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140

The smallest squares containing k 140's :
140625 = 375²,
140351409 = 11847²,
140140540581409 = 11838097².

A Cubic Polynomial : (X + 105²)(X + 140²)(X + 288²) = X³ + 337²X² + 52500²X + 4233600².

A, B, C, A + B, B + C, and C + A are squares, where A = 140², B = 480², C = 693².

140²± 3 are primes.

(1² + 2)(4² + 2)(19² + 2) = (140² + 2),
(3² + 2)(5² + 2)(8² + 2) = (2² - 2)(99² - 2) = (140² - 2),
(11² + 8)(12² + 8) = (140² + 8).

Komachi equation: 140² = - 9² + 8² * 7² + 6² + 5² + 4² * 32² + 10².

140² + 141² + 142² + ... + 428² = 5032².

(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13) = 140²,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11 + 12 + 13)(14) = 140².

(1³ + 2³ + ... + 15³)(16³ + 17³ + ... + 119³)(120³ + 121³ + ... + 140³) = 5837832000².

140 = (1² + 2² + 3² + 4² + 5² + 6² + 7²).

140² = 19600 appears in the decimal expressions of π and e:
  π = 3.14159•••19600••• (from the 21337th digit),
  e = 2.71828•••19600••• (from the 1954th digit)
  19600 is the ninth 5-digit square in the expression of e.

Page of Squares : First Upload May 31, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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141

The smallest squares containing k 141's :
14161 = 119²,
14114152809 = 118803²,
645814114114129 = 25412873².

141² = 19881, a square with 3 kinds of digits.

141² = 19881, 19 + 8 + 8 + 1 = 6²,
141² = 19881, 19 + 881 = 30².

141² is the fourth square which is the sum of 6 fifth powers : 1,1,4,4,4,7

141^k + 1645^k + 2773^k + 4277^k are squares for k = 1,2,3 (94², 5358², 322514²).

Komachi equations:
141² = 123² + 4² - 56² - 7² + 89² = 987² / 6² * 54² / 3² / 21²,
141² = 9³ + 8³ * 7³ + 6³ - 54³ - 3³ * 2³ + 10³ = 9³ + 8³ * 7³ - 6³ - 54³ + 3³ * 2³ + 10³
  = 9³ + 8³ * 7³ - 6³ * 54³ / 3³ / 2³ + 10³.

Two 3-by-3 magic squares consisting of different squares with constant 141²:

(2² + 9)(5² + 9)(6² + 9) = (141² + 9),
(2² + 9)(39² + 9) = (5² + 9)(24² + 9) = (11² + 9)(12² + 9) = (141² + 9),
(1² + 9)(2² + 9)(12² + 9) = (2² + 9)(5² + 9)(6² + 9) = (141² + 9).

(1)(2 + ... + 78)(79 + ... + 141) = 4620²,
(1 + ... + 6)(7 + ... + 20)(21 + ... + 141) = 6237²,
(1 + ... + 14)(15 + ... + 20)(21 + ... + 141) = 10395²,
(1 + ... + 14)(15 + ... + 50)(51 + ... + 141) = 32760²,
(1 + ... + 27)(28 + 29)(30 + ... + 141) = 14364²,
(1 + ... + 27)(28 + ... + 105)(106 + ... + 141) = 93366²,
(1 + ... + 64)(65 + ... + 78)(79 + ... + 141) = 120120,
(1 + ... + 91)(92 + ... + 118)(119 + ... + 141) = 188370².

(1³ + ... + 33³)(34³ + ... + 101³)(102³ + ... + 141³) = 24658104240^2.

141² = 19881 appears in the decimal expression of π:
  π = 3.14159•••19881••• (from the 1535th digit),
  (19881 is the fourth 5-digit square in the expression of π.)

Page of Squares : First Upload May 31, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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142

The smallest squares containing k 142's :
114244 = 338²,
1429142416 = 37804²,
1311427142631424 = 36213632².

1 / 142 = 0.00704225352112..., 7² + 0² + 4² + 2² + 2² + 5² + 3² + 5² + 2² + 1² + 1² + 2² = 142.

142² = 20164, a square with different digits.

142² = 20164, 2³ + 0³ + 1³ + 6³ + 4³ = 17²,
142² = 20164, 20³ + 1³ + 6³ + 4³ = 91².

142² = 2! + 2! + 7! + 7! + 7! + 7!

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

Komachi Square Sum : 142² = 1² + 3² + 5² + 64² + 87² + 92² = 2² + 3² + 5² + 61² + 87² + 94² = 3² + 5² + 7² + 9² + 68² + 124².

(3² - 4)(5² - 4)(14² - 4) = (142² - 4).

142² + 143² + 144² + ... + 3287² = 108823².

(1 + 2 + 3 + 4)(5 + 6 + 7)(8 + ... + 142) = 1350²,
(1 + 2 + 3 + 4)(5 + ... + 85)(86 + ... + 142) = 15390²,
(1 + ... + 5)(6 + ... + 42)(43 + ... + 142) = 11100²,
(1 + ... + 9)(10 + ... + 85)(86 + ... + 142) = 32490²,
(1 + ... + 27)(28 + ... + 97)(98 + ... + 142) = 94500²,
(1 + ... + 27)(28 + ... + 107)(108 + ... + 142) = 94500².

Page of Squares : First Upload May 31, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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143

The smallest squares containing k 143's :
143641 = 379²,
51431436225 = 226785²,
143371431435961 = 11973781².

2² + 9² + 16² + 23² + 30² + 37 ² + 44² + 51² + 58² + 65² + 72² = 143².

(1² + 2² + 3² + ... + 143²) = 984984. which consists of 3 kinds of digits.

1716^k + 2184^k + 6097^k + 10452^k are squares for k = 1,2,3 (143², 12415², 1176409²).
3553^k + 4554^k + 5742^k + 6600^k are squares for k = 1,2,3 (143², 10483², 784927²).

143² = 3⁰ + 3² + 3³ + 3⁶ + 3⁹.

Komachi Fraction : 143² = 9570132/468.

143 is the sum of a prime and a square in 5 ways :
2² + 139, 4² + 127, 6² + 107, 8² + 79, 10² + 43.

143² = 7² + 8² + 9² + ... + 39²,
143² = 38² + 39² + 40² + ... + 48².

A 3-by-3 magic square consisting of different squares with constant 143²:

(1 + ... + 4)(5 + ... + 115)(116 + ... + 143) = 15540²,
(1 + ... + 9)(10 + ... + 24)(25 + ... + 143) = 10710²,
(1 + ... + 9)(10 + ... + 111)(112 + ... + 143) = 33660²,
(1 + ... + 17)(18 + ... + 24)(25 + ... + 143) = 14994²,
(1 + ... + 17)(18 + ... + 102)(103 + ... + 143) = 62730²,
(1 + ... + 21)(22 + ... + 98)(99 + ... + 143) = 76230²,
(1 + ... + 31)(32 + ... + 61)(62 + ... + 143) = 76260²,
(1 + ... + 31)(32 + ... + 73)(74 + ... + 143) = 91140²,
(1 + ... + 31)(32 + ... + 112)(113 + ... + 143) = 107136²,
(1 + ... + 39)(40 + ... + 129)(130 + ... + 143) = 106470².

143² = 20449 appears in the decimal expressions of π and e:
  π = 3.14159•••20449••• (from the 59662nd digit),
  e = 2.71828•••20449••• (from the 333rd digit)
  (20449 is the third 5-digit square in the expression of e.)

Page of Squares : First Upload May 31, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

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144

The square of 12.

The smallest squares containing k 144's :
144 = 12²,
36144144 = 6012²,
10144301440144 = 3185012².

The squares which begin with 144 and end in 144 are
144288144 = 12012²,   1443088144 = 37988²,   1444912144 = 38012²,
14402880144 = 120012²,   144011142144 = 379488²,...

A reversible square, 441 = 21².

144² = 20736, 2 + 0 + 73 + 6 = 9²,
144² = 20736, 20 + 7 + 3 + 6 = 6².

The first exchangeable square : 12² = 144, 441 = 21².

144² is the 8th square which is the sum of 9 sixth powers.

144² = 20736, a zigzag square with different digits.

144² = (1 + 2 + 3)² + (4 + 5 + 6)² + (7 + 8 + 9)² + ... + (25 + 26 + 27)².

144² = (2² + 8)(4² + 8)(8² + 8) = (2² - 1)(5² - 1)(17² - 1).

Cubic polynomials:
  (X + 44²)(X + 57²)(X + 144²) = X³ + 161²X² + 10668²X + 361152²,
  (X + 87²)(X + 144²)(X + 364²) = X³ + 401²X² + 62508²X + 4560192².

A Fibonacci square.

Komachi equations:
144² = 9876 + 543 * 2 * 10,
144² = 12² / 3² * 4² * 56² / 7² / 8² * 9² = 12² / 3² * 4² / 56² * 7² * 8² * 9².

121² + 122² + 123² + ... + 144² = 650².

(1 + ... + 8)(9 + ... + 144) = 612²,
(1 + ... + 14)(15 + ... + 135)(136 + ... + 144) = 34650²,
(1 + ... + 20)(21 + ... + 119)(120 + ... + 144) = 69300²,
(1 + ... + 24)(25 + ... + 105)(106 + ... + 144) = 87750²,
(1 + ... + 25)(26 + ... + 79)(80 + ... + 144) = 81900².

144² = 20736 appears in the decimal expression of e:
  e = 2.71828•••20736••• (from the 57966th digit)

Page of Squares : First Upload May 31, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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145

The smallest squares containing k 145's :
145161 = 381²,
11451456 = 3384²,
1452814514566416 = 38115804².

(129 / 145)² = 0.791486325... (Komachic).

145³ - 144³ + 143³ - 142³ + ... + 1³ = 1241².

(1² + 2² + 3² + ... + 145²) = 1026745, which consists of different digits,
(the first 7-digit integer taht is the sum of the consecutive squares : 1² + 2² + 3² + ... + n²).

Loop of length 8 by the function f(N) = ... + c² + b² + a² for N = ... + 10²c + 10b + a:
145 -- 42 -- 20 -- 4 -- 16 -- 37 -- 58 -- 145

42^k + 66^k + 108^k + 145^k are squares for k = 1,2,3 (19², 197², 2161²).
609^k + 4524^k + 6960^k + 8932^k are squares for k = 1,2,3 (145², 12209², 1068911²).

Komachi Fraction : 2081475/396 = (145/2)².

Komachi equations:
145² = 1 - 2 - 34 + 5 * 6 * 78 * 9 = 9 + 87 + 654 * 32 + 1,
145² = 9² + 87² + 6² * 5² * 4² - 32² - 1² = 9² * 87² * 6² / 54² / 3² / 2² * 10²
  = 9² * 87² / 6² / 54² * 3² * 2² * 10².

A 3-by-3 magic square consisting of different squares with constant 145²:

(7² + 8)(19² + 8) = (145² + 8).

(1 + 2)(3 + 4 + 5 + ... + 46)(47 + 48 + 49 + ... + 145) = 5544²,
(1 + 2 + 3 + ... + 49)(50 + 51 + 52 + ... + 97)(98 + 99 + 100 + ... + 145) = 158760²,
(1 + 2 + 3 + ... + 65)(66 + 67 + 68 + ... + 129)(130 + 131 + 132 + ... + 145) = 171600².

145² = 21025 appears in the decimal expressions of π and e:
  π = 3.14159•••21025••• (from the 12924th digit),
  e = 2.71828•••21025••• (from the 119850th digit)

Page of Squares : First Upload June 7, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

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146

The smallest squares containing k 146's :
14641 = 121²,
14614633881 = 120891²,
146146314631801 = 12089099².

146² = 21316, a zigzag square pegged by 1.

146^k + 154^k + 654^k + 1546^k are squares for k = 1,2,3 (50², 1692², 63100²).

146²± 3 are primes.

Komachi Fraction : 567/1342908 = (3/146)².

(3² + 2)(44² + 2) = (146² + 2),
(3² + 2)(6² + 2)(7² + 2) = (146² + 2),
(6² + 4)(23² + 4) = (146² + 4),
(1² + 4)(2² + 4)(23² + 4) = (1² + 4)(3² + 4)(18² + 4) = (146² + 4).

146² + 147² + 148² + ... + 3578² = 123588².

(1)(2 + 3 + ... + 109)(110 + 111 + ... + 146) = 5328²,
(1 + 2 + 3)(4 + 5 + ... + 29)(30 + 31 + ... + 146) = 5148²,
(1 + 2 + 3)(4 + 5 + ... + 48)(49 + 50 + ... + 146) = 8190²,
(1 + 2 + 3)(4 + 5 + ... + 113)(114 + 115 + ... + 146) = 12870²,
(1 + 2 + ... + 12)(13 + 14 + ... + 65)(66 + 67 + ... + 146) = 37206²,
(1 + 2 + ... + 13)(14 + 15 + ... + 133)(134 + 135 + ... + 146) = 38220²,
(1 + 2 + ... + 14)(15 + 16 + ... + 83)(84 + 85 + ... + 146) = 50715²,
(1 + 2 + ... + 24)(25 + 26 + ... + 96)(97 + 98 + ... + 146) = 89100²,
(1 + 2 + ... + 27)(28 + 29 + ... + 125)(126 + 127 + ... + 146) = 89964²,
(1 + 2 + ... + 35)(36 + 37 + ... + 48)(49 + 50 + ... + 146) = 57330²,
(1 + 2 + ... + 36)(37 + 38 + ... + 109)(110 + 111 + ... + 146) = 129648²,
(1 + 2 + ... + 40)(41 + 42 + ... + 58)(59 + 60 + ... + 146) = 81180²,
(1 + 2 + ... + 41)(42 + 43 + ... + 105)(106 + 107 + ... + 146) = 144648²,
(1 + 2 + ... + 62)(63 + 64 + ... + 84)(85 + 86 + ... + 146) = 150381²,
(1 + 2 + ... + 116)(117 + 118 + ... + 143)(144 + 145 + 146) = 101790².

146² = 21316 appears in the decimal expression of e:
  e = 2.71828•••21316••• (from the 128657th digit)

Page of Squares : First Upload June 7, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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147

The smallest squares containing k 147's :
147456 = 384²,
14757147441 = 121479²,
147147234941476 = 12130426².

147² = 21609, a zigzag square with different digits.

147² = (4² + 5)(32² + 5).

147² = 21609, 21 + 6 + 0 + 9 = 6²,
147² = 21609, 216 + 0 + 9 = 15².

Cubic Polynomial : (X + 112²)(X + 147²)(X + 396²) = X³ + 437²X² + 75012²X + 6519744².

Komachi equations:
147² = 9 + 8 - 7 - 6 + 5 * 4321 = - 9 * 8 + 76 + 5 * 4321,
147² = - 9² * 8² * 7² + 6² * 5² / 4² / 3² * 210²,
147² = 9⁵ - 8⁵ - 7⁵ + 6⁵ + 5⁵ + 4⁵ + 3⁵ - 2⁵ - 1⁵.

Three 3-by-3 magic squares consisting of different squares with constant 141²:

(12² - 9)(13² - 9) = (147² - 9).

(1² + 2² + 3² + ... + 50²) + (1² + 2² + 3² + ... + 145²) = (1² + 2² + 3² + ... + 147²).

52² + 53² + 54² + ... + 147² = 1012².

(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 147 ) = 2160²,
(1 + 2 + 3 + 4)( 5 + 6 + ... + 52)(53 + 54 + ... + 147 ) = 11400²,
(1 + 2 + 3 + ... + 17)(18 + 19 + ... + 22)(23 + 24 + ... + 147 ) = 12750²,
(1 + 2 + 3 + ... + 80)(81 + 82 + ... + 120)(121 + 122 + ... + 147 ) = 217080².

147² = 21609 appears in the decimal expressions of π and e:
  π = 3.14159•••21609••• (from the 53688th digit),
  e = 2.71828•••21609••• (from the 831st digit)
  (21609 is the fifth 5-digit square in the expression of e.)

Page of Squares : First Upload June 7, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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148

The smallest squares containing k 148's :
14884 = 122²,
14814801 = 3849²,
148694148148441 = 12194021².

148 = (1² + 2² + 3² + ... + 55²) / (1² + 2² + 3² + ... + 10²).

148² = 21904, a zigzag square with different digits.

148² = 21904, 2 + 1 + 9 + 0 + 4 = 4²,
148² = 21904, 2 + 19 + 0 + 4 = 5²,
148² = 21904, 2 + 190 + 4 = 14².

Komachi Fraction : 3267/591408 = (11/148)².

Komachi equation: 148² = 9³ * 8³ - 76³ + 5³ + 43³ + 2³ * 10³.

(5² - 8)(36² - 8) = (12² - 8)(13² - 8) = (148² - 8),
(4² - 8)(5² - 8)(13² - 8) = (148² - 8).

(1)(2 + 3 + ... + 148) = 105²,
(1 + 2 + 3)(4 + 5 + ... + 60)(61 + 62 + ... + 148) = 10032²,
(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + 22 + ... + 148) = 6552²,
(1 + 2 + ... + 8)(9 + 10 + ... + 71)(72 + 73 + ... + 148) = 27720²,
(1 + 2 + ... + 14)(15 + 16 + ... + 20)(21 + 22 + ... + 148) = 10920²,
(1 + 2 + ... + 45)(46 + 47 + ... + 58)(59 + 60 + ... + 148) = 80730²,
(1 + 2 + ... + 45)(46 + 47 + ... + 115)(116 + 117 + ... + 148) = 159390²,
(1 + 2 + ... + 49)(50 + 51 + ... + 148) = 3465²,
(1 + 2 + ... + 64)(65 + 66 + ... + 103)(104 + 105 + ... + 148) = 196560²,
(1 + 2 + ... + 64)(65 + 66 + ... + 124)(125 + 126 + ... + 148) = 196560².

(1³ + 2³ + ... + 20³)(21³ + 22³ + ... + 148³) = 2315040²,

148² = 21904 appears in the decimal expression of e:
  e = 2.71828•••21904••• (from the 17487th digit)

Page of Squares : First Upload June 7, 2004 ; Last Revised April 20, 2010
by Yoshio Mimura, Kobe, Japan

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149

The smallest squares containing k 149's :
114921 = 339²,
114957614916 = 339054²,
1492481491491904 = 38632648².

149² = 22201, a square with 3 kinds of digits.

149² = 22201, 2³ + 2³ + 2³ + 0³ + 1³ = 5²,
149² = 22201, 2⁴ + 2⁴ + 2⁴ + 0⁴ + 1⁴ = 7²,
149² = 22201, 2 + 2 + 20 + 1 = 5²,
149² = 22201, 2 + 22 + 0 + 1 = 5²,
149² = 22201, 22 + 2 + 0 + 1 = 5².

(1² + 2² + 3² + ... + 149²) = 1113775, which consists of odd digits,
(the second 7-digit sum which consists of odd digits, cf. 174)

Komachi equations:
149² = 9² - 8² + 76² + 5² + 4² * 32² - 1² = 9² / 87² / 6² / 5² * 43210².

A 3-by-3 magic square consisting of different squares with constant 149²:

(12² - 7)(13² - 7) = (149² - 7),
(4² - 7)(5² - 7)(12² - 7) = (149² - 7).

(1 + 2)(3 + 4 + ... + 11)(12 + 13 + ... + 149) = 1449²,
(1 + 2)(3 + 4 + ... + 21)(22 + 23 + ... + 149) = 2736²,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 149) = 4095²,
(1 + 2)(3 + 4 + ... + 92)(93 + 94 + ... + 149) = 9405²,
(1 + 2)(3 + 4 + ... + 123)(124 + 125 + ... + 149) = 9009²,
(1 + 2 + 3)(4 + ... + 122)(123 + 124 + ... + 149) = 12852²,
(1 + 2 + ... + 6)(7 + 8 + ... + 32)(33 + ... + 149) = 10647²,
(1 + 2 + ... + 11)(12 + 13 + ... + 32)(33 + ... + 149) = 18018²,
(1 + 2 + ... + 17)(18 + 19 + ... + 122)(123 + ... + 149) = 64260²,
(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + ... + 149) = 51051²,
(1 + 2 + ... + 18)(19 + 20 + ... + 92)(93 + ... + 149) = 69597²,
(1 + 2 + ... + 21)(22 + 23 + ... + 32)(33 + ... + 149) = 27027²,
(1 + 2 + ... + 21)(22 + 23 + ... + 147)(148 + 149) = 27027²,
(1 + 2 + ... + 27)(28 + 29 + ... + 32)(33 + ... + 149) = 24570²,
(1 + 2 + ... + 27)(28 + 29 + ... + 126)(127 + ... + 149) = 95634²,
(1 + 2 + ... + 34)(35 + 36 + ... + 84)(85 + ... + 149) = 116025²,
(1 + 2 + ... + 47)(48 + 49 + ... + 132)(133 + ... + 149) = 143820²,
(1 + 2 + ... + 48)(49 + 50 + ... + 147)(148 + 149) = 58212²,
(1 + 2 + ... + 50)(51 + 52 + ... + 84)(85 + ... + 149) = 149175²,
(1 + 2 + ... + 53)(54 + 55 + ... + 62)(63 + ... + 149) = 82998²,
(1 + 2 + ... + 57)(58 + 59 + ... + 111)(112 + ... + 149) = 193401²,
(1 + 2 + ... + 67)(68 + 69 + ... + 118)(119 + ... + 149) = 211854²,
(1 + 2 + ... + 72)(73 + 74 + ... + 146)(147 + 148 + 149) = 97236²,
(1 + 2 + ... + 74)(75 )(76 + 77 + ... + 149) = 41625²,
(1 + 2 + ... + 74)(75 + 76 + ... + 110)(111 + ... + 149) = 216450²,
(1 + 2 + ... + 75)(76 + 77 + ... + 95)(96 + ... + 149) = 179550²,
(1 + 2 + ... + 81)(82 + 83 + ... + 122)(123 + ... + 149) = 225828².

(1³ + 2³ + ... + 80³)(81³ + 82 + ... + 149³) = 34651800²,
(1³ + 2³ + ... + 110³)(111³ + 112³ + ... + 149³) = 57142800².

149² = 22201 appears in the decimal expression of π:
  π = 3.14159•••22201••• (from the 10427th digit).

Page of Squares : First Upload June 7, 2004 ; Last Revised April 20, 2010
by Yoshio Mimura, Kobe, Japan

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