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170

The smallest squares containing k 170's :
170569 = 413²,
17081705809 = 130697²,
1704017017027081 = 41279741².

The square root of 170 is 13.0384048..., 13² = 0² + 3² + 8² + 4² + 0² + 4² + 8².

170² = (1² + 1)(2² + 1)(4² + 1)(13² + 1)
= (1² + 1)(3² + 1)(38² + 1) = (3² + 1)(4² + 1)(13² + 1).

170² = (1² + 1)(2² + 1)(4² + 1)(13² + 1) = (1² + 1)(3² + 1)(38² + 1) = (1² + 4)(76² + 4)
= (1² + 4)(8² + 4)(9² + 4) = (3² + 1)(4² + 1)(13² + 1) = (5² + 9)(29² + 9).

(2² + 8)(49² + 8) = (170² + 8).

170^k + 370^k + 830^k + 1130^k are squares for k = 1,2,3 (50², 1460², 45500²).
46^k + 170^k + 202^k + 258^k are squares for k = 1,2,3 (26², 372², 5516²).
49^k + 98^k + 170^k + 212^k are squares for k = 1,2,3 (23², 293², 3937²).

Komachi equations:
170² = 1² * 2² + 34² * 5² - 6² + 7² + 8² - 9² = - 1² * 2² + 34² * 5² + 6² - 7² - 8² + 9².

170² + 171² + 172² + ... + 466² = 5676².

(1 + 2 + ... + 5)(6 + 7 + ... + 104)(105 + 106 + ... + 170) = 27225²,
(1 + 2 + ... + 9)(10 + 11 + ... + 15)(16 + 17 + ... + 170) = 6975²,
(1 + 2 + ... + 9)(10 + 11 + ... + 104)(105 + 106 + ... + 170) = 47025²,
(1 + 2 + ... + 9)(10 + 11 + ... + 159)(160 + 161 + ... + 170) = 32175²,
(1 + 2 + ... + 21)(22 + 23 + ... + 77)(78 + 79 + ... + 170) = 85932²,
(1 + 2 + ... + 25)(26 + 27 + ... + 79)(80 + 81 + ... + 170) = 102375²,
(1 + 2 + ... + 25)(26 + 27 + ... + 90)(91 + 92 + ... + 170) = 113100²,
(1 + 2 + ... + 25)(26 + 27 + ... + 144)(145 + 146 + ... + 170) = 116025²,
(1 + 2 + ... + 26)(27)(28 + 29 + ... + 170) = 11583²,
(1 + 2 + ... + 27)(28 + 29 + ... + 126)(127 + 128 + ... + 170) = 137214²,
(1 + 2 + ... + 32)(33 + 34 + ... + 71)(72 + 73 + ... + 170) = 113256²,
(1 + 2 + ... + 32)(33 + 34 + ... + 77)(78 + 79 + ... + 170) = 122760²,
(1 + 2 + ... + 47)(48 + 49 + ... + 123)(124 + 125 + ... + 170) = 225036²,
(1 + 2 + ... + 49)(50 + 51 + ... + 71)(72 + 73 + ... + 170) = 139755²,
(1 + 2 + ... + 90)(91 + 92 + ... + 104)(105 + 106 + ... + 170) = 225225²,
(1 + 2 + ... + 117)(118)(119 + 120 + ... + 170) = 78234²,
(1 + 2 + ... + 152)(153 + 154 + ... + 170) = 5814².

(1³ + 2³ + ... + 152³)(153³ + 154³ + ... + 170³) = 101407788².

170² = 28900 appears in the decimal expressions of π and e:
  π = 3.14159•••28900••• (from the 39660th digit),
  e = 2.71828•••28900••• (from the 14989th digit)

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

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171

The smallest squares containing k 171's :
17161 = 131²,
191711716 = 13846²,
17117127171225 = 4137285².

171² = 29241, a zigzag square.

171² = 29241, 2 + 9 + 24 + 1 = 6²,
171² = 29241, 29 + 2 + 4 + 1 = 6².

171² + 172² + 173² + ... + 180² = 181² + 182² + 183² + ... + 189².

A cubic polynomial : (X + 64²)(X + 171²)(X + 432²) = X³ + 469²X² + 79632²X + 4727808².

(1² + 9)(54² + 9) = (171² + 9),
(1² + 9)(54² + 9) = (171² + 9),
(2² + 9)(4² + 9)(9² + 9) = (171² + 9).

1281² = 50² + 51² + 52² + ... + 171².

266^k + 6232^k + 10070^k + 12673^k are squares for k = 1,2,3 (171², 17347², 1816191²).
1330^k + 8417^k + 8626^k + 10868^k are squares for k = 1,2,3 (171², 16283², 1588761²).
2812^k + 4940^k + 8056^k + 13433^k are squares for k = 1,2,3 (171², 16663², 1757709²).

Komachi equation: 171² = 9 + 87 * 6 / 5 * 4 / 3 * 210.

Four 3-by-3 magic squares consisting of different squares with constant 171²:

(1)(2 + 3 + ... + 52)(53 + 54 + ... + 171) = 4284²,
(1 + 2 + 3)(4 + 5 + ... + 108)(109 + 110 + ... + 171) = 17640²,
(1 + 2 + ... + 15)(16 + 17 + ... + 24)(25 + 26 + ... + 171) = 17640²,
(1 + 2 + ... + 15)(16 + 17 + ... + 80)(81 + 82 + ... + 171) = 65520²,
(1 + 2 + ... + 15)(16 + 17 + ... + 83)(84 + 85 + ... + 171) = 67320²,
(1 + 2 + ... + 15)(16 + 17 + ... + 108)(109 + 110 + ... + 171) = 78120²,
(1 + 2 + ... + 17)(18 + 19 + ... + 52)(53 + 54 + ... + 171) = 49980²,
(1 + 2 + ... + 23)(24 + 25 + ... + 161)(162 + 163 + ... + 171) = 76590²,
(1 + 2 + ... + 49)(50 + 51 + 52)(53 + 54 + ... + 171) = 49980²,
(1 + 2 + ... + 80)(81 + 82 + ... + 143)(144 + 145 + ... + 171) = 317520²,
(1 + 2 + ... + 93)(94 + 95 + ... + 140)(141 + 142 + ... + 171) = 340938².

1³ + 2³ + 3³ + 4³ + ... + 18³ = 171².

(1³ + 2³ + ... + 15³)(16³ + 17³ + ... + 75³)(76³ + 77³ + ... + 171³) = 4929724800²,
(1³ + 2³ + ... + 49³)(50³ + 51³ + ... + 129³)(130³ + 131³ + ... + 171³) = 122763301500².

171² = 29241 appears in the decimal expression of e:
  e = 2.71828•••29241••• (from the 4444th digit)

Page of Squares : First Upload July 12, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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172

The smallest squares containing k 172's :
172225 = 415²,
317231721 = 17811²,
172831172161729 = 13146527².

172² = 29584, a zigzag square with different digits.

172²± 3 are primes.

172² = 29584, 2 + 9 + 5 + 84 = 10².

34^k + 82^k + 98^k + 110^k are squares for k = 1,2,3 (18², 172², 1692²).

Komachi equation: 172² = 987 * 6 * 5 - 4 - 32 + 10.

(4² + 8)(35² + 8) = (172² + 8).

172² = 27² + 29² + 31² + 33² + 35² + 37² + ... + 57².

172² + 173² + 174² + ... + 710² = 10857².

(1 + 2 + ... + 9)(10 + 11 + ... + 22)(23 + 24 + ... + 172) = 11700²,
(1 + 2 + ... + 18)(19 + 20 + ... + 37)(38 + 39 + ... + 172) = 35910²,
(1 + 2 + ... + 25)(26 + 27 + ... + 91)(92 + 93 + ... + 172) = 115830²,
(1 + 2 + ... + 25)(26 + 27 + ... + 100)(101 + 102 + ... + 172) = 122850²,
(1 + 2 + ... + 25)(26 + 27 + ... + 142)(143 + 144 + ... + 172) = 122850²,
(1 + 2 + ... + 27)(28 + 29 + ... + 147)(148 + 149 + ... + 172) = 126000²,
(1 + 2 + ... + 32)(33 + 34 + ... + 47)(48 + 49 + ... + 172) = 66000²,
(1 + 2 + ... + 39)(40 + 41 + ... + 55)(56 + 57 + ... + 172) = 88920²,
(1 + 2 + ... + 48)(49 + 50 + ... + 87)(88 + 89 + ... + 172) = 185640²,
(1 + 2 + ... + 109)(110 + 111 + ... + 154)(155 + 156 + ... + 172) = 323730²,
(1 + 2 + ... + 114)(115)(116 + 117 + ... + 172) = 78660².

172² = 29584 appears in the decimal expression of e:
  e = 2.71828•••29584••• (from the 93771st digit)

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

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173

The smallest squares containing k 173's :
173056 = 416²,
5217317361 = 72231²,
1173963173817316 = 34263146².

173² = 29929, with 2 kinds of digits.

173² = 29929, 2 + 9 + 9 * 2 * 9 = 2 + 9 * 9 * 2 + 9 = 2 * 9 * 9 + 2 + 9 = 173.

173² = 29929, 2 + 9 + 9 + 29 = 7²,
173² = 29929, 29 + 9 + 2 + 9 = 7².

Komachi Fraction : 173² = 9158274/306.

173^k + 245^k + 397^k + 629^k are squares for k = 1,2,3 (38², 802², 18202²).

A 3-by-3 magic square consisting of different squares with constant 173²:

(3² - 4)(5² - 4)(17² - 4) = (173² - 4),
(13² - 9)(14² - 9) = (173² - 9).

1055² = 124² + 125² + 126² + ... + 173².

(1 + 2 + ... + 8)(9 + 10 + ... + 96)(97 + 98 + ... + 173) = 41580²,
(1 + 2 + ... + 34)(35 + 36 + ... + 169)(170 + 171 + 172 + 173) = 74970².

173² = 29929 appears in the decimal expression of π and e:
  π = 3.14159•••29929••• (from the 1101706th digit)
  e = 2.71828•••29929••• (from the 134592nd digit)

Page of Squares : First Upload July 12, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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174

The smallest squares containing k 174's :
17424 = 132²,
17417400625 = 131975²,
117471174681744 = 10838412².

174² = 30276, a square with different digits.

174² = (1² + 5)(71² + 5).

58^k + 7018^k + 10034^k + 13166^k are squares for k = 1,2,3 (174², 17980², 1907388²).

Komachi equations:
174² = - 1 * 234 + 5 * 678 * 9,
174² = 9² * 87² / 6² * 5² * 4² / 3² * 2² / 10² = 9² * 87² / 6² / 5² * 4² / 3² / 2² * 10².

1² + 2² + 3² + ... + 174² = 1771175, which consists of 3 kinds of odd digits (the 2nd 7-digit sum. Cf. 149).

174² = 30276, 3 + 0 + 2 + 76 = 9²,
174² = 30276, 3 + 0 + 27 + 6 = 6².

(6² - 8)(33² - 8) = (13² - 8)(14² - 8) = (174² - 8).

(1 + 2 + ... + 14)(15 + 16 + ... + 174) = 1260².

(1 + 2 + ... + 168)(169 + 170 + 171 + 172 + 173 + 174) = 3822².

174² = 30276 appears in the decimal expressions of π and e:
  π = 3.14159•••30276••• (from the 9520th digit),
  e = 2.71828•••30276••• (from the 3275th digit)

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

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175

The smallest squares containing k 175's :
175561 = 419²,
1751757316 = 41854²,
261751757417536 = 16178744².

175² = 30625, a zigzag square with different digits.

175² = 30625, 3 + 0 + 6 + 2 + 5 = 4².

Komachi equations:
175² = 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.

Three 3-by-3 magic squares consisting of different squares with constant 175²:

(1² + 2² + 3² + ... + 175²) = 1801800, which consists of 3 kinds of digits.

(1² + 2)(9² + 2)(11² + 2) = (175² + 2),
(1² + 3)(4² + 3)(20² + 3) = (175² + 3),
(13² - 7)(14² - 7) = (175² - 7),
(4² - 7)(5² - 7)(14² - 7) = (175² - 7).

175² + 176² + 177² + ... + 512² = 6565²,
175² + 176² + 177² + ... + 5503² = 235717².

(1 + 2 + ... + 23)(24 + 25 + ... + 123)(124 + 125 + ... + 175) = 125580²,
(1 + 2 + ... + 49)(50 + 51 + ... + 112)(113 + 114 + ... + 175) = 238140²,
(1 + 2 + ... + 55)(56 + 57 + ... + 175) = 4620²,
(1 + 2 + ... + 92)(93 + 94 + ... + 123)(124 + 125 + ... + 175) = 333684²,
(1 + 2 + ... + 147)(148)(149 + 150 + ... + 175) = 83916².

175² = 30625 appears in the decimal expression of e:
  e = 2.71828•••30625••• (from the 115429th digit)

Page of Squares : First Upload July 20, 2004 ; Last Revised April 23, 2010
by Yoshio Mimura, Kobe, Japan

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176

The smallest squares containing k 176's :
1764 = 42²,
176252176 = 13276²,
11762711761761 = 3429681².

The squares which begin with 176 and end in 176 are
176252176 = 13276²,   17615660176 = 132724²,   17629466176 = 132776²,
176168236176 = 419724²,   176211890176 = 419776²,...

176² = 30976, a square with different digits.

176² = 30976, 3 + 0 + 9 + 7 + 6 = 5².

176² = (2² + 7)(3² + 7)(13² + 7) = (2² + 7)(5² + 7)(9² + 7) = (2² + 7)(53² + 7) = (5² + 7)(31² + 7).

2090^k + 8074^k + 8558^k + 12254^k are squares for k = 1,2,3 (176², 17116², 1732720²).
5698^k + 6622^k + 8338^k + 10318^k are squares for k = 1,2,3 (176², 15884², 1467488²).

(13² - 6)(14² - 6) = (176² - 6),
(4² - 6)(5² - 6)(13² - 6) = (176² - 6).

(1 + 2 + ... + 23)(24 + 25 + ... + 68)(69 + 70 + ... + 176) = 86940²,
(1 + 2 + ... + 28)(29 + 30 + ... + 55)(56 + 57 + ... + 176) = 80388²,
(1 + 2 + ... + 42)(43 + 44 + ... + 47)(48 + 49 + ... + 176) = 54180²,
(1 + 2 + ... + 42)(43 + 44 + ... + 167)(168 + 169 + ... + 176) = 135450²,
(1 + 2 + ... + 104)(105 + 106 + ... + 111)(112 + 113 + ... + 176) = 196560².

(1³ + 2³ + ... + 55³)(56³ + 57³ + ... + 175³)(176³) = 55097011200².

176² = 30976 appears in the decimal expressions of π and e:
  π = 3.14159•••30976••• (from the 27784th digit),
  e = 2.71828•••30976••• (from the 108287th digit)

Page of Squares : First Upload July 20, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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177

The smallest squares containing k 177's :
177241 = 421²,
10177177924 = 100882²,
17751778917796 = 4213286².

177² = 31329, 31 * 3 * 2 - 9 = 177.

177² = 31329, a zigzag square.

177² = 31329, 3 + 1 + 3 + 29 = 6²,
177² = 31329, 3 + 132 + 9 = 12²,
177² = 31329, 313 + 2 + 9 = 18².

177² = 7⁴ + 8⁴ + 8⁴ + 12⁴.

48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).

Komachi Fraction : 177² = 2537649/81.

Komachi equations:
177² = 9 + 87 / 6 * 5 * 432 * 1 = 9 + 87 / 6 * 5 * 432 / 1.

Three 3-by-3 magic squares consisting of different squares with constant 177²:

(13² - 5)(14² - 5) = (177² - 5).

(1 + 2 + ... + 25)(26 + 27 + ... + 82)(83 + 84 + ... + 177) = 111150²,
(1 + 2 + ... + 44)(45 + 46 + ... + 65)(66 + 67 + ... + 177) = 124740².

(1³ + 2³ + ... + 117³)(118³ + 119³ + ... + 177³) = 97746480².

177² = 31329 appears in the decimal expression of e:
  e = 2.71828•••31329••• (from the 17900th digit)

Page of Squares : First Upload July 20, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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178

The smallest squares containing k 178's :
178084 = 422²,
17844417889 = 133583²,
1781787178310841 = 42211221².

178² = 31684, a square with different digits.

178² = 31684, 31 + 6 + 8 + 4 = 7²,
178² = 31684, 31 + 6 + 84 = 11²,
178² = 31684, 316 + 84 = 20².

178² = 1⁴ + 7⁴ + 11⁴ + 11⁴.

178^k + 242^k + 494^k + 850^k are squares for k = 1,2,3 (42², 1028², 27468²).

Komachi Fraction : (19/178)² = 6498/570312.

Komachi equations:
178² = 12² / 3² * 4² / 56² * 7² * 89²,
178² = 9⁵ - 8⁵ + 7⁵ - 6⁵ + 5⁵ + 4⁵ - 3⁵ * 2⁵ - 1⁵.

(13² - 4)(14² - 4) = (178² - 4).

178² + 179² + 180² + ... + 361² = 3726².

(1 + 2)(3 + 4 + ... + 46)(47 + 48 + ... + 178) = 6930²,
(1 + 2 + 3)(4 + 5 + ... + 16)(17 + 18 + ... + 178) = 3510²,
(1 + 2 + ... + 28)(29 + 30 + ... + 115)(116 + 117 + ... + 178) = 153468²,
(1 + 2 + ... + 80)(81 + 82 + ... + 161)(162 + 163 + ... + 178) = 302940².

178² = 31684 appears in the decimal expression of e:
  e = 2.71828•••31684••• (from the 105398th digit)

Page of Squares : First Upload July 20, 2004 ; Last Revised February 15, 2011
by Yoshio Mimura, Kobe, Japan

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179

The smallest squares containing k 179's :
17956 = 134²,
17917964164 = 133858²,
317981793217936 = 17832044².

179² = 32041, a square with different digits.

179² = 32041, 3³ + 2³ + 0³ + 4³ + 1³ = 10²,
179² = 32041, 3 + 20 + 41 = 8²,
179² = 32041, 320 + 41 = 19².

179² = 25³ + 2⁵ + 4⁷.

(1² + 2² + 3² + ... + 179²) = 1927830, which consists of different digits.

Komachi equations:
179² = 9² - 8² + 76² + 54² * 3² + 2² */ 1² = 9² - 8² + 76² + 54² * 3² + 2² / 1².

Komachi Square Sum : 179² = 2² + 4² + 38² + 79² + 156².

Two 3-by-3 magic squares consisting of different squares with constant 179²:

(13² - 3)(14² - 3) = (179² - 3),
(3² + 4)(5² + 4)(9² + 4) = (179² + 4),
(4² + 5)(39² + 5) = (179² + 5),
(10² - 9)(19² - 9) = (179² - 9).

(1 + 2 + ... + 18)(19 + 20 + ... + 27)(28 + 29 + ... + 179) = 23598²,
(1 + 2 + ... + 38)(39 + 40 + ... + 132)(133 + 134 + ... + 179) = 208962²,
(1 + 2 + ... + 49)(50 + 51 + ... + 54)(55 + 56 + ... + 179) = 68250²,
(1 + 2 + ... + 62)(63 + 64 + ... + 161)(162 + 163 + ... + 179) = 257796²,
(1 + 2 + ... + 71)(72 + 73 + ... + 104)(105 + 106 + ... + 179) = 281160²,
(1 + 2 + ... + 87)(88 + 89 + ... + 168)(169 + 170 + ... + 179) = 275616².

179² = 32041 appears in the decimal expression of e:
  e = 2.71828•••32041••• (from the 114044th digit)

Page of Squares : First Upload July 20, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

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