---

130

The smallest squares containing k 130's :
130321 = 361²,
61301303281 = 247591²,
130592213013025 = 11427695².

130 = (1² + 2² + 3² + ... + 12²) / (1² + 2²).

130² is the second square which is the sum of 9 seventh powers.

(130²)² = 285610000, 2 + 8 + 5 + 6 + 1 + 0 + 0 + 0 + 0 = 130.

130² = 16900, (16 and 900 are squares).

130² = (1² + 1)(2² + 1)(5² + 1)(8² + 1) = (1² + 1)(5² + 1)(18² + 1) = (1² + 4)(3² + 4)(16² + 4)
= (1² + 9)(2² + 9)(11² + 9) = (1² + 9)(41² + 9) = (3² + 1)(5² + 1)(8² + 1) = (3² + 4)(36² + 4)
= (4² + 4)(29 ² + 4).

(11² - 2)(12² - 2) = (130² - 2).

130^k + 1937^k + 4264^k + 7358^k are squares for k = 1,2,3 (117², 8723², 695097²).
22^k + 73^k + 130^k + 136^k are squares for k = 1,2,3 (19², 203², 2261²).
14^k + 58^k + 122^k + 130^k are squares for k = 1,2,3 (18², 188², 2052²).
130^k + 1794^k + 6994^k + 7982^k are squares for k = 1,2,3 (130², 10764², 925444²).
730^k + 3970^k + 4190^k + 8010^k are squares for k = 1,2,3 (130², 9900², 806500²).
793^k + 1417^k + 2977^k + 11713^k are squares for k = 1,2,3 (130², 12194², 1279330²).
1326^k + 2730^k + 5226^k + 7618^k are squares for k = 1,2,3 (130², 9724², 779428²).
1330^k + 3610^k + 5690^k + 6270^k are squares for k = 1,2,3 (130², 9300², 692900²).
1378^k + 3354^k + 4446^k + 7722^k are squares for k = 1,2,3 (130², 9620², 767260²).

Komachi equations:
130² = 9² * 8² + 7² + 6² * 54² / 3² + 2² - 1² = 98² / 7² / 6² * 54² + 32² */ 1²
  = 9² * 8² * 7² * 65² / 4² / 3² / 21².

(1 + 2 + ... + 11)(12 + 13 + ... + 22)(23 + 24 + ... + 130) = 10098²,
(1 + 2 + ... + 22)(23 + 24 + ... + 76)(77 + 78 + ... + 130) = 61479².

130² = 16900 appears in the decimal expression of e:
  e = 2.71828•••16900••• (from the 58310th digit)

Page of Squares : First Upload May 17, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

131

The smallest squares containing k 131's :
21316 = 146²,
513113104 = 22652²,
613113113176641 = 24761121².

131² = 17161, a zigzag square with 3 kinds of digits.

131² = 17161, 1 + 7 + 1 + 6 + 1 = 4²,
131² = 17161, 1 + 7 + 16 + 1 = 5²,
131² = 17161, 1 + 7 + 161 = 13²,
131² = 17161, 17 + 1 + 6 + 1 = 5².

131² = 17161, a square pegged by 1.

The sum of the consecutive primes 3, 5, 7, .., 131 is 43².

A 3-by-3 magic square consisting of different squares with constant 131²:

(11² - 1)(12² - 1) = (131² - 1),
(3² - 1)(4² - 1)(12² - 1) = (131² - 1).

131² + 132² + 133² + ... + 852² = 14345².

(1 + 2 + ... + 6)(7 + 8 + ... + 110)(111 + 112 + ... + 131) = 18018²,
(1 + 2 + ... + 8)(9 + 10 + ... + 110)(111 + 112 + ... + 131) = 23562²,
(1 + 2 + ... + 86)(87 + 88 + ... + 129)(130 + 131) = 67338².

131² = 17161 appears in the decimal expression of π:
  π = 3.14159•••17161••• (from the 74951st digit).

Page of Squares : First Upload May 17, 2004 ; Last Revised January 7, 2009
by Yoshio Mimura, Kobe, Japan

---

132

The smallest squares containing k 132's :
13225 = 115²,
13213225 = 3635²,
76132413201321 = 8725389².

132² = 17424, 17 * 4 * 2 - 4 = 132.

132² = 28² + 64² + 112² : 211² + 46² + 82² = 231².

132² = (2² + 8)(38² + 8) = (2² + 8)(5² + 8)(6² + 8).

132² = 17424, 1 + 7 + 4 + 24 = 6²,
132² = 17424, 1 + 74 + 2 + 4 = 9²,
132² = 17424, 17 + 424 = 21².

132² is the tenth square which is the sum of 10 fifth powers.

Komachi equations:
132² = 9 + 87 - 6 + 54 * 321,
132² = - 9³ + 8³ * 7³ - 6³ - 54³ + 3³ * 2³ + 1³ = - 9³ + 8³ * 7³ - 6³ * 54³ / 3³ / 2³ + 1³
 = - 9³ + 8³ * 7³ + 6³ - 54³ - 3³ * 2³ + 1³.

(4² + 1)(32² + 1) = (132² + 1),
(3² + 6)(34² + 6) = (132² + 6).

132² + 133² + 134² + ... + 41340² = 4852918²,
132² + 133² + 134² + ... + 430² = 5083².

(1 + 2 + ... + 32)(33) = 132².

(1³ + 2³ + ... + 60³)(61³ + 62³ + ... + 75³)(76³ + 77³ + ... + 132³) = 33195614400².

132² = 17424 appears in the decimal expression of e:
  e = 2.71828•••17424••• (from the 76456th digit)

Page of Squares : First Upload May 17, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

---

133

The smallest squares containing k 133's :
133225 = 365²,
1330133841 = 36471²,
246133613313316 = 15688646².

133² = 17689, a square with different digits.

133² = 27² + 88² + 96² = 69² + 88² + 72².

228^k + 4104^k + 5700^k + 7657^k are squares for k = 1,2,3 (133², 10393², 838603²).
969^k + 5016^k + 5092^k + 6612^k are squares for k = 1,2,3 (133², 9785², 740411²).

Komachi equation: 133² = 98² * 76² * 5² / 4² * 3² / 210².

A 3-by-3 magic square consisting of different squares with constant 133²:

(11² + 1)(12² + 1) = (133² + 1).

(1 + 2 + ... + 64)(65 + 66 + ... + 100)(101 + 102 + ... + 133) = 154440²,
(1 + 2 + ... + 110)(111)(112 + 113 + ... + 133) = 42735².

133² = 17689 appears in the decimal expression of e:
  e = 2.71828•••17689••• (from the 84117th digit)

Page of Squares : First Upload May 17, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

134

The smallest squares containing k 134's :
13456 = 116²,
1134881344 = 33688²,
391341134134336 = 19782344².

134² = 17956, a square with different digits.

134² = 17956, a square with odd digits except the last digit 6.

134² = 10³ + 11³ + 25³.

13² + 43² + 73² + 103² = 134².

Komachi equation: 134² = - 9³ + 8³ + 7³ + 6³ * 5³ + 4³ + 3³ - 21³.

(2² - 1)(4² - 1)(20² - 1) = (134² - 1),
(11² + 2)(12² + 2) = (134² + 2),
(2² + 7)(4² + 7)(8² + 7) = (134² + 7).

(1 + 2 + ... + 35)(36 + 37 + ... + 90)(91 + 92 + ... + 134) = 103950²,
(1 + 2 + ... + 71)(72 + 73 + ... + 78)(79 + 80 + ... + 134) = 89460².

134² = 17956 appears in the decimal expression of e:
  e = 2.71828•••17956••• (from the 39417th digit)

Page of Squares : First Upload May 17, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

135

The smallest squares containing k 135's :
113569 = 337²,
13513597504 = 116248²,
13581355013521 = 3685289².

135² = 18225, 1 + 8 + 2 + 25 = 6²,
135² = 18225, 1 + 8 + 22 + 5 = 6².

(1² + 2² + 3² + ... + 12²) + (1² + 2² + 3² + ... + 37²) = 135².

The sum of any two integers in the set {135, 1890, 3586, 8514} is a square.

1065^k + 3360^k + 6180^k + 7620^k are squares for k = 1,2,3 (135², 10425², 847125²).
2460^k + 2490^k + 3930^k + 9345^k are squares for k = 1,2,3 (135², 10725², 952425²).

Komachi Fraction : (135/2)² = 1804275/396.

Komachi equations:
135² = 1² * 2² + 3² * 45² - 6² + 7² + 8² - 9² = - 1² * 2² + 3² * 45² + 6² - 7² - 8² + 9²
  = 9² * 8² / 7² / 6² * 5² / 4² * 3² * 21² = 9² / 8² * 7² * 6² * 5² * 4² * 3² / 21²
  = 9² / 8² / 7² * 6² * 5² * 4² / 3² * 21².

135 and 136 are consecutive integers having square factors (the 10th case).

Two 3-by-3 magic squares consisting of different squares with constant 135²:

(2² + 3)(51² + 3) = (11² + 3)(12² + 3) = (135² + 3).

(1 + 2)(3)(4 + 5 + 6)(7 + 8)(9) = 135²,
(1)(2 + 3 + ... + 7)(8 + 9 + ... + 37) = 135²,
(1 + 2 + ... + 7)(8 + 9 + ... + 18)(19 + 20 + ... + 135) = 6006²,
(1 + 2 + ... + 7)(8 + 9 + ... + 72)(73 + 74 + ... + 135) = 21840²,
(1 + 2 + ... + 9)(10 + 11 + ... + 14)(15 + 16 + ... + 135) = 4950².

135² = 18225 appears in the decimal expression of e:
  e = 2.71828•••18225••• (from the 81092nd digit)

Page of Squares : First Upload May 24, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

---

136

The smallest squares containing k 136's :
1369 = 37²,
1364711364 = 36942²,
50901361361361 = 7134519².

The squares which begin with 136 and end in 136 are
1364859136 = 36944²,   13675899136 = 116944²,   136119675136 = 368944²,
136202331136 = 369056²,   136488869136 = 369444²,...

(63 / 136)² = 0.214586937... (Komachic).

136² = 18496, a zigzag square with different digits.

136² = 18496, 1 + 8 + 49 + 6 = 8²,
136² = 18496, 1 + 84 + 9 + 6 = 10².

136² + 137² + 138² + ... + 144² = 145² + 146² + 147² + ... + 152².

81^k + 136^k + 304^k + 704^k are squares for k = 1,2,3 (35², 783², 19495²).
22^k + 73^k + 130^k + 136^k are squares for k = 1,2,3 (19², 203², 2261²).
51^k + 1683^k + 6069^k + 10693^k are squares for k = 1,2,3 (136², 12410², 1204552²).
578^k + 2754^k + 6358^k + 8806^k are squares for k = 1,2,3 (136², 11220², 980288²).
1666^k + 2210^k + 5270^k + 9350^k are squares for k = 1,2,3 (136², 11084², 989536²).

Komachi Fraction : 136² = 5160384/279.

(11² + 4)(12² + 4) = (136² + 4).

87² + 88² + 89² + ... + 136² = 795².

136² = 15 x 16 + 17 x 18 + 19 x 20 + 21 x 22 + ... + 47 x 48.

(1 + 2 + 3)(4 + 5 + ... + 15)(16 + 17 + ... + 136) = 2508²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 28)(29 + 30 + ... + 136) = 5940²,
(1 + 2 + ... + 6)(7 + 8 + ... + 97)(98 + 99 + ... + 136) = 21294²,
(1 + 2 + ... + 7)(8 + 9 + 10)(11 + 12 + ... + 136) = 2646²,
(1 + 2 + ... + 7)(8 + 9 + ... + 55)(56 + 57 + ... + 136) = 18144²,
(1 + 2 + ... + 7)(8 + 9 + ... + 97)(98 + 99 + ... + 136) = 24570²,
(1 + 2 + ... + 11)(12 + 13 + ... + 48)(49 + 50 + ... + 136) = 24420²,
(1 + 2 + ... + 23)(24)(25 + 26 + ... + 136) = 7728²,
(1 + 2 + ... + 38)(39 + 40 + ... + 129)(130 + 131 + ... + 136) = 72618²,
(1 + 2 + ... + 74)(75 + 76 + ... + 111)(112 + 113 + ... + 136) = 172050²,
(1 + 2 + ... + 96)(97)(98 + 99 + ... + 136) = 45396².

(1³ + 2³ + ... + 119³)(120³ + 121³ + ... + 136³) = 42725760².

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ + 11³ + 12³ + 13³ + 14³ + 15³ + 16³ = 136².

136² = 18496 appears in the decimal expression of e:
  e = 2.71828•••18496••• (from the 100741st digit)

Page of Squares : First Upload May 24, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

---

137

The smallest squares containing k 137's :
137641 = 371²,
1378191376 = 37124²,
1371377541373225 = 37032115².

1 / 137 = 0.00729..., 729 = 27².

137² = 2⁴ + 2⁴ + 8⁴ + 11⁴.

137² = 18769, a square with different digits.

137² = 18769, 187 + 69 = 16².

A 3-by-3 magic square consisting of different squares with constant 137²:

(11² + 5)(12² + 5) = (137² + 5),
(1² + 5)(4² + 5)(12² + 5) = (2² + 5)(3² + 5)(12² + 5) = (137² + 5).

137² + 138² + 139² + ... + 232² = 1828².

(1 + 2 + ... + 18)(19 + 20 + ... + 57)(58 + 59 + ... + 137) = 44460²,
(1 + 2 + ... + 25)(26 + 27 + ... + 38)(39 + 40 + ... + 137) = 34320²,
(1 + 2 + ... + 32)(33 + 34 + ... + 87)(88 + 89 + ... + 137) = 99000²,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 137) = 127050²,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 137) = 40590².

137² = 18769 appears in the decimal expression of π:
  π = 3.14159•••18769••• (from the 24348th digit).

Page of Squares : First Upload May 24, 2004 ; Last Revised January 7, 2009
by Yoshio Mimura, Kobe, Japan

---

138

The smallest squares containing k 138's :
138384 = 372²,
1381385889 = 37167²,
1381138813832976 = 37163676².

11² + 43² + 75² + 107² = 138².

(11² + 6)(12² + 6) = (138² + 6),
(2² + 6)(3² + 6)(11² + 6) = (138² + 6).

138² = 17² + 18² + 19² + ... + 39².

138^k + 417^k + 582^k + 888^k are squares for k = 1,2,3 (45², 1149², 31185²).
138^k + 570^k + 1086^k + 1122^k are squares for k = 1,2,3 (54², 1668², 53676²).
138^k + 713^k + 5152^k + 7222^k are squares for k = 1,2,3 (115², 8901², 716795²).

(1 + 2 + 3 + 4)(5 + 6 + ... + 129)(130 + 131 + ... + 138) = 10050²,
(1 + 2 + ... + 13)(14 + 15 + ... + 21)(22 + 23 + ... + 138) = 10920²,
(1 + 2 + ... + 13)(14 + 15 + ... + 43)(44 + 45 + ... + 138) = 25935²,
(1 + 2 + ... + 13)(14 + 15 + ... + 103)(104 + 105 + ... + 138) = 45045²,
(1 + 2 + ... + 14)(15 + 16 + ... + 104)(105 + 106 + ... + 138) = 48195²,
(1 + 2 + ... + 17)(18 + 19 + ... + 136)(137 + 138) = 19635²,
(1 + 2 + ... + 26)(27 + 28 + ... + 117)(118 + 119 + ... + 138) = 78624²,
(1 + 2 + ... + 75)(76 + 77 + ... + 113)(114 + 115 + ... + 138) = 179550²,
(1 + 2 + ... + 76)(77 + 78 + ... + 127)(128 + 129 + ... + 138) = 149226²,
(1 + 2 + ... + 110)(111 + 112 + ... + 120)(121 + 122 + ... + 138) = 128205²,
(1 + 2 + ... + 112)(113)(114 + 115 + ... + 138) = 47460².

138² = 19044 appears in the decimal expression of e:
  e = 2.71828•••19044••• (from the 7205th digit)

Page of Squares : First Upload May 24, 2004 ; Last Revised February 10, 2011
by Yoshio Mimura, Kobe, Japan

---

139

The smallest squares containing k 139's :
13924 = 118²,
1397413924 = 37382²,
613904139313921 = 24777089².

139² = 19321, 1 + 9 + 3 + 2 + 1 = 4²,
139² = 19321, 19 + 3 + 2 + 1 = 5²,
139² = 19321, 193 + 2 + 1 = 14².

Komachi equations:
139² = 1 * 23 * 4 * 5 * 6 * 7 - 8 + 9,
139² = 98² + 76² + 54² + 32² + 1².

A 3-by-3 magic square consisting of different squares with constant 139²:

(4² - 7)(6² - 7)(9² - 7) = (139² - 7),
(11² + 7)(12² + 7) = (139² + 7),
(1² + 7)(3² + 7)(12² + 7) = (139² + 7).

(1 + 2 + ... + 49)(50 + 51 + ... + 76)(77 + 78 + ... + 139) = 119070².

139² = 19321 appears in the decimal expression of π:
  π = 3.14159•••19321••• (from the 18470th digit).

Page of Squares : First Upload May 24, 2004 ; Last Revised April 16, 2010
by Yoshio Mimura, Kobe, Japan

---