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180

The smallest squares containing k 180's :
91809 = 303²,
180821809 = 13447²,
180591804018025 = 13438445².

The sum of the divisors of 180² is 341².

The integral triangle of sides 104, 657, 697 has square area 180².

180² = 10³ + 24³ + 26³.

180² = (7² - 1)(26² - 1) = (1² + 7)(65² + 7).

(13² - 2)(14² - 2) = (180² - 2),
(4² - 5)(6² - 5)(10² - 5) = (180² - 5).

(1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9)(10) = 180²,
(1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + 10 + 11) = 180²,
(1)(2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11) = 180²,
(1 + 2)(3)(4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12) = 180²,
(1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13) = 180²,
(1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14)(15) = 180²,
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 17) = 180²,
(1 + 2 + ... + 9)(10 + 11 + ... + 180) = 855²,
(1 + 2 + ... + 10)(11 + 12 + ... + 44)(45 + 46 + ... + 180) = 28050²,
(1 + 2 + ... + 17)(18 + 19 + ... + 142)(143 + 144 + ... + 180) = 96900²,
(1 + 2 + ... + 32)(33 + 34 + ... + 143)(144 + 145 + ... + 180) = 175824²,
(1 + 2 + ... + 63)(64 + 65 + ... + 119)(120 + 121 + ... + 180) = 307440²,
(1 + 2 + ... + 98)(99 + 100 + ... + 147)(148 + 149 + ... + 180) = 397782².

Page of Squares : First Upload July 26, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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181

The smallest squares containing k 181's :
181476 = 426²,
7318118116 = 85546²,
181515181181049 = 13472757².

181² = 32761, a square with different digits.

181² = 32761, 32 + 7 + 61 = 10².

181³ - 180³ + 179³ - 178³ + ... + 1³ = 1729².

Komachi equations:
181² = 9 * 8 * 7 * 65 - 4 + 3 + 2 */ 1 = 9 * 8 * 7 * 65 - 4 + 3 * 2 - 1
= 9 * 8 * 7 * 65 - 4 - 3 - 2 + 10.

A 3-by-3 magic square consisting of different squares with constant 181²:

(3² - 1)(64² - 1) = (13² - 1)( 14² - 1) = (181² - 1),
(2² - 1)(4² - 1)(27² - 1) = (181² - 1),
(3² - 7)(128² - 7) = (181² - 7),
(1² + 7)(5² + 7)(11² + 7) = (181² + 7).

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

(1² + 2² + 3² + ... + 181²) = 1992991, which consists of 3 kinds of digits and the first 7-digit palindromic sum.

(1)(2 + 3 + ... + 16)(17 + 18 + ... + 181) = 1485²,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 16)(17 + 18 + ... + 181) = 5445²,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 60)(61 + 62 + ... + 181) = 19965²,
(1 + 2 + ... + 7)(8 + 9 + ... + 34)(35 + 36 + ... + 181) = 15876²,
(1 + 2 + ... + 30)(31 + 32 + ... + 128)(129 + 130 + ... + 181) = 172515²,
(1 + 2 + ... + 31)(32 + 33 + ... + 61)(62 + 63 + ... + 181) = 100440²,
(1 + 2 + ... + 45)(46 + 47 + ... + 71)(72 + 73 + ... + 181) = 148005²,
(1 + 2 + ... + 45)(46 + 47 + ... + 115)(116 + 117 + ... + 181) = 239085²,
(1 + 2 + ... + 152)(153 + 154 + ... + 170)(171 + 172 + ... + 181) = 255816².

181² = 32761 appears in the decimal expression of e:
  e = 2.71828•••32761••• (from the 1698th digit)
  (32761 is the eighth 5-digit square in the expression of e.)

Page of Squares : First Upload July 26, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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182

The smallest squares containing k 182's :
18225 = 135²,
1829101824 = 42768²,
182182182310144 = 13497488².

182 = (1² + 2² + 3² + ... + 195²) / (1² + 2² + 3² + ... + 34²).

182² = 33124, a square every digit of which is non-zero and smaller than 5.

182² = 33124, 33 + 12 + 4 = 7².

182² is the 7th square which is the sum of 7 fifth powers : (3,4,4,5,5,6,7).

182² = 33124, 3 * 31 * 2 - 4 = 182.

82^k + 182^k + 722^k + 1318^k are squares for k = 1,2,3 (48², 1516², 51696²).
1794^k + 3250^k + 12402^k + 15678^k are squares for k = 1,2,3 (182², 20332², 2408588²).
1898^k + 4862^k + 5954^k + 20410^k are squares for k = 1,2,3 (182², 21892², 2972372²).
5330^k + 6682^k + 7514^k + 13598^k are squares for k = 1,2,3 (182², 17732², 1840748²).

Komachi equation: 182² = 1² / 2² * 3² / 4² * 56² * 78² / 9².

182 is the second integer which is the sum of a square and a prime in 6 ways :
  1² + 181, 3² + 173, 5² + 157, 9² + 101, 11² + 61, 13² + 13.

182² = 25² + 26² + 27² + ... + 48².

(1 + 2 + 3 + 4)(5 + 6 + ... + 157)(158 + 159 + ... + 182) = 22950²,
(1 + 2 + ... + 11)(12 + 13 + ... + 87)(88 + 89 + ... + 182) = 56430²,
(1 + 2 + ... + 17)(18 + 19 + ... + 157)(158 + 159 + ... + 182) = 89250²,
(1 + 2 + ... + 24)(25 + 26 + ... + 47)(48 + 49 + ... + 182) = 62100²,
(1 + 2 + ... + 99)(100 + 101 + ... + 149)(150 + 151 + ... + 182) = 410850².

(1³ + 2³ + ... + 77³)(78³ + 79³ + ... + 182³) = 49189140².

182² = 33124 appears in the decimal expression of e:
  e = 2.71828•••33124••• (from the 66855th digit)

Page of Squares : First Upload July 26, 2004 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

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183

The smallest squares containing k 183's :
118336 = 344²,
21831835536 = 147756²,
1758183183183025 = 41930695².

183184 = 428².

Komachi equations:
183² = - 9 - 87 * 6 + 54 * 3 * 210,
183² = 9² + 87² + 6² + 54² * 3² - 21² = - 98² + 7² * 6² * 5² + 4² - 32² + 1²
= - 9² - 87² - 6² - 54² - 3² + 210²,
183² = - 9³ + 8³ + 7³ - 6³ - 5³ - 4³ + 32³ + 10³.

A cubic polynomial :
  (X + 183²)(X + 288²)(X + 3416²) = X³ + 3433²X² + 1166808²X + 180036864².

Three 3-by-3 magic squares consisting of different squares with constant 183²:

(13² + 1)(14² + 1) = (183² + 1),
(3² + 1)(4² + 1)(14² + 1) = (183² + 1),
(1² + 1)(2² + 1)(4² + 1)(14² + 1) = (183² + 1),
(1² + 6)(7² + 6)(9² + 6) = (183² + 6).

183² + 184² + 185² + ... + 1240² = 25185²,
183² + 184² + 185² + ... + 3438² = 116402²,
183² + 184² + 185² + ... + 57303² = 7919743²,
183² + 184² + 185² + ... + 589² = 8140².

(1 + 2 + 3)(4 + 5 + ... + 156)(157 + 158 + ... + 183) = 18360²,
(1 + 2 + ... + 6)(7 + 8 + ... + 111)(112 + 113 + ... + 183) = 37170²,
(1 + 2 + ... + 7)(8 + 9 + ... + 14)(15 + 16 + ... + 183) = 6006²,
(1 + 2 + ... + 14)(15 + 16 + ... + 113)(114 + 115 + ... + 183) = 83160²,
(1 + 2 + ... + 33)(34 + 35 + ... + 47)(48 + 49 + ... + 183) = 70686²,
(1 + 2 + ... + 35)(36 + 37 + ... + 75)(76 + 77 + ... + 183) = 139860²,
(1 + 2 + ... + 40)(41 + 42 + ... + 103)(104 + 105 + ... + 183) = 206640²,
(1 + 2 + ... + 108)(109 + 110 + ... + 143)(144 + 145 + ... + 183) = 412020²,
(1 + 2 + ... + 125)(126 + 127 + ... + 154)(155 + 156 + ... + 183) = 395850².

183² = 33489 appears in the decimal expression of π:
  π = 3.14159•••33489••• (from the 14252nd digit).

Page of Squares : First Upload July 26, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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184

The smallest squares containing k 184's :
1849 = 43²,
184199184 = 13572²,
1844921841841 = 1358279².

The squares which begin with 184 and end in 184 are
184199184 = 13572²,   1842813184 = 42928²,   18476421184 = 135928²,
184102781184 = 429072²,   184408407184 = 429428²,...

184¹ + 345¹ = 23², 184² + 345² = 391², 184³ + 345³ = 6877²  (See 23).

184² = (1² + 7)(65² + 7).

184² = 2⁵ + 2⁵ + 4⁵ + 8⁵.

183184 = 428².

184² = 33856, 3 + 3 + 8 + 5 + 6 = 5².

184² is the fourth square which is the sum of 4 fifth powers : 2⁴ + 2⁴ + 4⁴ + 8⁴.

713^k + 5359^k + 10511^k + 17273^k are squares for k = 1,2,3 (184², 20930², 2543432²).

(13² + 2)(14² + 2) = (184² + 2),
(1² + 2)(4² + 2)(25² + 2) = (1² + 2)(8² + 2)(13² + 2) = (1² + 2)(3² + 2)(32² + 2)
= (3² + 2)(4² + 2)(13² + 2) = (184² + 2),
(1² + 2)(3² + 2)(5² + 2)(6² + 2) = (1² + 2)(2² + 2)(3² + 2)(13² + 2) = (184² + 2).

(1 + 2)(3 + 4 + ... + 167)(168 + 169 + ... + 184) = 11220²,
(1 + 2 + ... + 32)(33 + 34 + ... + 156)(157 + 158 + ... + 184) = 171864²,
(1 + 2 + ... + 33)(34 + 35 + ... + 167)(168 + 169 + ... + 184) = 150348²,
(1 + 2 + ... + 49)(50 + 51 + ... + 166)(167 + 168 + ... + 184) = 221130²,
(1 + 2 + ... + 62)(63)(64 + 65 + ... + 184) = 42966².

184² = 33856 appears in the decimal expression of e:
  e = 2.71828•••33856••• (from the 13223th digit)

Page of Squares : First Upload July 26, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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185

The smallest squares containing k 185's :
185761 = 431²,
18518582889 = 136083²,
241851858218569 = 15551587².

185² = 34225, 3 + 4 + 2 + 2 + 5 = 4².

3441^k + 6290^k + 10138^k + 14356^k are squares for k = 1,2,3 (185², 18981², 2071297²).

Komachi equation: 185² = 1³ * 2³ * 3³ + 4³ * 56³ / 7³ + 8³ + 9³.

Two 3-by-3 magic squares consisting of different squares with constant 185²:

(1² + 2² + 3² + ... + 34²) + (1² + 2² + 3² + ... + 39²) = 185².

(13² + 3)(14² + 3) = (185² + 3).

(1 + 2 + ... + 30)(31 + 32 + ... + 185) = 2790²,
(1 + 2 + ... + 33)(34 + 35 + ... + 66)(67 + 68 + ... + 185) = 117810².

(1³ + 2³ + ... + 147³)(148³ + 149³ + ... + 182³)(183³ + 184³ + 185³) = 592974791640².

185² = 34225 appears in the decimal expressions of π and e:
  π = 3.14159•••34225••• (from the 22625th digit),
  e = 2.71828•••34225••• (from the 1245th digit)
  (34225 is the seventh 5-digit square in the expression of e.)

Page of Squares : First Upload July 26, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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186

The smallest squares containing k 186's :
186624 = 432²,
186186025 = 13645²,
327318618618601 = 18091949².

186² = 34596, a square with different digits.

186² = 34596, 3 + 45 + 96 = 12²,
186² = 34596, 345 + 96 = 21².

186^k + 234^k + 258^k + 478^k are squares for k = 1,2,3 (34², 620², 12068²).
48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).

Komachi Fraction : 186² = 2179548/63.

Komachi equations:
186² = 12 + 34567 + 8 + 9,
186² = - 9² * 8² + 7² - 65² - 4² * 3² + 210².

(13² + 4)(14² + 4) = (186² + 4).
(1² + 4)(6² + 4)(13² + 4) = (186² + 4).

(1 + 2 + ... + 5)(6 + 7 + ... + 174)(175 + 176 + ... + 186) = 22230²,
(1 + 2 + ... + 11)(12 + 13 + ... + 65)(66 + 67 + ... + 186) = 45738²,
(1 + 2 + ... + 26)(27 + 28 + ... + 168)(169 + 170 + ... + 186) = 124605²,
(1 + 2 + ... + 35)(36 + 37 + ... + 156)(157 + 158 + ... + 186) = 194040²,
(1 + 2 + ... + 37)(38)(39 + 40 + ... + 186) = 21090²,
(1 + 2 + ... + 38)(39 + 40 + ... + 56)(57 + 58 + ... + 186) = 100035²,
(1 + 2 + ... + 45)(46 + 47 + 48)(49 + 50 + ... + 186) = 48645²,
(1 + 2 + ... + 86)(87 + 88 + ... + 128)(129 + 130 + ... + 186) = 392805²,
(1 + 2 + ... + 122)(123 + 124 + ... + 182)(183 + 184 + ... + 186) = 225090²,
(1 + 2 + ... + 125)(126 + 127 + ... + 179)(180 + 181 + ... + 186) = 288225²,
(1 + 2 + ... + 161)(162 + 163 + ... + 181)(182 + 183 + ... + 186) = 202860².

(1³ + 2³ + ... + 21³)(22³ + 23³ + ... + 33³)(34³ + 35³ + ... + 186³) = 2052721440².

186² = 34596 appears in the decimal expressions of π and e:
  π = 3.14159•••34596••• (from the 82162nd digit),
  e = 2.71828•••34596••• (from the 37139th digit)

Page of Squares : First Upload July 26, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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187

The smallest squares containing k 187's :
18769 = 137²,
18734491876 = 136874²,
318718718763025 = 17852695².

187² = 34969, 3 + 49 + 69 = 11²,
187² = 34969, 3⁵ + 49⁵ + 69⁵ = 42971².

If A = 187², B = 1020², and C = 1584², then A + B = 1037², B + C = 1884² and C + A = 1595².

Komachi equation: 187² = - 1² + 234² * 5² / 6² - 7² * 8² + 9².

A 3-by-3 magic square consisting of different squares with constant 187²:

(13² + 5)(14² + 5) = (187² + 5).

(1 + 2 + ... + 38)(39 + 40 + ... + 57)(58 + 59 + ... + 187) = 103740².

187² = 34969 appears in the decimal expression of e:
  e = 2.71828•••34969••• (from the 36141st digit)

Page of Squares : First Upload July 26, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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188

The smallest squares containing k 188's :
11881 = 109²,
18883431889 = 137417²,
1881881881489 = 1371817².

188² = 35344, a square with 3 kinds of digits.

188² = 16³ + 19³ + 29³ = 14³ + 22³ + 28³.

188² = 35344, 3 + 53 + 4 + 4 = 8²,
188² = 35344, 3 + 53 + 44 = 10²,
188² = 35344, 353 + 4 + 4 = 19².

10^k + 74^k + 98^k + 142^k are squares for k = 1,2,3 (18², 188², 2052²).
14^k + 58^k + 122^k + 130^k are squares for k = 1,2,3 (18², 188², 2052²).

Komachi equation: 188² = 987² * 6² * 5² * 4² / 3² / 210².

188² is the second square which is sum of 2 cubs in 2 ways :
  188² = 14³ + 22³ + 28³ = 16³ + 19³ + 29³.

If n ≥ 188, then n is the sum of k distinct nonzero squares with some k ≤ 6.

The third integer which is the sum of a square and a prime in 6 ways :
  188 = 3² + 179 = 5² + 163 = 7² + 139 = 9² + 107 = 11² + 67 = 13² + 19.

(13² + 6)(14² + 6) = (188² + 6).

(1 + 2 + ... + 8)(9 + 10 + ... + 72)(73 + 74 + ... + 188) = 37584²,
(1 + 2 + ... + 10)(11 + 12 + ... + 43)(44 + 45 + ... + 188) = 28710²,
(1 + 2 + ... + 10)(11 + 12 + ... + 45)(46 + 47 + ... + 188) = 30030²,
(1 + 2 + ... + 10)(11 + 12 + ... + 99)(100 + 101 + ... + 188) = 58740²,
(1 + 2 + ... + 18)(19 + 20 + ... + 171)(172 + 173 + ... + 188) = 87210²,
(1 + 2 + ... + 23)(24 + 25 + ... + 45)(46 + 47 + ... + 188) = 59202²,
(1 + 2 + ... + 32)(33 + 34 + ... + 45)(46 + 47 + ... + 188) = 66924²,
(1 + 2 + ... + 48)(49 + 50 + ... + 72)(73 + 74 + ... + 188) = 160776²,
(1 + 2 + ... + 55)(56 + 57 + ... + 153)(154 + 155 + ... + 188) = 307230²,
(1 + 2 + ... + 170)(171)(172 + 173 + ... + 188) = 87210².

Page of Squares : First Upload July 26, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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189

The smallest squares containing k 189's :
189225 = 435²,
18918902116 = 137546²,
189118989218916 = 13752054².

1 / 189 = 0.00529..., 529 = 23².

189² = 35721, a square with different digits.

189² = 35721, 3 + 5 + 7 + 21 = 6²,
189² = 35721, 3 + 5 + 72 + 1 = 9²,
189² = 35721, 3 + 5 + 721 = 27²,
189² = 35721, 3 + 57 + 21 = 9²,
189² = 35721, 3³ + 57³ + 21³ = 441²,
189² = 35721, 3 + 572 + 1 = 24².

41³ - 40³ + 39³ - 38³ + .. + 1³ = 189².

1617^k + 8106^k + 10290^k + 15708^k are squares for k = 1,2,3 (189², 20517², 2345679²).
2793^k + 4578^k + 13062^k + 15288^k are squares for k = 1,2,3 (189², 20811², 2432997²).
4998^k + 6468^k + 10542^k + 13713^k are squares for k = 1,2,3 (189², 19131², 2036097²).
4998^k + 6888^k + 8106^k + 15729^k are squares for k = 1,2,3 (189², 19635², 2208087²).

Komachi Fraction : 576/321489 = (8/189)².

Komachi equations:
189² = 1² * 2² * 3² * 4² * 567² / 8² / 9² = 1² / 2² * 3² / 4² * 567² * 8² / 9²
= 98² / 7² / 6² * 54² * 3² / 2² */ 1² = 9² * 8² * 7² * 6² * 5² * 4² / 32² / 10²
= 9² / 8² * 7² * 6² * 5² / 4² * 32² / 10².

Four 3-by-3 magic squares consisting of different squares with constant 189²:

(13² + 7)(14² + 7) = (189² + 7),
(2² + 7)(3² + 7)(14² + 7) = (189² + 7).

(1 + 2)(3 + 4)(5 + 6 + ... + 58) = 189²,
(1 + 2 + ... + 17)(18 + 19 + ... + 68)(69 + 70 + ... + 189) = 72369².

189² = 35721 appears in the decimal expression of e:
  e = 2.71828•••35721••• (from the 69165th digit)

Page of Squares : First Upload July 26, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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