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110

The smallest squares containing k 110's :
11025 = 105²,
11023110081 = 104991²,
11051110111041 = 3324321².

110 = (1² + 2² + 3² + ... + 275²) / (1² + 2² + 3² + ... + 57²).

110² is the 10th square which is the sum of 7 fourth powers [1,2,4,4,4,5,5].

110² = (2² + 6)(4² + 6)(7² + 6).

34^k + 82^k + 98^k + 110^k are squares for k = 1,2,3 (18², 172², 1692²).
510^k + 690^k + 1330^k + 9570^k are squares for k = 1,2,3 (110², 9700², 937700²).
517^k + 869^k + 3245^k + 7469^k are squares for k = 1,2,3 (110², 8206², 672034²).
830^k + 2010^k + 4570^k + 4690^k are squares for k = 1,2,3 (110², 6900², 455300²).
1265^k + 1529^k + 3817^k + 5489^k are squares for k = 1,2,3 (110², 6974², 476014²).

Komachi equations:
110² = - 987 + 6543 * 2 + 1 = - 98 + 76 / 5 / 4 * 3210,
110² = 1² * 2² + 3² * 45² + 6² - 78² - 9².

(1 + 2 + 3)(4 + 5 + ... + 60)(61 + 62 + ... + 110) = 6840²,
(1 + 2 + ... + 4)(5 + 6 + ... + 10)(11 + 12 + ... + 110) = 1650²,
(1 + 2 + ... + 4)(5 + 6 + ... + 85)(86 + 87 + ... + 110) = 9450²,
(1 + 2 + ... + 9)(10)(11 + 12 + ... + 110) = 1650²,
(1 + 2 + ... + 9)(10 + 11 + ... + 85)(86 + 87 + ... + 110) = 19950²,
(1 + 2 + ... + 27)(28 + 29 + ... + 42)(43 + 44 + ... + 110) = 32130²,
(1 + 2 + ... + 27)(28 + 29 + ... + 71)(72 + 73 + ... + 110) = 54054²,
(1 + 2 + ... + 28)(29 + 30 + ... + 92)(93 + 94 + ... + 110) = 53592²,
(1 + 2 + ... + 32)(33 + 34 + ... + 110) = 1716²,
(1 + 2 + ... + 48)(49 + 50 + ... + 101)(102 + 103 + ... + 110) = 66780²,
(1 + 2 + ... + 49)(50 + 51 + ... + 97)(98 + 99 + ... + 110) = 76440².

110² = 12100 appears in the decimal expressions of π and e:
  π = 3.14159•••12100••• (from the 91463rd digit),
  e = 2.71828•••12100••• (from the 1027th digit)
  12100 is the sixth 5-digit square in the expression of e.

Page of Squares : First Upload April 12, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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111

The smallest squares containing k 111's :
111556 = 334²,
11112111396 = 105414²,
111111500411136 = 10540944².

111² = 12321 is a palindromic square with three kinds of digits.

111² = 12321, 1 + 2 + 3 + 2 + 1 = 3²,
111² = 12321, 1 + 2 + 32 + 1 = 6²,
111² = 12321, 1 + 2 + 321 = 18²,
111² = 12321, 12³ + 3³ + 2³ + 1³ = 42²,
111² = 12321, 12 + 3 + 21 = 6²,
111² = 12321, 123 + 21 = 12².

9³ + 43³ + 77³ + 111³ = 1380².

Komachi equations:
111² = 9876 * 5 / 4 - 3 - 21 = 9 + 8 + 76 * 54 * 3 + 2 - 10
  = 9 - 8 + 76 * 54 * 3 - 2 + 10,
111² = - 9² + 8² + 76² + 54² * 3² / 2² + 1²,
111² = 1³ + 23³ - 4³ - 5³ + 6³ + 7³ + 8³ - 9³ = - 1³ + 23³ - 4³ - 5³ - 6³ + 7³ - 8³ + 9³.

A 3-by-3 magic square consisting of different squares with constant 111²:

(10² + 1)(11² + 1) = (111² + 1),
(3² + 3)(32² + 3) = (111² + 3),
(1² + 4)(5² + 4)(9² + 4) = (111² + 4),
(4² + 7)(23² + 7) = (111² + 7),
(3² + 9)(26² + 9) = (111² + 9).

679² = 15² + 16² + 17² + ... + 111².

(1² + 2² + 3²)(4² + 5² + ... + 35²)(36² + 37² + ... + 111²) = 305368²,
(1² + 2² + ... + 7²)(8² + 9² + ... + 14²)(15² + 16² + ... + 111²) = 237650²,
(1² + ² + ... + 62²)(63² + 64² + ... + 94²)(95² + 96² + ... + 111²) = 54228300².

(1³ + 2² + ... + 27³)(28³ + 29³ + ... + 104³)(105³ + 106³ + ... + 111³) = 6117149808²,
(1³ + 2² + ... + 63³)(64³ + 65³ + ... + 111³) = 11854080².

Page of Squares : First Upload April 12, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

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112

The smallest squares containing k 112's :
11236 = 106²,
61121124 = 7818²,
17061121121121 = 4130511².

112² = 12544,1 + 2 + 5 + 4 + 4 = 4²,
112² = 12544, 12 + 5 + 4 + 4 = 5²,
112² = 12544, 125 + 44 = 13².

112² is the second square which is the sum of 6 squares.

112² = 12544 is a reversible square, 44521 = 211².

112² = 12544, where 1, 25 and 4 are squares.

112² = 12544, 1 * 2 * 54 + 4 = 112.

112²± 3 are primes.

112² = 4⁴ + 8⁴ + 8⁴ + 8⁴.

A cubic polynomial : X³ + 112²X² + 147²X + 396² = (X + 437²)(X + 75012²)(X + 6519744²).

938^k + 1274^k + 3430^k + 6902^k are squares for k = 1,2,3 (112², 7868², 609952²).

Komachi equations:
112² = 12 - 3 + 4 * 56 * 7 * 8 - 9 = - 12 + 3 + 4 * 56 * 7 * 8 + 9
  = - 98 + 7 * 6 + 5 * 4 * 3 * 210,
112² = 98² - 7² + 6² + 54² + 3² * 2² + 1² = - 9² * 8² * 7² / 6² + 5² * 4² / 3² * 21²
  = - 98² / 7² * 6² + 5² * 4² / 3² * 21² = 9² * 8² * 7² / 6² * 5² * 4² / 3² * 2² / 10²
  = 9² * 8² * 7² / 6² / 5² * 4² / 3² / 2² * 10² = 98² / 7² * 6² * 5² * 4² / 3² * 2² / 10²
  = 98² / 7² * 6² / 5² * 4² / 3² / 2² * 10² = 98² / 7² / 6² / 5² * 4² * 3² * 2² * 10².

(10² + 2)(11² + 2) = (112² + 2)

112² + 113² + 114² + ... + 305² = 3007²,
112² + 113² + 114² + ... + 35080² = 3793482².

(1 + 2 + ... + 7)(8 + 9 + ... + 112) = 420²,
(1 + 2 + ... + 32)(33 + 34 + ... + 87)(88 + 89 + ... + 112) = 66000².

112² = 12544 appears in the decimal expression of π:
  π = 3.14159•••12544••• (from the 52032nd digit).

Page of Squares : First Upload April 12, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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113

The smallest squares containing k 113's :
113569 = 337²,
4113811321 = 64139²,
113113711353049 = 10635493².

113² = 12769 is a square with different digits.

113² = 12769 is a reversible square, 96721 = 311².

113² = 12769, 1 + 2 + 7 + 6 + 9 = 5²,
113² = 12769, 127 + 69 = 14².

1³ - 2³ + 3³ - 4³ + ... +113³ = 855².

(1² + 2² + 3² + ... + 113²) = 487369, which consists of different diggts.

113^k + 124^k + 262^k + 590^k are squares for k = 1,2,3 (33², 667², 15057²).

Komachi equations:
113² = 12 / 3 * 456 * 7 - 8 + 9 = 9 - 8 - 7 - 65 + 4 * 3210,
113² = - 1² + 23² * 4² + 5² / 6² * 78² + 9².

A 3-by-3 magic square consisting of different squares with constant 113²:

(10² + 3)(11² + 3) = (113² + 3).

(1 + 2 + ... + 6)(7 + 8 + ... + 110)(111 + 112 + 113) = 6552²,
(1 + 2 + ... + 7)(8 + 9 + ... + 14)(15 + 16 + ... + 113) = 3696²,
(1 + 2 + ... + 8)(9 + 10 + ... + 110)(111 + 112 + 113) = 8568²,
(1 + 2 + ... + 11)(12 + 13 + ... + 33)(34 + 35 + ... + 113) = 13860²,
(1 + 2 + ... + 13)(14 + 15 + ... + 61)(62 + 63 + ... + 113) = 27300²,
(1 + 2 + ... + 15)(16 + 17 + ... + 33)(34 + 35 + ... + 113) = 17640²,
(1 + 2 + ... + 28)(29 + 30 + ... + 55)(56 + 57 + ... + 113) = 47502²,
(1 + 2 + ... + 36)(37 + 38 + ... + 71)(72 + 73 + ... + 113) = 69930².

(1³ + 2³ + ... + 5³)(6³ + 7³ + ... + 38³)(39³ + 40³ + ... + 113³) = 71101800².

113² = 12769 appears in the decimal expression of e:
  e = 2.71828•••12769••• (from the 119470th digit)

Page of Squares : First Upload April 12, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

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114

The smallest squares containing k 114's :
11449 = 107²,
114511401 = 10701²,
51145611411456 = 7151616².

12996(= 114²) -- 1296(= 36²) -- 196(= 14²) -- 16(= 4²) -- 1(= 1²), a chain of squares.

114² = 12996, 12 + 9 + 9 + 6 = 6²,
114² = 12996, 129 + 9 + 6 = 12²,
114² = 12996, 129 + 96 = 15².

114² = 12996, 1 * 2 * 9 + 96 = 129 - 9 - 6 = 114.

Komach equations:
114² = - 98 + 7 + 6543 * 2 + 1 = 98 - 7 + 65 + 4 * 3210,
114² = 1² + 23² * 4² + 5² + 67² - 8² + 9² = 98² - 7² + 65² - 4² / 3² * 21²
  = 9² * 8² * 76² * 5² / 4² / 3² / 2² / 10² = 9² / 8² * 76² * 5² * 4² / 3² * 2² / 10²
  = 9² / 8² * 76² / 5² * 4² / 3² / 2² * 10² = 98² + 7² + 6² + 54² - 3² + 2² * 10².

(10² + 4)(11² + 4) = (114² + 4),
(1² + 4)(3² + 4)(14² + 4) = (2² + 4)(3² + 4)(11² + 4) = (114² + 4).

(1 + 2)(3 + 4 + 5 + 6)(7 + 8 + ... + 114) = 594²,
(1 + 2 + 3)(4 + 5 + ... + 77)(78 + 79 + ... + 114) = 7992²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 29)(30 + 31 + ... + 114) = 5100²,
(1 + 2 + ... + 8)(9 + 10 + ... + 32)(33 + 34 + ... + 114) = 10332²,
(1 + 2 + ... + 27)(28 + 29 + ... + 98)(99 + 100 + ... + 114) = 53676²,
(1 + 2 + ... + 32)(33 + 34 + ... + 39)(40 + 41 + ... + 114) = 27720²,
(1 + 2 + ... + 62)(63 + 64 + ... + 93)(94 + 95 + ... + 114) = 101556².

(1³ + 2³ + ... + 29³)(30³ + 31³ + ... + 38³)(39³ + 40³ + ... + 114³) = 1699548480²,
(1³ + 2³ + ... + 56³)(57³ + 58³ + ... + 75³)(76³ + 77³ + ... + 114³) = 22245383160².

114² = 12996 appears in the decimal expressions of π and e:
  π = 3.14159•••12966••• (from the 4450th digit),
  (12996 is the seventh 5-digit square in the expression of π.)
  e = 2.71828•••12996••• (from the 57755th digit)

Page of Squares : First Upload April 12, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

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115

The smallest squares containing k 115's :
1156 = 34²,
6011521156 = 77534²,
11531158811536 = 3395756².

115² = 13225, 1³ + 3³ + 2³ + 2³ + 5³ = 13².

138^k + 713^k + 5152^k + 7222^k are squares for k = 1,2,3 (115², 8901², 716795²).

Komachi equation: 115² = 9² * 8² + 76² - 5² + 43² + 21².

115² = 1⁵ + 2⁵ + 3⁵ + 4⁵ + 4⁵ + 5⁵ + 6⁵, the sixth square.

(4² + 5)(25² + 5) = (10² + 5)(11² + 5) = (115² + 5),
(1² + 5)(4² + 5)(10² + 5) = (2² + 5)(3² + 5)(10² + 5) = (115² + 5),
(1² + 2² + ... + 11²)*(1² + 2² + ... + 14²) = (1² + 2² + ... + 115²),
(1² + 2² + ... + 11²)*(1² + 2² + ... + 11²)*(1² + 2² + ... + 14²) = (1² + 2² + ... + 105²).

(1² + 2² + 3² + ... + 17²) + (1² + 2² + 3² + ... + 32²) = (115²).

(1² + 2² + 3² + ... + 11²)(1² + 2² + 3² + ... + 14²) = (1² + 2² + 3² + ... + 115²).

115² + 116² + 117² + ... + 7265² = 357550².

(1 + 2 + ... + 10)(11 + 12 + ... + 109)(110 + 111 + ... + 115) = 14850²,
(1 + 2 + ... + 26)(27)(28 + 29 + ... + 115) = 7722²,
(1 + 2 + ... + 45)(46 + 47 + ... + 115) = 2415²,
(1 + 2 + ... + 72)(73)(74 + 75 + ... + 115) = 27594².

115² = 13225 appears in the decimal expression of e:
  e = 2.71828•••13225••• (from the 52141st digit)

Page of Squares : First Upload April 19, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

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116

The smallest squares containing k 116's :
2116 = 46²,
11607261169 = 107737²,
11651169851161 = 3413381².

The squares which begin with 116 and end in 116 are
11654066116 = 107954²,   11673938116 = 108046²,   116249630116 = 340954²,
116312374116 = 341046²,   116590834116 = 341454²,...

116² = 13456, a square with different and increasing digits.

116² = 13456, 1 + 3 + 4 + 56 = 8²,
116² = 13456, 13 + 45 + 6 = 8².

(1² + 2² + 3² + ... + 116²) = 527046, which consists of different digits.

116² = 4³ + 14³ + 22³.

2291^k + 2465^k + 2755^k + 5945^k are squares for k = 1,2,3 (116², 7366², 507964²).

Komachi Square Sum : 116² = 1² + 26² + 49² + 53² + 87² = 1² + 27² + 43² + 59² + 86².

(10² + 6)(11² + 6) = (116² + 6),
(3² + 8)(28² + 8) = (5² + 8)(20² + 8) = (116² + 8),
(3² + 8 )( 4² + 8 )( 5² + 8 ) = (116² + 8).

116 and 117 are consecutive integers having square factors (the 8th case).

655² = 67² + 68² + 69² + ... + 116²,
724² = 21² + 22² + 23² + ... + 116².

116² = 12 x 13 + 14 x 15 + 16 x 17 + 18 x 19 + ... + 42 x 43.

(1 + 2 + ... + 6)(7 + 8 + ... + 11)(12 + 13 + ... + 116) = 2520²,
(1 + 2 + ... + 7)(8 + 9 + ... + 100)(101 + 102 + ... + 116) = 15624²,
(1 + 2 + ... + 8)(9 + 10 + ... + 18)(19 + 20 + ... + 116) = 5670²,
(1 + 2 + ... + 9)(10 + 11)(12 + 13 + ... + 116) = 2520²,
(1 + 2 + ... + 17)(18 + 19 + ... + 84)(85 + 86 + ... + 116) = 41004²,
(1 + 2 + ... + 21)(22 + 23 + ... + 44)(45 + 46 + ... + 116) = 31878²,
(1 + 2 + ... + 27)(28 + 29 + ... + 107)(108 + 109 + ... + 116) = 45360²,
(1 + 2 + ... + 55)(56 + 57 + ... + 98)(99 + 100 + ... + 116) = 99330²,
(1 + 2 + ... + 63)(64 + 65 + ... + 95)(96 + 97 + ... + 116) = 106848².

116² = 13456 appears in the decimal expressions of π and e:
  π = 3.14159•••13456••• (from the 43453rd digit),
  e = 2.71828•••13456••• (from the 14908th digit)

Page of Squares : First Upload April 19, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

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117

The smallest squares containing k 117's :
117649 = 343²,
11711784841 = 108221²,
331171171175881 = 18198109².

117² = 13689, a square with different and increasing digits.

117²± 2 are primes (the 6th case).

117² = 13689, 1 + 3 + 68 + 9 = 9²,
117² = 13689, 13 + 6 + 8 + 9 = 6²,
117² = 13689, 136 + 89 = 15².

78^k + 1092^k + 1950^k + 10569^k are squares for k = 1,2,3 (117², 10803², 1090557²).
130^k + 1937^k + 4264^k + 7358^k are squares for k = 1,2,3 (117², 8723², 695097²).
390^k + 2262^k + 4329^k + 6708^k are squares for k = 1,2,3 (117², 8307², 628173²).
2080^k + 3497^k + 4004^k + 4108^k are squares for k = 1,2,3 (117², 7033², 430443²).

Komachi equations:
117² = 98² + 7² + 6² + 5² * 4² + 3² * 2² * 10² = 98² + 7² + 6² + 5² * 4² * 3² + 2² * 10²
  = 9² * 8² * 7² * 65² / 4² * 3² / 210².

52^k + 91^k + 117^k + 169^k + 247^k are squares (26², 338², 4732², 69290²) for k = 1,2,3,4.

A 3-by-3 magic square consisting of different squares with constant 117²:

(10² + 7)(11² + 7) = (117² + 7),
(1² + 7)(3² + 7)(10² + 7) = (117² + 7).

117² + 118² + 119² + ... + 1797² = 43993²,
117² + 118² + 119² + ... + 789² = 12787².

(1 + 2 + 3)(4 + 5 + ... + 53)(54 + 55 + ... + 117) = 6840²,
(1 + 2 + ... + 15)(16 + 17 + ... + 110)(111 + 112 + ... + 117) = 23940²,
(1 + 2 + ... + 17)(18 + 19 + ... + 32)(33 + 34 + ... + 117) = 19125²,
(1 + 2 + ... + 40)(41 + 42 + ... + 82)(83 + 84 + ... + 117) = 86100²,
(1 + 2 + ... + 69)(70 + 71 + ... + 92)(93 + 94 + ... + 117) = 108675²,
(1 + 2 + ... + 81)(82 + 83 + ... + 87)(88 + 89 + ... + 117) = 71955².

(1³ + 2³ + ... + 87³)(88³ + 89³ + ... + 101³)(102³ + 103³ + ... + 117³) = 60631523568².

117² = 13689 appears in the decimal expression of e:
  e = 2.71828•••13689••• (from the 7730th digit)

Page of Squares : First Upload April 19, 2004 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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118

The smallest squares containing k 118's :
11881 = 109²,
11812211856 = 108684²,
118091189118529 = 10866977².

The square root of 118 is 10.86..., 10² = 8² + 6².

118² = 13924, a square with different digits.

118² = 13924, 1 + 3 + 92 + 4 = 10²,
118² = 13924, 1 + 39 + 24 = 8²,
118² = 13924, 13² + 92² + 4² = 93².

Komachi equation: 118² = - 1² + 2² * 34² + 56² + 78² + 9².

(10² + 8)(11² + 8) = (118² + 8),
(1² + 8)(2² + 8)(11² + 8) = (118² + 8).

(1)(2 + 3 + ... + 37)(38 + 39 + ... + 118) = 2106²,
(1 + 2 + ... + 5)(6 + 7 + ... + 25)(26 + 27 + ... + 118) = 5580²,
(1 + 2 + ... + 12)(13 + 14 + ... + 37)(38 + 39 + ... + 118) = 17550²,
(1 + 2 + ... + 12)(13 + 14 + ... + 39)(40 + 41 + ... + 118) = 18486²,
(1 + 2 + ... + 63)(64 + 65 + ... + 112)(113 + 114 + ... + 118) = 77616².

(1² + 2² + ... + 13²)(14²)(15² + 16² + ... + 118²) = 298116².

(1³ + 2³ + ... + 8³)(9³ + 10³ + ... + 13³)(14³ + 15³ + ... + 118³) = 21122640².

118² = 13924 appears in the decimal expressions of π and e:
  π = 3.14159•••13924••• (from the 94645th digit),
  e = 2.71828•••13924••• (from the 106552nd digit)

Page of Squares : First Upload April 19, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

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119

The smallest squares containing k 119's :
119025 = 345²,
119311929 = 10923²,
1119191192119225 = 33454315².

119² = 14161, a zigzag square with 3 kinds of digits.

119² = 14161, 1 + 41 + 6 + 1 = 7².

119² = 14161, a square by pegged by 1.

119² = 14161 (1,4,16 and 1 are squares).

119² = (5 + 6 + 7 + 8 + 9 + 10 + 11)² + (12 + 13 + 14 + 15 + 16 + 17 + 18)².

Komachi Fractions : 892143/567 = (119/3)², 2039184/576 = (119/2)².

Komachi equation: 119² = 1⁴ - 2⁴ + 3⁴ * 4⁴ + 56⁴ / 7⁴ / 8⁴ - 9⁴.

Two 3-by-3 magic squares consisting of different squares with constant 119²:

(1² + 7)(42² + 7) = (119² + 7),
(2² + 7)(4² + 7)(7² + 7) = (119² + 7),
(10² + 9)(11² + 9) = (119² + 9),
(1² + 9)(2² + 9)(10² + 9) = (119² + 9).

119² + 120² = 169².

(1 + 2 + ... + 12)(13 + 14 + ... + 23)(24 + 25 + ... + 119) = 10296²,
(1 + 2 + ... + 84)(85 + 86 + ... + 119) = 3570².

119² = 14161 appears in the decimal expressions of π and e:
  π = 3.14159•••14161••• (from the 58851st digit),
  e = 2.71828•••14161••• (from the 24398th digit)

Page of Squares : First Upload April 19, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

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