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230

The smallest squares containing k 230's :
2304 = 48²,
23067230641 = 151879²,
162308823042304 = 12740048².

230² = 11³ + 17³ + 36³.

230³ = 12167000, 12² + 1² + 6² + 7² + 0² + 0² + 0² = 230.

230² = 36 x 37 + 37 x 38 + 38 x 39 + ... + 58 x 59.

230^k + 610^k + 665^k + 2720^k are squares for k = 1,2,3 (65², 2875², 143725²).
7498^k + 13662^k + 15778^k + 15962^k are squares for k = 1,2,3 (230², 27324², 3311540²).

Komachi equation: 230² = 9² * 8² + 7² * 6² * 5² + 4² + 3² * 2² * 10².

(1)(2 + 3 + ... + 30)(31 + 32 + ... + 230) = 3480²,
(1 + 2 + ... + 9)(10 + 11 + ... + 94)(95 + 96 + ... + 230) = 66300²,
(1 + 2 + ... + 49)(50 + 51 + ... + 174)(175 + 176 + ... + 230) = 441000²,
(1 + 2 + ... + 75)(76 + 77 + ... + 228)(229 + 230) = 174420².

Page of Squares : First Upload September 22, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

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231

The smallest squares containing k 231's :
23104 = 152²,
2231523121 = 47239²,
8231952316231449 = 90730107².

231 = (1² + 2² + 3² + ... + 27²) / (1² + 3² + 4²).

231² = 46² + 82² + 211² : 112² + 28² + 64² = 132².

154^k + 8624^k + 17633^k + 26950^k are squares for k = 1,2,3 (231², 33341², 5069295²).

Komachi equation: 231² = 1 / 2 * 3 * 456 * 78 + 9.

3-by-3 magic squares consisting of different squares with constant 231²:

(1)(2 + 3 + 4)(5 + 6)(7)(8 + 9 + 10 + 11 + 12 + 13 + 14) = 231².

(1 + 2 + ... + 23)(24)(25 + 26 + ... + 231) = 13248²,
(1 + 2 + ... + 71)(72 + 73 + ... + 123)(124 + 125 + ... + 231) = 498420².

231² = 53361 appears in the decimal expressions of π and e:
  π = 3.14159•••53361••• (from the 88722nd digit),
  e = 2.71828•••53361••• (from the 146317th digit).

Page of Squares : First Upload September 22, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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232

The smallest squares containing k 232's :
12321 = 111²,
1232782321 = 35111²,
62329272322321 = 7894889².

232² = 53824, a zigzag square with different digits.

232² = (2² + 4)(82² + 4).

232² = 53824, 5 + 38 + 2 + 4 = 7².

232, 233, and 234 is the smallest triple of consecutive numbers each of which is the sum of 2 squares. (4 - tupple not exist).

4930^k + 11078^k + 15254^k + 22562^k are squares for k = 1,2,3 (232², 29812², 4063712²).

Komachi equations:
232² = 1² * 234² - 5² * 6² - 7² - 8² + 9² = 98² + 7² * 6² * 5² - 4² + 3² * 2² + 10².

137² + 138² + 139² + ... + 232² = 1828².

232² + 233² + 234² + 235² + ... + 1289² = 26657².

(1 + 2 + ... + 5)(6 + 7 + ... + 15)(16 + 17 + ... + 232) = 6510²,
(1 + 2 + ... + 5)(6 + 7 + ... + 82)(83 + 84 + ... + 232) = 34650²,
(1 + 2 + ... + 7)(8 + 9 + ... + 82)(83 + 84 + ... + 232) = 47250²,
(1 + 2 + ... + 7)(8 + 9 + ... + 217)(218 + 219 + ... + 232) = 47250²,
(1 + 2 + ... + 11)(12)(13 + 14 + ... + 232) = 4620²,
(1 + 2 + ... + 14)(15)(16 + 17 + ... + 232) = 6510²,
(1 + 2 + ... + 14)(15 + 16 + ... + 217)(218 + 219 + ... + 232) = 91350²,
(1 + 2 + ... + 45)(46 + 47 + ... + 207)(208 + 209 + ... + 232) = 341550²,
(1 + 2 + ... + 150)(151)(152 + 153 + ... + 232) = 163080².

(1³ + 2³ + ... + 24³)(25³ + 26³ + ... + 40³)(41³ + 42³ + ... + 232³) = 6185088000².

232² = 53824 appears in the decimal expressions of π and e:
  π = 3.14159•••53824••• (from the 5538th digit),
  (53824 is the tenth 5-digit square in the expression of π.)
  e = 2.71828•••53824••• (from the 75094th digit).

Page of Squares : First Upload September 22, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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233

The smallest squares containing k 233's :
233289 = 483²,
233233984 = 15272²,
1775233423323361 = 42133519².

233² = 54289, a square with different digits.

233² = 54289, 5 + 4 + 2 + 89 = 10²,
233² = 54289, 5 + 42 + 8 + 9 = 8².

52^k + 190^k + 233^k + 254^k are squares for k = 1,2,3 (27², 397², 6003²).
67^k + 209^k + 233^k + 275^k are squares for k = 1,2,3 (28², 422², 6548²).

Komachi Fraction : 972/1465803 = (6/233)².

Komachi equations: 233² = 98² + 7² * 6² * 5² + 4² * 3² + 21².

233² + 234² + 235² + 236² + ... + 10866² = 653991².

3-by-3 magic squares consisting of different squares with constant 233²:

(1 + 2 + ... + 8)(9 + 10 + ... + 233) = 990²,
(1 + 2 + ... + 11)(12 + 13 + ... + 173)(174 + 175 + ... + 233) = 109890²,
(1 + 2 + ... + 39)(40 + 41 + ... + 129)(130 + 131 + ... + 233) = 334620²,
(1 + 2 + ... + 44)(45 + 46 + ... + 198)(199 + 200 + ... + 233) = 374220²,
(1 + 2 + ... + 60)(61 + 62 + ... + 71)(72 + 73 + ... + 233) = 181170²,
(1 + 2 + ... + 63)(64 + 65 + ... + 123)(124 + 125 + ... + 233) = 471240²,
(1 + 2 + ... + 121)(122 + 123 + ... + 193)(194 + 195 + ... + 233) = 845460²,
(1 + 2 + ... + 135)(136 + 137 + ... + 225)(226 + 227 + ... + 233) = 523260².

233² = 54289 appears in the decimal expression of π
  π = 3.14159•••54289••• (from the 13057th digit).

Page of Squares : First Upload September 22, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

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234

The smallest squares containing k 234's :
23409 = 153²,
142346234944 = 377288²,
23421234523401 = 4839549².

234² = 54756, a zigzag square.

234² = 54756, 5 + 47 * 5 - 6 = 5 * 47 + 5 - 6 = 234.

234² = (2² + 9)(3² + 9)(15² + 9).

The integral triangle of sides 40, 8749, 8787 has square area 234².

186^k + 234^k + 258^k + 478^k are squares for k = 1,2,3 (34², 620², 12068²).
273^k + 8853^k + 18213^k + 27417^k are squares for k = 1,2,3 (234², 34086², 5229198²).
858^k + 9438^k + 17394^k + 27066^k are squares for k = 1,2,3 (234², 33540², 5092308²).

Komachi equations:
234² = 9 * 8 * 765 - 4 - 32 * 10,
234² = 9² * 8² * 7² / 6² + 5² * 4² * 3² + 210² = 98² / 7² * 6² + 5² * 4² * 3² + 210².

234² = 54756, 5 + 4756 = 69².

234² + 235² + 236² + 237² + ... + 1194² = 23746².

(1 + 2)(3 + 4 + ... + 15)(16 + 17 + ... + 23) = 234²,
(1 + 2)(3 + 4 + ... + 10)(11 + 12 + ... + 28) = 234²,
(1 + 2 + ... + 12)(13 + 14 + ... + 39) = 234²,
(1 + 2)(3 + 4 + ... + 6)(7 + 8 + ... + 45) = 234²,
(1 + 2)(3 + 4 + ... + 171)(172 + 173 + ... + 234) = 23751²,
(1 + 2 + 3)(4 + 5 + ... + 122)(123 + 124 + ... + 234) = 29988²,
(1 + 2 + ... + 17)(18 + 19 + ... + 33)(34 + 35 + ... + 234) = 41004²,
(1 + 2 + ... + 17)(18 + 19 + ... + 122)(123 + 124 + ... + 234) = 149940²,
(1 + 2 + ... + 27)(28 + 29 + ... + 126)(127 + 128 + ... + 234) = 237006²,
(1 + 2 + ... + 44)(45 + 46 + ... + 69)(70 + 71 + ... + 234) = 188100²,
(1 + 2 + ... + 44)(45 + 46 + ... + 230)(231 + 232 + ... + 234) = 153450²,
(1 + 2 + ... + 81)(82 + 83 + ... + 122)(123 + 124 + ... + 234) = 526932²,
(1 + 2 + ... + 110)(111 + 112 + ... + 209)(210 + 211 + ... + 234) = 732600².

(1³ + 2³ + ... + 26³)(27³ + 28³ + ... + 89³)(90³ + 91³ + ... + 234³) = 38091853800².

234² = 54756 appears in the decimal expressions of π and e:
  π = 3.14159•••54756••• (from the 10653rd digit),
  e = 2.71828•••54756••• (from the 21583rd digit).

Page of Squares : First Upload September 22, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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235

The smallest squares containing k 235's :
235225 = 485²,
235223569 = 15337²,
1235235622354561 = 35145919².

235² = 55225, a square with 2 kinds of digits (2 and 5).

235² = 55225, 5 + 52 + 2 + 5 = 8²,
235² = 55225, 55 + 2 + 2 + 5 = 8².

235² = 55225, 5 + 5 + 225 = 235,
235² = 55225, 5 + 5 + 225 = 5 * 52 - 25 = 235.

4089^k + 9870^k + 15792^k + 25474^k are squares for k = 1,2,3 (235², 31819², 4636691²).

Komachi equations:
235² = 1⁵ - 2⁵ + 3⁵ + 4⁵ + 5⁵ + 6⁵ + 7⁵ - 8⁵ + 9⁵ = 9⁵ - 8⁵ + 7⁵ + 6⁵ + 5⁵ + 4⁵ + 3⁵ - 2⁵ + 1⁵.

235⁴ = 3049800625, 3² + 0² + 4² + 9² + 8² + 0² + 0² + 6² + 2² + 5² = 235.

3-by-3 magic squares consisting of different squares with constant 235²:

(1 + 2 + ... + 15)(16 + 17 + ... + 39)(40 + 41 + ... + 235) = 46200²,
(1 + 2 + ... + 23)(24 + 25 + ... + 97)(98 + 99 + ... + 235) = 168498²,
(1 + 2 + ... + 27)(28 + 29 + ... + 107)(108 + 109 + ... + 235) = 211680²,
(1 + 2 + ... + 95)(96 + 97 + ... + 114)(115 + 116 + ... + 235) = 438900²,
(1 + 2 + ... + 128)(129 + 130 + ... + 192)(193 + 194 + ... + 235) = 883392²,
(1 + 2 + ... + 128)(129 + 130 + ... + 214)(215 + 216 + ... + 235) = 758520².

235² = 55225 appears in the decimal expressions of π and e:
  π = 3.14159•••55225••• (from the 82332nd digit),
  e = 2.71828•••55225••• (from the 15458th digit).

Page of Squares : First Upload September 27, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

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236

The smallest squares containing k 236's :
11236 = 106²,
2362369 = 1537²,
132236062367236 = 11499394².

The squares which begin with 236 and end in 236 are
236975236 = 15394²,   2362543236 = 48606²,   23683363236 = 153894²,
236092979236 = 485894²,   236299043236 = 486106²,...

236² = 55696, a square with 3 kinds of digits.

5251^k + 7257^k + 15635^k + 27553^k are squares for k = 1,2,3 (236², 32922², 5026564²).

Komachi Square Sum : 236² = 5² + 7² + 8² + 9² + 46² + 231².

(1 + 2 + ... + 23)(24 + 25 + ... + 138)(139 + 140 + ... + 236) = 217350²,
(1 + 2 + ... + 30)(31 + 32 + ... + 155)(156 + 157 + ... + 236) = 292950²,
(1 + 2 + ... + 36)(37 + 38 + ... + 148)(149 + 150 + ... + 236) = 341880²,
(1 + 2 + ... + 49)(50 + 51 + ... + 137)(138 + 139 + ... + 236) = 431970²,
(1 + 2 + ... + 63)(64)(65 + 66 + ... + 236) = 57792²,
(1 + 2 + ... + 63)(64 + 65 + ... + 71)(72 + 73 + ... + 236) = 166320²,
(1 + 2 + ... + 136)(137)(138 + 139 + ... + 236) = 153714².

236² = 55696 appears in the decimal expressions of π and e:
  π = 3.14159•••55696••• (from the 48113rd digit),
  e = 2.71828•••55696••• (from the 18322nd digit).

Page of Squares : First Upload September 27, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

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237

The smallest squares containing k 237's :
23716 = 154²,
2372371849 = 48707²,
1125237237237136 = 33544556².

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

237²± 2 are primes (the 7th case).

(1² + 2² + 3² + ... + 124²) + (1² + 2² + 3² + ... + 225²) = (1² + 2² + 3² + ... + 237²).

3-by-3 magic squares consisting of different squares with constant 237²:

237² = 56169, 5 + 6 + 1 + 69 = 9²,
237² = 56169, 5 + 6 + 16 + 9 = 6²,
237² = 56169, 5 + 61 + 6 + 9 = 9²,
237² = 56169, 56 + 16 + 9 = 9²,
237² = 56169, 56 + 169 = 15²,
237² = 56169, 561 + 6 + 9 = 24²,
237² = 56169, 5616 + 9 = 75².

237² = 56169.

Komachi equation: 237² = 1 * 2 * 3 * 4 * 5 * 6 * 78 + 9.

(1 + 2 + ... + 8)(9 + 10 + ... + 32)(33 + 34 + ... + 237) = 22140²,
(1 + 2 + ... + 129)(130 + 131 + ... + 194)(195 + 196 + ... + 237) = 905580².

237² = 56169 appears in the decimal expression of e:
  e = 2.71828•••56169••• (from the 15812nd digit).

Page of Squares : First Upload September 27, 2004 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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238

The smallest squares containing k 238's :
238144 = 488²,
123832384 = 11128²,
82380238238736 = 9076356².

238² = 56644, a square with 3 kinds of digits.

238² = 56644, 5 + 6 + 6 + 4 + 4 = 5².

4165^k + 7973^k + 17017^k + 27489^k are squares for k = 1,2,3 (238², 33558², 5126282²).

Komachi equation: 238² = - 1² + 2² * 34² + 5² * 6² * 7² + 89².

(1 + 2 + 3)(4 + 5 + ... + 50)(51 + 52 + ... + 238) = 14382²,
(1 + 2 + ... + 8)(9 + 10 + ... + 198)(199 + 200 + ... + 238) = 78660²,
(1 + 2 + ... + 30)(31 + 32 + ... + 185)(186 + 187 + ... + 238) = 295740²,
(1 + 2 + ... + 47)(48 + 49 + 50)(51 + 52 + ... + 238) = 67116²,
(1 + 2 + ... + 127)(128 + 129 + ... + 142)(143 + 144 + ... + 238) = 548640².

(1³ + 2³ + ... + 56³)(57³ + 58³ + ... + 203³)(204³ + 205³ + ... + 238³) = 642412232280².

238² = 56644 appears in the decimal expression of π:
  π = 3.14159•••56644••• (from the 17844th digit).

Page of Squares : First Upload September 27, 2004 ; Last Revised February 22, 2011
by Yoshio Mimura, Kobe, Japan

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239

The smallest squares containing k 239's :
123904 = 352²,
22394523904 = 149648²,
239023937239201 = 15460399².

239 is the third prime for which the Legendre Symbol (a/239) = 1 for a = 1, 2, 3, 4, 5, 6.

239² is the 8th square which is the sum of 6 fifth powers : (1, 4, 6, 6, 6, 8).

239² = 57121, a zigzag square.

239² = 57121, 5 + 7 + 1 + 2 + 1 = 4²,
239² = 57121, 5 + 7 + 12 + 1 = 5².

if n > 239 then n² + 1 has a prime divisor > 16.

Komachi equation: 239² = - 98² + 7² + 65² * 4² - 32² + 10².

3-by-3 magic squares consisting of different squares with constant 239²:

(1 + 2 + ... + 13)(14 + 15 + ... + 85)(86 + 87 + ... + 239) = 90090².

239² = 57121 appears in the decimal expressions of π and e:
  π = 3.14159•••57121••• (from the 52264th digit),
  e = 2.71828•••57121••• (from the 3590th digit).

Page of Squares : First Upload September 27, 2004 ; Last Revised May 14, 2010
by Yoshio Mimura, Kobe, Japan

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