---

250

The smallest squares containing k 250's :
2500 = 50²,
622502500 = 24950²,
250250062500 = 500250².

250² = 5⁶ + 5⁶ + 5⁶ + 5⁶ is the sixth squares which is the sum of 4 sixth powers.

250² = (1² + 9)(79² + 9).

(1 + 2 + ... + 13)(14 + 15 + ... + 211)(212 + 213 + ... + 250) = 135135².

Komachi equation: 250² = - 9² * 8² + 7² + 65² * 4² + 3² * 2² - 1².

250² = 62500 appears in the decimal expressions of π and e:
  π = 3.14159•••62500••• (from the 75980th digit),
  e = 2.71828•••62500••• (from the 37540th digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

251

The smallest squares containing k 251's :
251001 = 501²,
251825161 = 15869²,
12512512513809 = 3537303².

251² = 4³ + 30³ + 33³.

Komachi Fraction : 504 / 7938126 = (2 / 251)².

251² = 63001, 63 + 0 + 0 + 1 = 8².

3-by-3 magic squares consisting of different squares with constant 251²:

(1 + 2 + ... + 23)(24 + 25 + ... + 185)(186 + 187 + ... + 251) = 259578²,
(1 + 2 + ... + 33)(34 + 35 + ... + 153)(154 + 155 + ... + 251) = 353430²,
(1 + 2 + ... + 48)(49 + 50 + ... + 111)(112 + 113 + ... + 251) = 388080²,
(1 + 2 + ... + 98)(99 + 100 + ... + 132)(133 + 134 + ... + 251) = 659736².

251² = 63001 appears in the decimal expression of e:
  e = 2.71828•••63001••• (from the 26394th digit)

Page of Squares : First Upload October 18, 2004 ; Last Revised January 20, 2009
by Yoshio Mimura, Kobe, Japan

---

252

The smallest squares containing k 252's :
25281 = 159²,
16252425225 = 127485²,
31725225225225 = 5632515².

252² = 63504, a zigzag square with different digits.

252² = (1³ + 1)(2³ + 1)(3³ + 1)(5³ + 1).

252² = 4⁴ + 8⁴ + 12⁴ + 14⁴ = 6⁴ + 12⁴ + 12⁴ + 12⁴.

The integral triangle of sides 305, 424, 567 (or 337, 441, 680) has square area 252².

Cubic Polynomial : (X + 192²)(X + 252²)(X + 301²) = X³ + 437²X² + 106932²X + 14563584².

Komachi equyations:
252² = 12 * 3 * 4 * 56 * 7 / 8 * 9,
252² = 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 / 5² / 4² * 3² * 2² * 10²
  = 9² / 8² * 7² * 6² / 5² * 4² / 3² * 2² * 10² = 98² / 7² * 6² * 5² * 4² * 3² / 2² / 10²
  = 98² / 7² * 6² / 5² / 4² * 3² * 2² * 10².

(1)(2)(3 + 4 + 5)(6)(7)(8 + 9 + 10 + 11 + 12 + 13) = 252²,
(1 + 2)(3 + 4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12 + 13 + 14 + 15) = 252²,
(1 + 2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15) = 252²,
(1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12)(13 + 14 + 15) = 252²,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15) = 252²,
(1)(2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)(12)(13 + 14 + 15) = 252²,
(1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + 9)(10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18) = 252².

(1 + 2 + ... + 104)(105 + 106 + ... + 189)(190 + 191 + ... + 252) = 974610².

252² = 63504 appears in the decimal expression of e:
  e = 2.71828•••63504••• (from the 26394th digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

253

The smallest squares containing k 253's :
125316 = 354²,
2535525316 = 50354²,
102536625325369 = 10126037².

253² = 64009, 6400 = 80², 9 = 3²,
253² = 64009, 64 = 8², 9 = 3².

253² = 1³ + 2³ + 40³ = 12³ + 25³ + 36³.

253² + 254² + 255² + ... + 264² = 265² + 266² + 267² + ... + 275².

1265^k + 7084^k + 16192^k + 39468^k are squares for k = 1,2,3 (253², 43263², 8129143²).
253^k + 295^k + 341^k + 407^k are squares for k = 1,2,3 (36², 658², 12204²).

Komachi Fractions : 3249 / 576081 = (19 / 253)².

253² = 27² + 28² + 29² + ... + 59².

253² + 254² + 255² + ... + 79213² = 12871767²,
253² + 254² + 255² + ... + 806² = 13019².

3-by-3 magic squares consisting of different squares with constant 253²:

(1)(2 + 3 + ... + 19)(20 + 21 + ... + 253) = 2457²,
(1)(2 + 3 + ... + 61)(62 + 63 + ... + 253) = 7560²,
(1 + 2 + ... + 6)(7 + 8 + ... + 19)(20 + 21 + ... + 253) = 10647²,
(1 + 2 + ... + 6)(7 + 8 + ... + 84)(85 + 86 + ... + 253) = 46137²,
(1 + 2 + ... + 6)(7 + 8 + ... + 201)(202 + 203 + ... + 253) = 70980²,
(1 + 2 + ... + 15)(16 + 17 + ... + 19)(20 + 21 + ... + 253) = 16380²,
(1 + 2 + ... + 15)(16 + 17 + ... + 201)(202 + 203 + ... + 253) = 169260²,
(1 + 2 + ... + 20)(21 + 22 + ... + 61)(62 + 63 + ... + 253) = 103320²,
(1 + 2 + ... + 20)(21 + 22 + ... + 84)(85 + 86 + ... + 253) = 141960²,
(1 + 2 + ... + 49)(50 + 51 + ... + 97)(98 + 99 + ... + 253) = 343980²,
(1 + 2 + ... + 53)(54 + 55 + ... + 106)(107 + 108 + ... + 253) = 400680²,
(1 + 2 + ... + 108)(109)(110 + 111 + ... + 253) = 129492²,
(1 + 2 + ... + 134)(135 + 136 + ... + 201)(202 + 203 + ... + 253) = 1097460²,
(1 + 2 + ... + 147)(148 + 149 + ... + 227)(228 + 229 + ... + 253) = 1010100².

253² = 64009 appears in the decimal expression of π:
  π = 3.14159•••64009••• (from the 12272nd digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

---

254

The smallest squares containing k 254's :
12544 = 112²,
42547725441 = 206271²,
254254792546161 = 15945369².

254² = 64516, a zigzag square.

254² = 64516, 6 * 45 - 16 = 254.

52^k + 190^k + 233^k + 254^k are squares for k = 1,2,3 (27², 397², 6003²).
254^k + 362^k + 394^k + 590^k are squares for k = 1,2,3 (40², 836², 18176²).

254² = 64516, 64 + 51 + 6 = 11².

(1 + 2 + ... + 9)(10 + 11 + ... + 34)(35 + 36 + ... + 254) = 28050².

(1³ + 2³ + ... + 20³)(21³ + 22³ + ... + 174³)(175³ + 176³ + ... + 254³) = 91378287000².

254² = 64516 appears in the decimal expressions of π and e:
  π = 3.14159•••64516••• (from the 33739th digit),
  e = 2.71828•••64516••• (from the 142420th digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

---

255

The smallest squares containing k 255's :
255025 = 505²,
255150255376 = 505124²,
2255371525525504 = 47490752².

Komachi Square Sum : 255² = 53² + 74² + 96² + 218².

3366^k + 7446^k + 15708^k + 38505^k are squares for k = 1,2,3 (255², 42381², 7836813²).

255² = 65025, 6 + 5 + 0 + 25 = 6²,
255² = 65025, 6 + 50 + 25 = 9².

255² = 14³ + 25³ + 36³.

255² = 40³ + 4⁵ + 1⁷.

255² + 256² + 257² + ... + 3279² = 108405²,
255² + 256² + 257² + ... + 783² = 12443²,
255² + 256² + 257² + ... + 976² = 17461².

3-by-3 magic squares consisting of different squares with constant 255²:

(1 + 2 + ... + 50)(51) = 255².

(1 + 2 + ... + 24)(25 + 26 + ... + 129)(130 + 131 + ... + 255) = 242550²,
(1 + 2 + ... + 44)(45 + 46 + ... + 244)(245 + 246 + ... + 255) = 280500²,
(1 + 2 + ... + 55)(56 + 57 + ... + 244)(245 + 246 + ... + 255) = 346500²,
(1 + 2 + ... + 99)(100 + 101 + ... + 164)(165 + 166 + ... + 255) = 900900²,
(1 + 2 + ... + 108)(109 + 110 + ... + 180)(181 + 182 + ... + 255) = 1000620².

255² = 65025 appears in the decimal expression of π:
  π = 3.14159•••65025••• (from the 76891st digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised February 23, 2011
by Yoshio Mimura, Kobe, Japan

---

256

The square of 16.

The smallest squares containing k 256's :
256 = 16²,
64256256 = 8016²,
25492562568256 = 5049016².

The squares which begin with 256 and end in 256 are
256512256 = 16016²,   25605120256 = 160016²,   256019808256 = 505984²,
256052192256 = 506016²,   256526042256 = 506484²,...

1 / 256 = 0.00390625, 39025 = 625².

256 = 16², a square with different and increasing digits.

256² = 65536, a square with 3 kinds of digits.

256² = (3² + 7)(5² + 7)(11² + 7).

256 is an exchangeable square : 625 = 25².

Komachi Fraction : 729 / 5308416 = (3 / 256)².

Komachi equations:
256² = 98 + 7 + 65432 - 1,
256² = 12⁴ * 3⁴ * 4⁴ * 56⁴ / 7⁴ / 8⁴ / 9⁴ = 12⁴ * 3⁴ * 4⁴ / 56⁴ * 7⁴ * 8⁴ / 9⁴,
256² = 12⁵ * 3⁵ / 4⁵ + 56⁵ / 7⁵ + 8⁵ - 9⁵ = - 12⁵ * 3⁵ / 4⁵ + 56⁵ / 7⁵ + 8⁵ + 9⁵.

256² = 65536, 6 + 5 + 5 + 3 + 6 = 5².

(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 256) = 7140²,
(1 + 2 + ... + 6)(7 + 8 + ... + 32)(33 + 34 + ... + 256) = 18564²,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + 34 + ... + 256) = 31416²,
(1 + 2 + ... + 11)(12 + 13 + ... + 212)(213 + 214 + ... + 256) = 123816²,
(1 + 2 + ... + 18)(19 + 20 + ... + 151)(152 + 153 + ... + 256) = 203490²,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + 34 + ... + 256) = 47124²,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + 34 + ... + 256) = 42840²,
(1 + 2 + ... + 32)(33 + 34 + ... + 47)(48 + 49 + ... + 256) = 100320²,
(1 + 2 + ... + 32)(33 + 34 + ... + 175)(176 + 177 + ... + 256) = 370656²,
(1 + 2 + ... + 35)(36 + 37 + ... + 100)(101 + 102 + ... + 256) = 278460²,
(1 + 2 + ... + 38)(39 + 40 + ... + 94)(95 + 96 + ... + 256) = 280098²,
(1 + 2 + ... + 38)(39 + 40 + ... + 104)(105 + 106 + ... + 256) = 309738²,
(1 + 2 + ... + 52)(53 + 54 + ... + 220)(221 + 222 + ... + 256) = 520884²,
(1 + 2 + ... + 55)(56 + 57 + ... + 139)(140 + 141 + ... + 256) = 540540²,
(1 + 2 + ... + 86)(87)(88 + 89 + ... + 256) = 97266².

256² = 65536 appears in the decimal expression of e:
  e = 2.71828•••65536••• (from the 9993rd digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

257

The smallest squares containing k 257's :
257049 = 507²,
3625725796 = 60214²,
325792572575625 = 18049725².

257² is the third square which is the sum of 9 seventh powers.

257² = 1⁴ + 4⁴ + 4⁴ + 16⁴ = 1⁸ + 2⁸ + 2⁸ + 4⁸.

257² = 66049, 6 + 6 + 0 + 4 + 9 = 5²,
257² = 66049, 6² + 6² + 0² + 4² + 9² = 13²,
257² = 66049, 6³ + 6³ + 0³ + 4³ + 9³ = 35²,
257² = 66049, 6^k + 6^k + 0^k + 4^k + 9^k = (2^k + 3^k)², in general.

3-by-3 magic squares consisting of different squares with constant 257²:

(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 257) = 3780²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 142)(143 + 144 + ... + 257) = 48300²,
(1 + 2 + ... + 26)(27 + 28 + ... + 117)(118 + 119 + ... + 257) = 245700²,
(1 + 2 + ... + 45)(46 + 47 + ... + 110)(111 + 112 + ... + 257) = 376740²,
(1 + 2 + ... + 140)(141 + 142 + ... + 210)(211 + 212 + ... + 257) = 1154790².

(1³ + 2³ + ... + 43³)(44³ + 45³ + ... + 85³)(86³ + 87³ + ... + 257³) = 110049773064²,
(1³ + 2³ + ... + 82³)(83³ + 84³ + ... + 122³)(123³ + 124³ + ... + 257³) = 734839736400².

257² = 66049 appears in the decimal expressions of π and e:
  π = 3.14159•••66049••• (from the 4830th digit),
  (66049 is the ninth 5-digit square in the expression of π.)
  e = 2.71828•••66049••• (from the 10723th digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised January 20, 2009
by Yoshio Mimura, Kobe, Japan

---

258

The smallest squares containing k 258's :
258064 = 508²,
25823525809 = 160697²,
125817925825881 = 11216859².

258² = (5³ + 4)(8³ + 4).

258² is the fourth square which is the sum of 9 eighth powers.

258² = 66564, a square with 3 kinds of digits.

258² = 66564 is an exchangeable square, 46656 = 216².

6^k + 12^k + 36^k + 129^k + 258^k is a square for k = 1,2,3,4 (21², 291², 4401², 68625²).

258² + 259² + 260² + ... + 906² = 15576².

154^k + 222^k + 258^k + 810^k are squares for k = 1,2,3 (38², 892², 23732²).
186^k + 234^k + 258^k + 478^k are squares for k = 1,2,3 (34², 620², 12068²).
46^k + 170^k + 202^k + 258^k are squares for k = 1,2,3 (26², 372², 5516²).
8170^k + 9374^k + 20210^k + 28810^k are squares for k = 1,2,3 (258², 37324², 5791068²).

258² = 66564, 6 + 6 + 5 + 64 = 9²,
258² = 66564, 6 + 6 + 564 = 24²,
258² = 66564, 6 + 65 + 6 + 4 = 9²,
258² = 66564, 66 + 5 + 6 + 4 = 9²,
258² = 66564, 665 + 64 = 27².

(1 + 2 + 3 + 4)(5 + 6 + ... + 249)(250 + 251 + ... + 258) = 26670²,
(1 + 2 + 3;4 + 5)(6 + 7 + ... + 93)(94 + 95 + ... + 258) = 43560²,
(1 + 2 + ... + 8)(9 + 10 + ... + 23)(24 + 25 + ... + 258) = 16920²,
(1 + 2 + ... + 8)(9 + 10 + ... + 93)(94 + 95 + ... + 258) = 67320²,
(1 + 2 + ... + 10)(11 + 12 + ... + 253)(254 + 255 + ... + 258) = 47520²,
(1 + 2 + ... + 18)(19 + 20 + ... + 30)(31 + 32 + ... + 258) = 40698²,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + ... + 258) = 56430²,
(1 + 2 + ... + 42)(43 + 44 + ... + 258) = 5418²,
(1 + 2 + ... + 45)(46 + 47 + ... + 201)(202 + 203 + ... + 258) = 511290²,
(1 + 2 + ... + 50)(51 + 52 + ... + 102)(103 + 104 + ... + 258) = 377910²,
(1 + 2 + ... + 76)(77 + 78 + ... + 209)(210 + 211 + ... + 258) = 798798²,
(1 + 2 + ... + 121)(122 + 123 + ... + 182)(183 + 184 + ... + 258) = 1070916²,
(1 + 2 + ... + 208)(209)(210 + 211 + ... + 258) = 228228².

(1³ + 2³ + ... + 20³)(21³ + 22³ + ... + 41³)(42³ + 43³ + ... + 258³) = 5856656400².

258² = 66564 appears in the decimal expressions of π and e:
  π = 3.14159•••66564••• (from the 10164th digit),
  e = 2.71828•••66564••• (from the 26890th digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

259

The smallest squares containing k 259's :
25921 = 161²,
4259259169 = 65263²,
125926259125969 = 11221687².

259² = 67081, a zigzag square with different digits.

1² + 2² + 3² + 4² + 5² + ... + 259² = 5824910, different digits.

Komachi Fraction : 5184 / 603729 = (24 / 259)².

Komachi Square Sum : 259² = 1² + 5² + 6² + 7² + 89² + 243² = 1² + 6² + 7² + 9² + 83² + 245².

1397² = 227² + 228² + 229² + 230² + 231² + ... + 259².

3-by-3 magic squares consisting of different squares with constant 259²:

(1 + 2 + ... + 31)(32 + 33 + ... + 112)(113 + 114 + ... + 259) = 281232²,
(1 + 2 + ... + 141)(142 + 143 + ... + 212)(213 + 214 + ... + 259) = 1181298²,
(1 + 2 + ... + 168)(169 + 170 + ... + 181)(182 + 183 + ... + 259) = 745290².

(1² + 2² + ... + 6²)(7² + 8² + ... + 259²) = 23023².

(1³ + 2³ + ... + 224³)(225³ + 226³ + ... + 259³) = 562716000².

259² = 67081 appears in the decimal expression of e:
  e = 2.71828•••67081••• (from the 14323rd digit).

Page of Squares : First Upload October 18, 2004 ; Last Revised January 20, 2009
by Yoshio Mimura, Kobe, Japan

---