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150

The smallest squares containing k 150's :
61504 = 248²,
150111504 = 12252²,
15015051505476 = 3874926².

(2² + 4)(53² + 4) = (150² + 4),
(12² - 6)(13² - 6) = (150² - 6).

150² = (1² + 9)(4² + 9)(9² + 9) = (2 ² + 6)(3² + 6)(12² + 6) = (1⁴ + 9)(2⁴ + 9)(3⁴ + 9).

150² = 12 x 13 + 14 x 15 + 16 x 17 + 18 x 19 + ... + 50 x 51.

150² = 7 x 8 x 9 + 8 x 9 x 10 + 9 x 10 x 11 + 16 x 17 x 18.

Komachi equations:
150² = 9² + 87² + 6² * 5² * 4² + 3² + 21² = 9² * 8² / 7² / 6² * 5² / 4² / 3² * 210².

(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12 + 13) = 150²,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14) = 150²,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14) = 150²,
(1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + ... + 17) = 150²,
(1 + 2 + 3 + 4)(5)(6 + 7 + ... + 30) = 150²,
(1 + 2 )(3 + 4 + ... + 122) = 150²,
(1 + 2 + 3)(4 + 5 + ... + 21)(22 + 23 + ... + 150) = 3870²,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 150) = 4410²,
(1 + 2 + ... + 80)(81 + 82 + ... + 149)(150) = 62100².

150² = 22500 appears in the decimal expression of π:
  π = 3.14159•••22500••• (from the 82334th digit).

Page of Squares : First Upload June 14, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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151

The smallest squares containing k 151's :
15129 = 123²,
15166415104 = 123152²,
15181512151716 = 3896346².

151² = 22801, 2³ + 2³ + 8³ + 0³ + 1³ = 23².

Komachi equations:
151² = 12 * 34 * 56 - 7 * 8 + 9 = 9 - 8 + 76 * 5 * 4 * 3 / 2 * 10
  = - 9 - 8 + 7 * 6 * 543 + 2 + 10,
151² = 1² * 23² + 4² * 5² * 6² - 7² + 89² = 98² + 76² + 5² + 43² * 2² */ 1².

Two 3-by-3 magic squares consisting of different squares with constant 151²:

(4² - 1)(39² - 1) = (151² - 1),
(12² - 5)(13² - 5) = (151² - 5),
(3² - 7)(6² - 7)(20² - 7) = (151² - 7).

(1 + 2 + 3 + 4)(5 + 6 + ... + 16)(17 + ... + 151) = 3780²,
(1 + 2 + ... + 13)(14 + 15 + ... + 112)(113 + 114 + ... + 151) = 54054²,
(1 + 2 + ... + 17)(18 + 19 + ... + 49)(50 + 51 + ... + 151) = 41004²,
(1 + 2 + ... + 32)(33 + 34 + ... + 52)(53 + 54 + ... + 151) = 67320²,
(1 + 2 + ... + 32)(33 + 34 + ... + 55)(56 + 57 + ... + 151) = 72864²,
(1 + 2 + ... + 32)(33 + 34 + ... + 123)(124 + 125 + ... + 151) = 120120²,
(1 + 2 + ... + 47)(48 + 49 + ... + 104)(105 + 106 + ... + 151) = 171456²,
(1 + 2 + ... + 55)(56 + 57 + ... + 88)(89 + 90 + ... + 151) = 166320².

151² = 22801 appears in the decimal expression of e:
  e = 2.71828•••22801••• (from the 74156th digit).

Page of Squares : First Upload June 14, 2004 ; Last Revised April 20, 2010
by Yoshio Mimura, Kobe, Japan

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152

The smallest squares containing k 152's :
1521 = 39²,
152152225 = 12335²,
15221522781529 = 3901477².

152² = 23104, a square with different digits.

152² = 23104, 2³ + 3³ + 1³ + 0³ + 4³ = 10².

152² = 28³ + 4⁵ + 2⁷.

152^k + 209^k + 3344^k + 5320^k are squares for k = 1,2,3 (95², 6289², 433561²).
627^k + 3667^k + 8037^k + 10773^k are squares for k = 1,2,3 (152², 13946², 1348696²).

Komachi equations:
152² = - 12 + 345 * 67 - 8 + 9,
152² = 123² + 4² + 5² - 6² + 7² + 89² = 9² * 8² * 76² / 54² * 3² / 2² */ 1²
152² = 98² - 7² * 6² * 5² + 4² * 3² * 2² * 10²,
152² = - 98³ / 7³ + 6³ * 5³ + 4³ - 3³ * 2³ - 10³.

(3² - 4)(68² - 4) = (12² - 4)(13² - 4) = (152² - 4),
(3² - 4)(8² - 4)(9² - 4) = (152² - 4).

(1 + 2 + ... + 7)(8 + 9 + ... + 27)(28 + 29 + ... + 152) = 10500².

152² = 23104 appears in the decimal expressions of π and e:
  π = 3.14159•••23104••• (from the 61787th digit),
  e = 2.71828•••23104••• (from the 97917th digit).

Page of Squares : First Upload June 14, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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153

The smallest squares containing k 153's :
15376 = 124²,
6521531536 = 80756²,
15315396153121 = 3913489².

153² = 23409, a square with different digits.

153² = (3² + 8)(37² + 8).

153² = 23409, 23 + 4 + 0 + 9 = 6².

153² = 1³ + 18³ + 26³, the 7th square which is the sum of 3 cubes.

The 2nd square which is the sum of 4 fourth powers in 2 ways :
  153² = 2⁴ + 4⁴ + 7⁴ + 12⁴ = 3⁴ + 6⁴ + 6⁴ + 12⁴.

204^k + 3162^k + 7242^k + 12801^k are squares for k = 1,2,3 (153², 15045², 1584009²).
2550^k + 2652^k + 6681^k + 11526^k are squares for k = 1,2,3 (153², 13821², 1365525²).

Komachi equations:
153² = 1 * 23456 - 7 * 8 + 9,
153² = 1² / 2² * 34² * 56² / 7² / 8² * 9² = 1² / 2² * 34² / 56² * 7² * 8² * 9².

A 3-by-3 magic square consisting of different squares with constant 153²:

153² = 2⁴ + 4⁴ + 7⁴ + 12⁴ = 3⁴ + 6⁴ + 6⁴ + 12⁴.

(12² - 3)(13² - 3) = (153² - 3).

(1 + 2 + ... + 63)(64 + 65 + ... + 91)(92 + 93 + ... + 153) = 182280².

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ + 11³ + 12³ + 13³ + 14³ + 15³ + 16³ + 17³ = 153².

(1³ + 2³ + ... + 10³)(11³ + 12³ + ... + 21³)(22³ + 23³ + ... + 153³) = 145345200².

153² = 23409 appears in the decimal expression of e:
  e = 2.71828•••23409••• (from the 60119th digit).

Page of Squares : First Upload June 14, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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154

The smallest squares containing k 154's :
154449 = 393²,
1543154089 = 39283²,
1545864154915441 = 39317479².

154 = (1² + 2² + 3² + ... + 104²) / (1² + 2² + 3² + ... + 19²).

154² = 23716, a square with different digits.

154² = 23716, 23 * 7 - 1 - 6 = 154.

154² = 23716, 23 + 71 + 6 = 10².

154^k + 222^k + 258^k + 810^k are squares for k = 1,2,3 (38², 892², 23732²).
154^k + 490^k + 1708^k + 3577^k are squares for k = 1,2,3 (77², 3997², 225547²).
154^k + 814^k + 1298^k + 2090^k are squares for k = 1,2,3 (66², 2596², 108900²).
154^k + 3038^k + 3626^k + 9058^k are squares for k = 1,2,3 (126², 10220², 904932²).
154^k + 8624^k + 17633^k + 26950^k are squares for k = 1,2,3 (231², 33341², 5069295²).
146^k + 154^k + 654^k + 1546^k are squares for k = 1,2,3 (50², 1692², 63100²).
638^k + 4378^k + 7106^k + 11594^k are squares for k = 1,2,3 (154², 14300², 1414732²).

Komachi Fraction : 154² = 8751204/369.

Komachi equation: 154² = 12 + 34 + 5 * 6 * 789.

(12² - 2)(13² - 2) = (154² - 2),
(2² + 9)(4² + 9)(8² + 9) = (154² + 9).

(1)(2 + 3 + ... + 7)(8 + 9 + ... + 154) = 567²,
(1)(2 + 3 + ... + 37)(38 + 39 + ... + 154) = 2808²,
(1)(2 + 3 + ... + 79)(80 + 81 + ... + 154) = 5265²,
(1)(2 + 3 + ... + 145)(146 + 147 + ... + 154) = 3780²,
(1)(2 + 3 + ... + 151)(152 + 153 + 154) = 2295²,
(1 + 2)(3 + 4 + ... + 7)(8 + ... + 154) = 945²,
(1 + 2)(3 + 4 + ... + 51)(52 + 53 + ... + 154) = 6489²,
(1 + 2 + ... + 6)(7)(8 + 9 + ... + 154 ) = 1323²,
(1 + 2 + ... + 6)(7 + 8 + ... + 104 )( 105 + 106 + ... + 154) = 27195²,
(1 + 2 + ... + 9)(10 + 11 + ... + 90 )( 91 + 92 + ... + 154) = 37800²,
(1 + 2 + ... + 12)(13 + 14 + ... + 37 )( 38 + 39 + ... + 154) = 23400²,
(1 + 2 + ... + 12)(13 + 14 + ... + 51 )( 52 + 53 + ... + 154) = 32136²,
(1 + 2 + ... + 16)(17 + 18 + ... + 52 )( 53 + 54 + ... + 154) = 42228²,
(1 + 2 + ... + 17)(18 + 19 + ... + 103 )( 104 + 105 + ... + 154) = 72369²,
(1 + 2 + ... + 26)(27 + 28 + ... + 51 )( 52 + 53 + ... + 154) = 60255²,
(1 + 2 + ... + 26)(27 + 28 + ... + 79 )( 80 + 81 + ... + 154) = 93015²,
(1 + 2 + ... + 37)(38 + 39 + ... + 73 )( 74 + 75 + ... + 154) = 113886²,
(1 + 2 + ... + 48)(49 + 50 + 51 )(52 + 53 + ... + 154) = 43260²,
(1 + 2 + ... + 48)(49 + 50 + ... + 101)(102 + 103 + ... + 154) = 178080²,
(1 + 2 + ... + 48)(49 + 50 + ... + 145)(146 + 147 + ... + 154) = 122220²,
(1 + 2 + ... + 50)(51)(52 + 53 + ... + 154) = 26265²,
(1 + 2 + ... + 50)(51 + 52 + ... + 151)(152 + 153 + 154) = 77265²,
(1 + 2 + ... + 64)(65 + 66 + ... + 79)(80 + 81 + ... + 154) = 140400²,
(1 + 2 + ... + 66)(67 + 68 + ... + 87)(88 + 89 + ... + 154) = 170247²,
(1 + 2 + ... + 78)(79)(80 + 81 + ... + 154) = 46215²,
(1 + 2 + ... + 97)(98 + 99 + ... + 136)(137 + 138 + ... + 154) = 238329².

154² = 23716 appears in the decimal expressions of π and e:
  π = 3.14159•••23716••• (from the 29008th digit),
  e = 2.71828•••23716••• (from the 46473rd digit).

Page of Squares : First Upload June 14, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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155

The smallest squares containing k 155's :
111556 = 334²,
15541611556 = 124666²,
315515500715536 = 17762756².

1 / 155 = 0.0064516..., 64516 = 254².

155² = 24025, 2² + 4² + 0² + 2² + 5² = 7²,
155² = 24025, 2 + 40 + 2 + 5 = 7²,
155² = 24025, 24 + 0 + 25 = 7².

155² = 24025, 2 * 40 * 2 - 5 = 155.

(1² + 2² + 3² + ... + 8²) + (1² + 2² + 3² + ... + 41²) = 155².

496^k + 5270^k + 7161^k + 11098^k are squares for k = 1,2,3 (155², 14229², 1371347²).
1488^k + 2356^k + 9145^k + 11036^k are squares for k = 1,2,3 (155², 14601², 1457837²).

Komach Fractions : (3/155)² = 486/1297350,  (4/155)² = 864/1297350.

Two 3-by-3 magic squares consisting of different squares with constant 155²:

(12² - 1)(13² - 1) = (155² - 1).

(1 + 2 + ... + 30)(31 + 32 + ... + 155) = 2325².

155² = 24025 appears in the decimal expression of e:
  e = 2.71828•••24025••• (from the 88742nd digit).

Page of Squares : First Upload June 21, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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156

The smallest squares containing k 156's :
1156 = 34²,
1562937156 = 39534²,
15611562615609 = 3951147².

The squares which begin with 156 and end in 156 are
1562937156 = 39534²,   15616501156 = 124966²,   15633501156 = 125034²,
156051861156 = 395034²,   156393357156 = 395466²,...

1 / 156 = 0.0064102, 6² + 4² + 10² + 2² = 156.

156² = 24336, 2 + 43 + 36 = 9²,
156² = 24336, 2 + 433 + 6 = 21²,
156² = 24336, 24 + 3 + 3 + 6 = 6².

156² is the 6th square which is the sum of 8 fifth powers : (3, 3, 4, 4, 5, 5, 6, 6).

156² = 14 x 15 x 16 + 16 x 17 x 18 + 18 x 19 x 20 + 20 x 21 x 22.

156² = (3² + 3)(45² + 3) = (3² + 3)(6² + 3)(7² + 3) = (1³ + 1)(23³ + 1).

Komachi equation:
156² = 12 * 345 * 6 - 7 * 8 * 9,
156² = 12² + 3² * 45² - 6² + 78² - 9².

(1 + 2 + ... + 27)(28 + 29 + ... + 35)(36 + 37 + ... + 156) = 33264²,
(1 + 2 + ... + 34)(35 + 36 + ... + 149)(150 + 151 + ... + 156) = 82110²,
(1 + 2 + ... + 121)(122 + 123 + ... + 148)(149 + 150 + ... + 156) = 181170².

(1³ + 2² + ... + 143²)(144² + 144² + ... + 156²) = 68262480².

156² = 24336 appears in the decimal expression of e:
  e = 2.71828•••24336••• (from the 144793rd digit).

Page of Squares : First Upload June 21, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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157

The smallest squares containing k 157's :
157609 = 397²,
21157157025 = 145455²,
157415716157764 = 12546542².

157² = 24649, 2 + 4 + 6 + 4 + 9 = 5²,
157² = 24649, 24⁴ + 6⁴ + 4⁴ + 9⁴ = 583².

157² = 24649, 24 * 6 + 4 + 9 = 157.

157² = 24649, a square pegged by 4.

157² is the 5th square which is the sum of 6 fifth powers : (1, 1, 2, 2, 6, 7).

(12² + 1)(13² + 1) = (157² + 1),
(2² + 1)(4² + 1)(17² + 1) = (3² + 1)(4² + 1)(12² + 1) = (157² + 1),
(1² + 1)(2² + 1)(4² + 1)(12² + 1) = (157² + 1),
(3² + 2)(5² + 2)(9² + 2) = (157² + 2),
(7² - 9)(25² - 9) = (157² - 9).

(1)(2 + 3 + 4 + ... + 76)(77 + 78 + ... + 157) = 5265²,
(1 + 2)(3 + 4 + ... + 122)(123 + 124 + ... + 157) = 10500²,
(1 + 2 + ... + 14)(15 + ... + 139)(140 + 141 + ... + 157) = 51975²,
(1 + 2 + ... + 17)(18 + ... + 67)(68 + 69 + ... + 157) = 57375²,
(1 + 2 + ... + 18)(19 + ... + 62)(63 + 64 + ... + 157) = 56430²,
(1 + 2 + ... + 24)(25 + ... + 31)(32 + 33 + ... + 157) = 26460²,
(1 + 2 + ... + 24)(25 + ... + 32)(33 + 34 + ... + 157) = 28500²,
(1 + 2 + ... + 24)(25 + ... + 66)(67 + 68 + ... + 157) = 76440²,
(1 + 2 + ... + 24)(25 + ... + 122)(123 + 124 + ... + 157) = 102900²,
(1 + 2 + ... + 25)(26 + ... + 52)(53 + 54 + ... + 157) = 61425²,
(1 + 2 + ... + 25)(26 + ... + 76)(77 + 78 + ... + 157) = 89505²,
(1 + 2 + ... + 47)(48 + ... + 77)(78 + 79 + ... + 157) = 141000²,
(1 + 2 + ... + 57)(58 + ... + 132)(133 + 134 + ... + 157) = 206625².

157² = 24649 appears in the decimal expressions of π and e:
  π = 3.14159•••24649••• (from the 65799th digit),
  e = 2.71828•••24649••• (from the 149866th digit).

Page of Squares : First Upload June 21, 2004 ; Last Revised May 29, 2006
by Yoshio Mimura, Kobe, Japan

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158

The smallest squares containing k 158's :
15876 = 126²,
15815829121 = 125761²,
158651582915809 = 12595697².

158² = 24964, 2 + 4 + 9 + 6 + 4 = 5²,
158² = 24964, 24⁴ + 9⁴ + 6⁴ + 4⁴ = 583²,
158² = 24964, 2496 + 4 = 50².

(12² + 2)(13² + 2) = (158² + 2),
(6² - 4)(28² - 4) = (158² - 4).

158² = 20² + 21² + 22² + ... + 43²,
158² + 159² + 160² + ... + 4551² = 177281².

(1 + 2)(3 + 4 + ... + 66)(67 + 68 + ... + 158) = 8280²,
(1 + 2 + ... + 6)(7 + 8 + ... + 149)(150 + 151 + ... + 158) = 18018²,
(1 + 2 + ... + 9)(10 + 11 + ... + 30)(31 + 32 + ... + 158) = 15120²,
(1 + 2 + ... + 9)(10 + 11 + ... + 130)(131 + 132 + ... + 158) = 39270²,
(1 + 2 + ... + 11)(12 + 13 + ... + 33)(34 + 35 + ... + 158) = 19800²,
(1 + 2 + ... + 11)(12 + 13 + ... + 39)(40 + 41 + ... + 158) = 23562²,
(1 + 2 + ... + 11)(12 + 13 + ... + 149)(150 + 151 + ... + 158) = 31878²,
(1 + 2 + ... + 13)(14 + 15 + ... + 130)(131 + 132 + ... + 158) = 55692²,
(1 + 2 + ... + 13)(14 + 15 + ... + 142)(143 + 144 + ... + 158) = 46956²,
(1 + 2 + ... + 15)(16 + 17 + ... + 33)(34 + 35 + ... + 158) = 25200²,
(1 + 2 + ... + 36)(37)(38 + 39 + ... + 158) = 17094²,
(1 + 2 + ... + 36)(37 + 38 + ... + 84)(85 + 86 + ... + 158) = 131868²,
(1 + 2 + ... + 38)(39 + 40 + ... + 101)(102 + 103 + ... + 158) = 155610²,
(1 + 2 + ... + 44)(45 + 46 + ... + 50)(51 + 52 + ... + 158) = 56430²,
(1 + 2 + ... + 44)(45 + 46 + ... + 131)(132 + 133 + ... + 158) = 172260²,
(1 + 2 + ... + 54)(55 + 56 + ... + 125)(126 + 127 + ... + 158) = 210870²,
(1 + 2 + ... + 62)(63 + 64 + 65)(66 + 67 + ... + 158) = 62496²,
(1 + 2 + ... + 63)(64 + 65 + ... + 84)(85 + 86 + ... + 158) = 167832²,
(1 + 2 + ... + 86)(87 + 88 + ... + 129)(130 + 131 + ... + 158) = 269352²,
(1 + 2 + ... + 125)(126 + 127 + ... + 130)(131 + 132 + ... + 158) = 142800².

(1² + 2² + ... + 14²)(15²+ 16² + ... + 87²)(88²+ 89² + ... + 158²) = 15782235²,
(1³ + 2³ + ... + 21³)(22³ + 23³ + ... + 33³)(34³ + 35³ + ... + 158³) = 1481911200².

158² = 24964 appears in the decimal expressions of π and e:
  π = 3.14159•••24964••• (from the 21082nd digit),
  e = 2.71828•••22500••• (from the 13536th digit).

Page of Squares : First Upload June 21, 2004 ; Last Revised May 29, 2006
by Yoshio Mimura, Kobe, Japan

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159

The smallest squares containing k 159's :
159201 = 399²,
115904159809 = 340447²,
159815941599889 = 12641833².

159² = 25281, a zigzag square.

159² = 25281, 2 * 5 * 2 * 8 - 1 = 2 - 5 + 2 * 81 = 159.

159² = 25281, 2 + 5 + 28 + 1 = 6²,
159² = 25281, 25 + 2 + 8 + 1 = 6².

Komachi equation: 159² = 9⁴ + 8⁴ + 7⁴ + 6⁴ - 5⁴ + 4⁴ + 3⁴ * 2⁴ + 10⁴.

Two 3-by-3 magic squares consisting of different squares with constant 159²:

(5² + 3)(30² + 3) = (12² + 3)(13² + 3) = (159² + 3),
(1² + 3)(2² + 3)(30² + 3) = (159² + 3).

(1 + 2 + ... + 27)(28 + 29 + ... + 71)(72 + 73 + ... + 159) = 91476².

159² = 25281 appears in the decimal expression of π:
  π = 3.14159•••25281••• (from the 48832nd digit).

Page of Squares : First Upload June 21, 2004 ; Last Revised April 20, 2010
by Yoshio Mimura, Kobe, Japan

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