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210

The smallest squares containing k 210's :
12100 = 110²,
210221001 = 14499²,
62210921012100 = 7887390².

210² = 44100, (4 and 100 are squares).

210² = (1² + 6)(2² + 6)(3² + 6)(6² + 6) = (1² + 6)(6² + 6)(12² + 6)
= (3² + 5)(5² + 5)(10² + 5) = (3² + 6)(6² + 6)(8² + 6).

210² + 211² + 212² + ... + 220² = 221² + 222² + 223² + ... + 230².

The integral triangle of sides 255, 353, 392 (or 73, 1274, 1299) has square area 210².

Komachi equations:
210² = 9 + 876 + 5 + 43210,
210² = 98² / 7² / 6² * 54² / 3² / 2² * 10².

210^k + 241^k + 538^k + 692^k are squares for k = 1,2,3 (41², 933², 22591²).
210^k + 606^k + 762^k + 1338^k are squares for k = 1,2,3 (54², 1668², 55404²).
210^k + 1470^k + 14490^k + 27930^k are squares for k = 1,2,3 (210², 31500², 4983300²).
66^k + 210^k + 1230^k + 1410^k are squares for k = 1,2,3 (54², 1884², 68364²).
18^k + 84^k + 129^k + 210^k are squares for k = 1,2,3 (21², 261², 3465²).
882^k + 2478^k + 16338^k + 24402^k are squares for k = 1,2,3 (210², 29484², 4348260²).

210² + 211² + 212² + 213² + ... + 585² = 7990²,
210² + 211² + 212² + 213² + ... + 4603² = 180323².

(1)(2)(3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11) = 210²,
(1)(2)(3 + 4)(5)(6 + 7 + 8 + 9)(10 + 11) = 210²,
(1 + 2 + 3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11 + 12) = 210²,
(1 + 2)(3 + 4)(5)(6)(7 + 8 + 9 + 10 + 11 + 12 + 13) = 210²,
(1)(2)(3 + 4 + 5 + 6 + 7 + 8 + 9)(10 + 11)(12 + 13) = 210²,
(1 + 2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10 + 11 + 12 + 13 + 14) = 210²,
(1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + 11 + 12 + 13)(14) = 210²,
(1)(2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13)(14) = 210²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 10)(11 + 12 + ... + 17) = 210²,
(1 + 2 + ... + 9)(10)(11 + 12 + ... + 17) = 210².

(1 + 2 + ... + 7)(8 + 9 + ... + 181)(182 + 183 + ... + 210) = 51156²,
(1 + 2 + ... + 9)(10 + 11 + ... + 34)(35 + 36 + ... + 210) = 23100²,
(1 + 2 + ... + 49)(50 + 51 + ... + 203)(204 + 205 + ... + 210) = 185955².

1³ + 2³ + 3³ + 4³ + 5³ + ... + 19³ + 20³ = 210².

(1³)(2³ + 3³)(4³ + 5³ + 6³ + 7³ + 8³) = 210².

Page of Squares : First Upload April 27, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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211

The smallest squares containing k 211's :
2116 = 46²,
12110122116 = 110046²,
2110021101421156 = 45934966².

211² = 44521, a reversible square (12544 = 112²).

211² = 44521, 4 + 4 + 5 + 2 + 1 = 4²,
211² = 44521, 4 + 4 + 521 = 23².

211² = 44521, 4 + 4 * 52 - 1 = 211.

Komachi equation: 211² = 9² - 8² + 7² - 6² + 5² * 4² - 3² + 210².

3-by-3 magic squares consisting of different squares with constant 211²:

(1 + 2+ ... + 8)(9 + 10 + ... + 19)(20 + 21 + 22 + ... + 211) = 11088²,
(1 + 2 + ... + 9)(10 + 11 + 12 + ... + 94)(95 + 96 + ... + 211) = 59670²,
(1 + 2 + ... + 27)(28 + 29 + ... + 188)(189 + 190 + ... + 211) = 173880²,
(1 + 2 + ... + 36)(37 + 38 + ... + 148)(149 + 150 + ... + 211) = 279720²,
(1 + 2 + ... + 38)(39 + 40 + ... + 113)(114 + 115 + ... + 211) = 259350²,
(1 + 2 + ... + 66)(67 + 68 + ... + 123)(124 + 125 + ... + 211) = 420090²,
(1 + 2 + ... + 75)(76 + 77 + ... + 92)(93 + 94 + ... + 211) = 271320²,
(1 + 2 + ... + 87)(88 + 89 + ... + 165)(166 + 167 + ... + 211) = 572286².

211² = 44521 appears in the decimal expressions of π and e:
  π = 3.14159•••44521••• (from the 7461st digit),
  e = 2.71828•••44521••• (from the 8703rd digit).

Page of Squares : First Upload April 27, 2004 ; Last Revised May 7, 2010
by Yoshio Mimura, Kobe, Japan

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212

The smallest squares containing k 212's :
142129 = 377²,
121242121 = 11011²,
212621221282369 = 14581537².

The square root of 212 is 14.5602197..., 14² = 5² + 6² + 0² + 2² + 1² + 9² + 7².

212² = 44944, a palindromic square with 2 kinds of digits (4 and 9).

212² = 44944, 4 + 4 + 9 + 4 + 4 = 5².

212² = 44944, 4 * 49 + 4 * 4 = 212.

212² = 44944 = 44944 (4, 9 and 49 are squares).

98^k + 212^k + 305^k + 346^k are squares for k = 1,2,3 (31², 517², 8959²).
49^k + 98^k + 170^k + 212^k are squares for k = 1,2,3 (23², 293², 3937²).

(1 + 2)(3 + 4 + ... + 12)(13 + 14 + ... + 212) = 2250²,
(1 + 2)(3 + 4 + ... + 42)(43 + 44 + ... + 212) = 7650²,
(1 + 2 + ... + 8)(9 + 10 + ... + 42)(43 + 44 + ... + 212) = 26010²,
(1 + 2 + ... + 14)(15 + 16 + ... + 84)(85 + 86 + ... + 212) = 83160²,
(1 + 2 + ... + 32)(33 + 34 + ... + 102)(103 + 104 + ... + 212) = 207900²,
(1 + 2 + ... + 80)(81 + 82 + ... + 84)(85 + 86 + ... + 212) = 142560²,
(1 + 2 + ... + 80)(81 + 82 + ... + 87)(88 + 89 + ... + 212) = 189000²,
(1 + 2 + ... + 80)(81 + 82 + ... + 195)(196 + 197 + ... + 212) = 422280²,
(1 + 2 + ... + 98)(99 + 100 + ... + 117)(118 + 119 + ... + 212) = 395010².

212² = 44944 appears in the decimal expressions of π and e:
  π = 3.14159•••44944••• (from the 1249th digit),
  (44944 is the second 5-digit square in the expression of π)
  e = 2.71828•••44944••• (from the 30590th digit).

Page of Squares : First Upload April 27, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

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213

The smallest squares containing k 213's :
21316 = 146²,
2132130625 = 46175²,
2132139213558649 = 46175093².

1 / 213 = 0.0046948..., 4² + 6² + 9² + 4² + 8² = 213.

213² = 14³ + 25³ + 30³.

213² = 45369, a square with different digits.

213² = 1! + 1! + 1! + 3! + 7! + 8! = 1! + 2! + 3! + 7! + 8!.

213² = 45369, 4 + 5 + 3 + 69 = 9².

1065^k + 8094^k + 14058^k + 22152^k are squares for k = 1,2,3 (213², 27477², 3765627²).

Komachi Fraction : 7569 / 408321 = (29/213)².

181² + 182² + 183² + 184² + ... + 213² = 1133².

3-by-3 magic squares consisting of different squares with constant 213²:

(1 + 2 + ... + 8)(9 + 10 + ... + 45)(46 + 47 + ... + 213) = 27972²,
(1 + 2 + ... + 9)(10 + 11 + ... + 24)(25 + 26 + ... + 213) = 16065²,
(1 + 2 + ... + 13)(14 + 15 + ... + 31)(32 + 33 + ... + 213) = 28665²,
(1 + 2 + ... + 13)(14 + 15 + ... + 91)(92 + 93 + ... + 213) = 83265²,
(1 + 2 + ... + 17)(18 + 19 + ... + 24)(25 + 26 + ... + 213) = 22491²,
(1 + 2 + ... + 17)(18 + 19 + ... + 94)(95 + 96 + ... + 213) = 109956²,
(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + 138 + ... + 213) = 137445²,
(1 + 2 + ... + 17)(18 + 19 + ... + 152)(153 + 154 + ... + 213) = 139995²,
(1 + 2 + ... + 17)(18 + 19 + ... + 186)(187 + 188 + ... + 213) = 119340²,
(1 + 2 + ... + 21)(22 + 23 + ... + 66)(67 + 68 + ... + 213) = 97020²,
(1 + 2 + ... + 21)(22 + 23 + ... + 147)(148 + 149 + ... + 213) = 171171²,
(1 + 2 + ... + 21)(22 + 23 + ... + 209)(210 + 211 + ... + 213) = 65142²,
(1 + 2 + ... + 22)(23 + 24 + ... + 154)(155 + 156 + ... + 213) = 179124²,
(1 + 2 + ... + 25)(26 + 27 + ... + 38)(39 + 40 + ... + 213) = 54600²,
(1 + 2 + ... + 26)(27 + 28 + ... + 143)(144 + 145 + ... + 213) = 208845²,
(1 + 2 + ... + 31)(32 + 33 + ... + 96)(97 + 98 + ... + 213) = 193440²,
(1 + 2 + ... + 32)(33 + 34 + ... + 114)(115 + 116 + ... + 213) = 227304²,
(1 + 2 + ... + 38)(39 + 40 + ... + 137)(138 + 139 + ... + 213) = 293436²,
(1 + 2 + ... + 45)(46 + 47 + ... + 75)(76 + 77 + ... + 213) = 193545²,
(1 + 2 + ... + 48)(49 + 50 + ... + 147)(148 + 149 + ... + 213) = 368676²,
(1 + 2 + ... + 64)(65 + 66 + ... + 91)(92 + 93 + ... + 213) = 285480²,
(1 + 2 + ... + 64)(65 + 66 + ... + 150)(151 + 152 + ... + 213) = 469560²,
(1 + 2 + ... + 80)(81 + 82 + ... + 149)(150 + 151 + ... + 213) = 546480²,
(1 + 2 + ... + 80)(81 + 82 + ... + 164)(165 + 166 + ... + 213) = 555660²,
(1 + 2 + ... + 80)(81 + 82 + ... + 185)(186 + 187 + ... + 213) = 502740²,
(1 + 2 + ... + 86)(87)(88 + 89 + ... + 213) = 78561²,
(1 + 2 + ... + 90)(91)(92 + 93 + ... + 213) = 83265²,
(1 + 2 + ... + 116)(117 + 118 + ... + 174)(175 + 176 + ... + 213) = 658242²,
(1 + 2 + ... + 147)(148 + 149 + ... + 185)(186 + 187 + ... + 213) = 620046².

(1² + 2² + ... + 95²)(96² + 97² + ... + 190²)(191² + 192² + ... + 213²) = 741186960².

213² = 45369 appears in the decimal expressions of π and e:
  π = 3.14159•••45369••• (from the 11261st digit),
  e = 2.71828•••45369••• (from the 54303rd digit).

Page of Squares : First Upload April 27, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

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214

The smallest squares containing k 214's :
214369 = 463²,
21489214464 = 146592²,
62142146214841 = 7883029².

214² = 45796, a square with different digits.

Komachi equations:
214² = - 9 - 8 + 7 * 6543 + 2 + 10,
214² = 9² + 8² - 7² + 6² * 5² * 4² / 3² + 210².

214² = 45796, 45 + 796 = 29².

214² + 215² + 216² + 217² + ... + 752² = 11781².

1² + 2² + 3² + 4² + 5² + ... + 214² = 3289715, different digits.

(1 + 2 + ... + 6)(7 + 8 + ... + 97)(98 + 99 + ... + 214) = 42588²,
(1 + 2 + ... + 7)(8 + 9 + ... + 16)(17 + 18 + ... + 214) = 8316²,
(1 + 2 + ... + 7)(8 + 9 + ... + 97)(98 + 99 + ... + 214) = 49140²,
(1 + 2 + ... + 7)(8 + 9 + ... + 118)(119 + 120 + ... + 214) = 55944²,
(1 + 2 + ... + 15)(16 + 17 + ... + 84)(85 + 86 + ... + 214) = 89700²,
(1 + 2 + ... + 20)(21 + 22 + ... + 114)(115 + 116 + ... + 214) = 148050²,
(1 + 2 + ... + 27)(28 + 29 + ... + 93)(94 + 95 + ... + 214) = 167706²,
(1 + 2 + ... + 32)(33 + 34 + ... + 110)(111 + 112 + ... + 214) = 223080²,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 214) = 228690²,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 214) = 73062²,
(1 + 2 + ... + 96)(97)(98 + 99 + ... + 214) = 90792²,
(1 + 2 + ... + 200)(201)(202 + 203 + ... + 214) = 104520².

214² = 45796 appears in the decimal expressions of π and e:
  π = 3.14159•••45796••• (from the 31817th digit),
  e = 2.71828•••45796••• (from the 102389th digit).

Page of Squares : First Upload April 30, 2004 ; Last Revised May 7, 2010
by Yoshio Mimura, Kobe, Japan

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215

The smallest squares containing k 215's :
215296 = 464²,
12154621504 = 110248²,
242152159215121 = 15561239².

215² - 1 = 4! + 5! + 6! + 7! + 8!

215² = 46225, 4⁴ + 6⁴ + 2⁴ + 2⁴ + 5⁴ = 47².

215² = 19³ + 27³ + 27³.

5289^k + 9202^k + 12986^k + 18748^k are squares for k = 1,2,3 (215², 25155², 3115565²).

Komachi equations:
215² = 9² - 8² - 7² + 6² + 5² * 43² - 2² * 1² = 9² - 8² - 7² + 6² + 5² * 43² - 2² / 1²
= - 9² + 8² + 7² - 6² + 5² * 43² + 2² * 1² = - 9² + 8² + 7² - 6² + 5² * 43² + 2² / 1².

3-by-3 magic squares consisting of different squares with constant 215²:

(1 + 2 + ... + 6)(7 + 8 + ... + 15)(16 + 17 + ... + 215) = 6930²,
(1 + 2 + ... + 8)(9 + 10 + ... + 152)(153 + 154 + ... + 215) = 69552²,
(1 + 2 + ... + 15)(16 + 17 + ... + 159)(160 + 161 + ... + 215) = 126000²,
(1 + 2 + ... + 39)(40 + 41 + ... + 59)(60 + 61 + ... + 215) = 128700²,
(1 + 2 + ... + 44)(45 + 46 + ... + 59)(60 + 61 + ... + 215) = 128700²,
(1 + 2 + ... + 56)(57 + 58 + ... + 209)(210 + 211 + ... + 215) = 203490²,
(1 + 2 + ... + 98)(99 + 100 + ... + 176)(177 + 178 + ... + 215) = 630630²,
(1 + 2 + ... + 99)(100 + 101 + ... + 180)(181 + 182 + ... + 215) = 623700²,
(1 + 2 + ... + 107)(108)(109 + 110 + ... + 215) = 104004²,
(1 + 2 + ... + 117)(118 + 119 + ... + 176)(177 + 178 + ... + 215) = 676494².

215² = 46225 appears in the decimal expression of e:
  e = 2.71828•••46225••• (from the 124945th digit).

Page of Squares : First Upload April 30, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

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216

The smallest squares containing k 216's :
9216 = 96²,
512162161 = 22631²,
21621607209216 = 4649904².

The squares which begin with 216 and end in 216 are
21637233216 = 147096²,   216135729216 = 464904²,   216314289216 = 465096²,
216600883216 = 465404²,   216779635216 = 465596²,...

216² is the 7th square which is the sum of 6 fifth powers : (6, 6, 6, 6, 6, 6).

216² = 46656, a square with 3 kinds of digits.

216² = (4² + 8)(44² + 8).

15² + 216 = 21², 15² - 216 = 3².

Komachi equations:
216² = 1² * 2² * 3² * 4² * 56² / 7² / 8² * 9² = 1² * 2² * 3² * 4² / 56² * 7² * 8² * 9²
  = 1² / 2² * 3² / 4² * 56² / 7² * 8² * 9² = 9² * 8² / 7² * 6² / 5² / 4² / 3² * 210²,
216² = 1⁶ / 2⁶ / 3⁶ * 4⁶ * 56⁶ / 7⁶ / 8⁶ * 9⁶ = 1⁶ / 2⁶ / 3⁶ * 4⁶ / 56⁶ * 7⁶ * 8⁶ * 9⁶
  = 9⁶ * 8⁶ * 7⁶ * 6⁶ * 5⁶ / 4⁶ / 3⁶ / 210⁶ = 9⁶ * 8⁶ / 7⁶ / 6⁶ / 5⁶ / 4⁶ / 3⁶ * 210⁶.

216² = 46656, 4 + 6 + 65 + 6 = 9²,
216² = 46656, 4 + 66 + 5 + 6 = 9².

216² = 4³ + 24³ + 32³ = 18³ + 24³ + 30³.

216² = 46656 is an exchangeable square (66564 = 258²).

216² + 217² + 218² + 219² + ... + 311² = 2596²,
216² + 217² + 218² + 219² + ... + 553² = 7293².

(1 + 2)(3)(4)(5 + 6 + 7)(8)(9) = 216²,
(1 + 2 + ... + 4)(5 + 6 + ... + 199)(200 + 201 + ... + 216) = 26520²,
(1 + 2 + ... + 8)(9 + 10 + ... + 72)(73 + 74 + ... + 216) = 44064²,
(1 + 2 + ... + 8)(9 + 10 + ... + 144)(145 + 146 + ... + 216) = 69768²,
(1 + 2 + ... + 13)(14 + 15 + ... + 160)(161 + 162 + ... + 216) = 110838²,
(1 + 2 + ... + 33)(34 + 35 + ... + 135)(136 + 137 + ... + 216) = 262548²,
(1 + 2 + ... + 40)(41 + 42 + ... + 175)(176 + 177 + ... + 216) = 309960²,
(1 + 2 + ... + 48)(49 + 50 + ... + 72)(73 + 74 + ... + 216) = 188496²,
(1 + 2 + ... + 48)(49 + 50 + ... + 207)(208 + 209 + ... + 216) = 213696².

216² = 46656 appears in the decimal expression of e:
  e = 2.71828•••46656••• (from the 44479th digit).

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

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217

The smallest squares containing k 217's :
217156 = 466²,
3217612176 = 56724²,
1217217932176 = 1103276².

1 / 217 = 0.004608294..., 4² + 6² + 0² + 8² + 2² + 9² + 4² = 217.

217² = 47089, a square with different digits.

217² = (2² + 3)(82² + 3).

217² = 47089, 4 + 7 + 0 + 89 = 10²,
217² = 47089, 47 + 0 + 8 + 9 = 8².

1² + 2² + 3² + 4² + 5² + ... + 217² = 3429685, different digits.

Komachi Fraction : 7569 / 423801 = (29/217)².

217^k + 218^k + 272^k + 518^k are squares for k = 1,2,3 ( 35², 661², 13405²).
682^k + 5084^k + 20553^k + 20770^k are squares for k = 1,2,3 ( 217², 29667², 4215907²).
7626^k + 9486^k + 11656^k + 18321^k are squares for k = 1,2,3 ( 217², 24893², 3005047²).

168² + 169² + 170² + 171² + 172² + ... + 217² = 1365².

3-by-3 magic squares consisting of different squares with constant 217²:

(1 + 2 + ... + 62)(63 + 64 + ... + 217) = 6510²,
(1 + 2 + ... + 63)(64 + 65 + ... + 112)(113 + 114 + ... + 217) = 388080².

217² = 47089 appears in the decimal expressions of π and e:
  π = 3.14159•••47089••• (from the 27889th digit),
  e = 2.71828•••47089••• (from the 34095th digit).

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

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218

The smallest squares containing k 218's :
121801 = 349²,
121842185481 = 349059²,
1621821821878801 = 40271849².

218² is the 8th square which is the sum of 8 fifth powers : (1, 3, 3, 3, 5, 5, 6, 8).

218²± 3 are primes.

Komachi equations:
218² = 9² + 8² - 7² + 654² / 3² + 2² - 10² = - 9² - 8² + 7² + 654² / 3² - 2² + 10².

10^k + 218^k + 542^k + 674^k are squares for k = 1,2,3 (38², 892², 21812²).
217^k + 218^k + 272^k + 518^k are squares for k = 1,2,3 (35², 661², 13405²).

(1 + 2 + ... + 9)(10 + 11 + ... + 199)(200 + 201 + ... + 218) = 59565²,
(1 + 2 + ... + 10)(11)(12 + 13 + ... + 218) = 3795²,
(1 + 2 + ... + 10)(11 + 12 + ... + 199)(200 + ... + 218) = 65835²,
(1 + 2 + ... + 13)(14 + 15)(16 + 17 + ... + 218) = 7917²,
(1 + 2 + ... + 13)(14 + 15 + ... + 28)(29 + 30 + ... + 218) = 25935²,
(1 + 2 + ... + 13)(14 + 15 + ... + 103)(104 + 105 + ... + 218) = 94185²,
(1 + 2 + ... + 14)(15 + 16 + ... + 104)(105 + 106 + ... + 218) = 101745²,
(1 + 2 + ... + 32)(33 + 34 + ... + 122)(123 + 124 + ... + 218) = 245520²,
(1 + 2 + ... + 34)(35 + 36 + ... + 119)(120 + 121 + ... + 218) = 255255²,
(1 + 2 + ... + 68)(69 + 70 + ... + 138)(139 + 140 + ... + 218) = 492660²,
(1 + 2 + ... + 80)(81 + 82 + ... + 149)(150 + 151 + ... + 218) = 571320²,
(1 + 2 + ... + 198)(199)(200 + 201 + ... + 218) = 124773².

218² = 47524 appears in the decimal expressions of π and e:
  π = 3.14159•••47524••• (from the 97772nd digit),
  e = 2.71828•••47524••• (from the 115294th digit).

Page of Squares : First Upload April 30, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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219

The smallest squares containing k 219's :
21904 = 148²,
1221921936 = 34956²,
2197219219204 = 1482302².

219² = 47961, a square with different digits.

Komachi equations:
219² = 9² * 876² / 54² * 3² / 2² * 1² = 9² * 876² / 54² * 3² / 2² / 1².

3-by-3 magic squares consisting of different squares with constant 219²:

219² = 47961, 4 + 7 + 9 + 61 = 9²,
219² = 47961, 4 + 79 + 61 = 12²,
219² = 47961, 47 + 96 + 1 = 12².

219² = 47961 appears in the decimal expressions of π and e:
  π = 3.14159•••47961••• (from the 20128th digit),
  e = 2.71828•••47961••• (from the 14640th digit).

9² + 23² + 37² + 51² + 65² + 79² + 93² + 107² + 121² + 135² + 149² + 163² + 177² + 191² + 205² + 219² = 524².

(1 + 2)(3 + 4 + ... + 123)(124 + 125 + ... + 219) = 19404²,
(1 + 2 + ... + 17)(18 + 19 + ... + 118)(119 + 120 + ... + 219) = 133926²,
(1 + 2 + ... + 45)(46 + 47 + ... + 69)(70 + 71 + ... + 219) = 175950²,
(1 + 2 + ... + 75)(76 + 77 + ... + 84)(85 + 86 + ... + 219) = 205200².

Page of Squares : First Upload April 30, 2004 ; Last Revised May 7, 2010
by Yoshio Mimura, Kobe, Japan

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