisting of different squares with constant 261²:

(1 + 2 + ... + 11)(12 + 13 + ... + 63)(64 + 65 + ... + 261) = 64350²,
(1 + 2 + ... + 11)(12 + 13 + ... + 186)(187 + 188 + ... + 261) = 138600²,
(1 + 2 + ... + 13)(14 + 15 + ... + 63)(64 + 65 + ... + 261) = 75075²,
(1 + 2 + ... + 13)(14 + 15 + ... + 79)(80 + 81 + ... + 261) = 93093²,
(1 + 2 + ... + 24)(25 + 26 + ... + 63)(64 + 65 + ... + 261) = 128700²,
(1 + 2 + ... + 26)(27)(28 + 29 + ... + 261) = 17901²,
(1 + 2 + ... + 27)(28 + 29 + ... + 170)(171 + 172 + ... + 261) = 324324²,
(1 + 2 + ... + 44)(45 + 46 + ... + 74)(75 + 76 + ... + 261) = 235620²,
(1 + 2 + ... + 44)(45 + 46 + ... + 99)(100 + 101 + ... + 261) = 338580²,
(1 + 2 + ... + 76)(77 + 78 + ... + 118)(119 + 120 + ... + 261) = 570570²,
(1 + 2 + ... + 90)(91 + 92 + ... + 118)(119 + 120 + ... + 261) = 570570²,
(1 + 2 + ... + 90)(91 + 92 + ... + 156)(157 + 158 + ... + 261) = 855855²,
(1 + 2 + ... + 90)(91 + 92 + ... + 233)(234 + 235 + ... + 261) = 810810²,
(1 + 2 + ... + 98)(99)(100 + 101 + ... + 261) = 118503²,
(1 + 2 + ... + 128)(129 + 130 + ... + 258)(259 + 260 + 261) = 402480².

261² = 68121 appears in the decimal expressions of π:
  π = 3.14159•••68121••• (from the 6983rd digit),
  e = 2.71828•••68121••• (from the 69625th digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

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262

The smallest squares containing k 262's :
26244 = 162²,
262926225 = 16215²,
172726226226225 = 13142535².

262² = 68644, a square with 3 kinds of even digits.

262² = 68644, 6 - 8 + 6 * 44 = 262.

262² = 4³ + 21³ + 39³.

113^k + 124^k + 262^k + 590^k are squares for k = 1,2,3 (33², 667², 15057²).

262² = 3⁰ + 3² + 3³ + 3⁴ + 3⁶ + 3⁷ + 3⁸ + 3¹⁰.

Komachi Square Sum : 262² = 1² + 8² + 53² + 69² + 247²7² + 8² + 53² + 61² + 249².

262² = 68644, 6 + 8 + 6 + 44 = 8²,
262² = 68644, 6 + 86 + 4 + 4 = 10².

262² + 263² + 264² + ... + 1967² = 50327².

(1)(2 + ... + 97)(98 + 99 + ... + 262) = 11880²,
(1 + 2 + ... + 32)(33 + 34 + ... + 87)(88 + 89 + ... + 262) = 231000²,
(1 + 2 + ... + 32)(33 + 34 + ... + 97)(98 + 99 + ... + 262) = 257400²,
(1 + 2 + ... + 33)(34 + 35 + ... + 77)(78 + 79 + ... + 262) = 207570²,
(1 + 2 + ... + 80)(81 + 82 + ... + 115)(116 + 117 + ... + 262) = 555660²,
(1 + 2 + ... + 216)(217)(218 + 219 + ... + 262) = 234360².

262² = 68644 appears in the decimal expressions of π:
  π = 3.14159•••68644••• (from the 52199th digit),
  e = 2.71828•••68644••• (from the 468th digit),
  (68644 is the fourth 5-digit square in the expression of e.)

Page of Squares : First Upload October 25, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

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263

The smallest squares containing k 263's :
263169 = 513²,
12632636025 = 112395²,
2630263232632464 = 51286092².

263² = 69169, a square with 3 kinds of digits.

Komachi equations:
263² = 1234 * 56 + 7 * 8 + 9 = 1234 * 56 - 7 + 8 * 9.

3-by-3 magic squares consisting of different squares with constant 263²:

(1 + 2 + ... + 9)(10 + 11 + ... + 256)(257 + 258 + ... + 263) = 51870²,
(1 + 2 + ... + 13)(14 + 15 + ... + 256)(257 + 258 + ... + 263) = 73710²,
(1 + 2 + ... + 18)(19 + 20 + 21)(22 + 23 + ... + 263) = 18810²,
(1 + 2 + ... + 18)(19 + 20 + ... + 170)(171 + 172 + ... + 263) = 222642²,
(1 + 2 + ... + 44)(45 + 46 + ... + 76)(77 + 78 + ... + 263) = 246840²,
(1 + 2 + ... + 120)(121 + 122 + ... + 165)(166 + 167 + ... + 263) = 990990²,
(1 + 2 + ... + 135)(136 + 137 + ... + 170)(171 + 172 + ... + 263) = 996030²,
(1 + 2 + ... + 188)(189 + 190 + ... + 216)(217 + 218 + ... + 263) = 1065960².

Page of Squares : First Upload October 25, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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264

The smallest squares containing k 264's :
43264 = 208²,
264452644 = 16262²,
264134264875264 = 16252208².

The squares which begin with 264 and end in 264 are
26473893264 = 162708²,   264409867264 = 514208²,   264496261264 = 514292²,
264924325264 = 514708²,   2641301043264 = 1625208²,...

264² is the 9 th square which is the sum of 8 fifth powers : (2,2,4,4,4,4,8,8).

264² = 10³ + 24³ + 38³.

264² = 69696, a zigzag and palindromic square.

264² = 69696 is a square consisting of only 2 kinds of digits.

264² = 69696.

The integral triangle of sides 130, 2057, 2169 has square area 264².

Cubic Polynomial :
(X + 264²)(X + 864²)(X + 1273²) = (X³ + 1561²X² + 1172472²X + 290366208²).

Komachi equation: 264² = 1² * 2² + 3² + 45² * 6² - 7² * 8² - 9².

264² = 69696, 6 + 9 + 6 + 9 + 6 = 6²,
264² = 69696, 69 + 69 + 6 = 12².

(1 + 2)(3)(4 + 5 + 6 + ... + 124) = 264²,
(1 + 2 + ... + 11)(12 + 13 + ... + 264) = 1518²,
(1 + 2 + ... + 56)(57)(58 + 59 + ... + 264) = 55062²,
(1 + 2 + ... + 171)(172)(173 + 174 + ... + 264) = 225492².

(1³ + 2³ + ... + 23³)(24³)(25³ + 26³ + ... + 264³) = 1135088640²,
(1³ + 2³ + ... + 54³)(55³ + 56³ + ... + 99³)(100³ + 101³ + ... + 264³) = 242817324750²,
(1³ + 2³ + ... + 60³)(61³ + 62³ + ... + 164³)(165³ + 166³ + ... + 264³) = 791350560000².

Page of Squares : First Upload October 25, 2004 ; Last Revised September 27, 2011
by Yoshio Mimura, Kobe, Japan

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265

The smallest squares containing k 265's :
26569 = 163²,
4265265481 = 65309²,
1265162653265625 = 35569125².

(191 / 265)² = 0.519487362... (Komachic).

265² is the 5th square which is the sum 5 fifth powers : 2⁵ + 3⁵ + 5⁵ + 6⁵ + 9⁵.

265² = 27² + 29² + 31² + 33² + 35² + 37² + 39² + ... + 75².

20^k + 260^k + 265^k + 680^k are squares for k = 1,2,3 (35², 775², 18725²).
8374^k + 11289^k + 22366^k + 28196^k are squares for k = 1,2,3 (265², 38637², 5969125²).

Komachi Fractions : 243 / 1896075 = 486 / 3792150 = (3 / 265)²,
  432 / 1896075 = 864/3792150 = (4 / 265)².

265² = 70225, 7 + 0 + 2 + 2 + 5 = 4²,
265² = 70225, 7³ + 0³ + 2³ + 2³ + 5³ = 22².

265³ - 264³ + 263³ - 262³ + 261³ - 260³ + ... + 1³ = 3059².

3-by-3 magic squares consisting of different squares with constant 265²:

(1 + 2 + ... + 49)(50 + 51 + ... + 238)(239 + 240 + ... + 265) = 476280²,
(1 + 2 + ... + 164)(165 + 166 + ... + 185)(186 + 187 + ... + 265) = 947100².

265² = 70225 appears in the decimal expression of e:
  e = 2.71828•••70225••• (from the 13051st digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

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266

The smallest squares containing k 266's :
266256 = 516²,
2662663201 = 51601²,
1113266266266025 = 33365645².

266² is the third square which is the sum of 7 sixth powers : (2,4,4,5,5,5,5).

266² = 70756, a zigzag square.

266² = (2² + 3)(4² + 3)(23² + 3) = (4² + 3)(61² + 3).

266^k + 6232^k + 10070^k + 12673^k are squares for k = 1,2,3 (171², 17347², 1816191²).
106^k + 266^k + 814^k + 1314^k are squares for k = 1,2,3 (50², 1572², 53180²).
22^k + 161^k + 266^k + 280^k are squares for k = 1,2,3 (27², 419², 6705²).
62^k + 73^k + 266^k + 440^k are squares for k = 1,2,3 (29², 523², 10229²).
10450^k + 14098^k + 19342^k + 26866^k are squares for k = 1,2,3 (266², 37468², 5529076²).

Komachi equation: 266² = 1² * 2² * 3² * 456² * 7² / 8² / 9².

266² = 70756, 7 + 0 + 7 + 5 + 6 = 5².

(1 + 2 + ... + 5)(6 + 7 + ... + 23)(24 + 25 + ... + 266) = 11745²,
(1 + 2 + ... + 10)(11 + 12 + ... + 21)(22 + 23 + ... + 266) = 18480²,
(1 + 2 + ... + 10)(11 + 12 + ... + 253)(254 + 255 + ... + 266) = 77220²,
(1 + 2 + ... + 22)(23 + 24 + ... + 32)(33 + 34 + ... + 266) = 49335²,
(1 + 2 + ... + 22)(23 + 24 + ... + 184)(185 + 186 + ... + 266) = 280071²,
(1 + 2 + ... + 26)(27 + 28 + ... + 71)(72 + 73 + ... + 266) = 159705²,
(1 + 2 + ... + 31)(32 + 33 + ... + 61)(62 + 63 + ... + 266) = 152520²,
(1 + 2 + ... + 31)(32 + 33 + ... + 136)(137 + 138 + ... + 266) = 338520²,
(1 + 2 + ... + 66)(67 + 68)(69 + 70 + ... + 266) = 99495²,
(1 + 2 + ... + 87)(88 + 89 + ... + 255)(256 + 257 + ... + 266) = 562716²,
(1 + 2 + ... + 98)(99 + 100 + ... + 244)(245 + 246 + ... + 266) = 826287²,
(1 + 2 + ... + 147)(148 + 149 + ... + 192)(193 + 194 + ... + 266) = 1188810².

266² = 70756 appears in the decimal expression of π:
  π = 3.14159•••67600••• (from the 34676th digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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267

The smallest squares containing k 267's :
126736 = 356²,
4092672676 = 63974²,
2220626702672676 = 47123526².

267² = 71289, a square with different square.

890^k + 12104^k + 22873^k + 35422^k are squares for k = 1,2,3 (267², 43877², 7627923²).

Komachi Fraction : 675 / 1924803 = (5 / 267)².

Komachi equations:
267² = 1² * 2² * 3² * 4² / 56² * 7² * 89² = 1² / 2² * 3² / 4² * 56² / 7² * 89².

267² = 71289, 7 + 12 + 8 + 9 = 6²,
267² = 71289, 7 + 128 + 9 = 12²,
267² = 71289, 7 + 1289 = 36²,
267² = 71289, 712 + 8 + 9 = 27².

3-by-3 magic squares consisting of different squares with constant 267²:

(1 + 2 + ... + 35)(36 + 37 + ... + 180)(181 + 182 + ... + 267) = 438480².

267² = 71289 appears in the decimal expressions of π:
  π = 3.14159•••71289••• (from the 38984th digit),
  e = 2.71828•••71289••• (from the 38677th digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

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268

The smallest squares containing k 268's :
26896 = 164²,
268402689 = 16383²,
202682687268889 = 14236667².

1 / 268 = 0.00373134328358...,
3² + 7² + 3² + 1² + 3² + 4² + 3² + 2² + 8² + 3² + 5² + 8² = 268.

268² = 71824, is a zigzag square with different digits.

268² = 71824, 7 + 18 + 24 = 7².

(1)(2 + 3 + ... + 25)(26 + 27 + ... + 268) = 3402²,
(1 + 2 + 3)(4 + 5 + ... + 115)(116 + 117 + ... + 268) = 34272²,
(1 + 2 + ... + 4)(5 + 6 + ... + 28)(29 + 30 + ... + 268) = 11880²,
(1 + 2 + ... + 8)(9 + 10 + ... + 25)(26 + 27 + ... + 268) = 19278²,
(1 + 2 + ... + 8)(9 + 10 + ... + 89)(90 + 91 + ... + 268) = 67662²,
(1 + 2 + ... + 15)(16 + 17 + ... + 55)(56 + 57 + ... + 268) = 76680²,
(1 + 2 + ... + 16)(17 + 18 + ... + 88)(89 + 90 + ... + 268) = 128520²,
(1 + 2 + ... + 16)(17 + 18 + ... + 115)(116 + 117 + ... + 268) = 161568²,
(1 + 2 + ... + 16)(17 + 18 + ... + 187)(188 + 189 + ... + 268) = 209304²,
(1 + 2 + ... + 26)(27 + 28 + ... + 91)(92 + 93 + ... + 268) = 207090²,
(1 + 2 + ... + 32)(33 + 34 + ... + 92)(93 + 94 + ... + 268) = 250800²,
(1 + 2 + ... + 53)(54 + 55 + ... + 106)(107 + 108 + ... + 268) = 429300²,
(1 + 2 + ... + 56)(57 + 58 + ... + 151)(152 + 153 + ... + 268) = 622440²,
(1 + 2 + ... + 111)(112 + 113 + ... + 231)(232 + 233 + ... + 268) = 1087800²,
(1 + 2 + ... + 116)(117 + 118 + ... + 260)(261 + 262 + ... + 268) = 624312²,
(1 + 2 + ... + 123)(124 + 125 + ... + 227)(228 + 229 + ... + 268) = 1189656²,
(1 + 2 + ... + 144)(145 + 146 + ... + 260)(261 + 262 + ... + 268) = 720360²,
(1 + 2 + ... + 146)(147 + 148 + ... + 219)(220 + 221 + ... + 268) = 1309182²,
(1 + 2 + ... + 160)(161 + 162 + ... + 214)(215 + 216 + ... + 268) = 1304100²,
(1 + 2 + ... + 168)(169 + 170 + ... + 263)(264 + 265 + ... + 268) = 622440²,
(1 + 2 + ... + 242)(243)(244 + 245 + ... + 268) = 213840².

(1² + 2² + ... + 15²)(16² + 17² + ... + 48²)(49² + 50² + ... + 268²) = 17104560².

268² = 71824 appears in the decimal expression of π:
  π = 3.14159•••71824••• (from the 60982nd digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised Jue 19, 2006
by Yoshio Mimura, Kobe, Japan

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269

The smallest squares containing k 269's :
269361 = 519²,
69926926969 = 264437²,
269226926994496 = 16408136².

269² = 72361, a square with different digits.

269² = 72361, 723 + 61 = 28².

269² + 270² + 271² + ... + 1885² = 47201².

Komachi equations:
269² = 12³ / 3³ / 4³ - 5³ - 67³ + 8³ * 9³ = - 12³ - 3³ - 4³ - 5³ + 6³ * 7³ - 8³ + 9³.

3-by-3 magic squares consisting of different squares with constant 269²:

(1 + 2 + ... + 24)(25 + 26 + ... + 45)(46 + 47 + ... + 269) = 88200²2,
(1 + 2 + ... + 25)(26 + 27 + ... + 35)(36 + 37 + ... + 269) = 59475²,
(1 + 2 + ... + 49)(50 + 51 + ... + 115)(116 + 117 + ... + 269) = 444675²,
(1 + 2 + ... + 52)(53 + 54 + ... + 207)(208 + 209 + ... + 269) = 640770²,
(1 + 2 + ... + 153)(154 + 155 + ... + 252)(253 + 254 + ... + 269) = 1024947²,
(1 + 2 + ... + 242)(243 + 244 + ... + 269) = 14256².

269² = 72361 appears in the decimal expressions of π:
  π = 3.14159•••72361••• (from the 52190th digit),
  e = 2.71828•••72361••• (from the 38677th digit).

Page of Squares : First Upload October 25, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan

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