---

190

The smallest squares containing k 190's :
19044 = 138²,
1190319001 = 34501²,
1901903019025 = 1379095².

10^k + 50^k + 190^k + 320^k + 330^k are squares (30², 500², 8700², 153800²) for k = 1, 2, 3, 4.

190² = 36100, with 36 = 6² and 100 = 10².

190^k + 314^k + 410^k + 530^k are squares for k = 1,2,3 (38², 764², 15988²).
190^k + 338^k + 786^k + 802^k are squares for k = 1,2,3 (46², 1188², 32356²).
52^k + 190^k + 233^k + 254^k are squares for k = 1,2,3 (27², 397², 6003²).
2242^k + 4978^k + 13262^k + 15618^k are squares for k = 1,2,3 (190², 21204², 2505340²).

Komachi Square Sum : 190² = 5² + 7² + 24² + 83² + 169².

(3² + 8)(46² + 8) = (13² + 8)(14² + 8) = (190² + 8),
(2² + 8)(3² + 8)(13² + 8) = (190² + 8).

1518² = 7² + 8² + 9² + 10² + 11² + ... + 190².

(1 + 2 + 3)(4 + 5 + ... + 115)(116 + 117 + ... + 190) = 21420²,
(1 + 2 + ... + 16)(17 + 18 + ... + 115)(116 + 117 + ... + 190) = 100980²,
(1 + 2 + ... + 29)(30 + 31 + ... + 145)(146 + 147 + ... + 190) = 182700²,
(1 + 2 + ... + 108)(109)(110 + 111 + ... + 190) = 88290².

1³ + 2³ + 3³ + 4³ + ... + 13³ + 14³ + 15³ + 16³ + 17³ + 18³ + 19³ = 190².

(1³ + 2³ + ... + 115³)(116³ + 117³ + ... + 145³)(146³ + 147³ + ... + 190³) = 807937767000².

190² = 36100 appears in the decimal expressions of π and e:
  π = 3.14159•••36100••• (from the 11051st digit),
  e = 2.71828•••36100••• (from the 71830th digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

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191

The smallest squares containing k 191's :
191844 = 438²,
19109191696 = 138236²,
191181919191025 = 13826855².

(1² + 2² + 3² + ... + 191²) = 2340896, which consists of different digits.

191 is the second prime for which the Legendre Symbol (a/191) = 1 for a = 1, 2, 3, 4, 5, 6.

191² = 36481, a zigzag square with different digits.

191² = 36481, 36 + 4 + 8 + 1 = 7²,
191² = 36481, 36 + 4 + 81 = 11².

191² = 36481 (36 = 6², 4 = 2², 81 = 9²).

Komachi equations:
191² = 9 * 8 * 76 * 5 * 4 / 3 + 2 - 1 = 9 + 8 * 76 * 5 * 4 * 3 + 2 - 10.

Three 3-by-3 magic squares consisting of different squares with constant 191²:

(5² - 1)(39² - 1) = (191² - 1),
(2² - 1)(3² - 1)(39² - 1) = (191² - 1),
(4² - 3)(53² - 3) = (191² - 3),
(4² - 3)(7² - 3)(8² - 3) = (191² - 3),
(13² + 9)(14² + 9) = (191² + 9).

(1)(2 + 3 + ... + 96)(97 + 98 + ... + 191) = 7980²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 123)(124 + 125 + ... + 191) = 28560²,
(1 + 2 + ... + 23)(24 + 25 + ... + 183)(184 + 185 + ... + 191) = 82800²,
(1 + 2 + ... + 39)(40 + 41 + ... + 74)(75 + 76 + ... + 191) = 155610²,
(1 + 2 + ... + 104)(105 + 106 + ... + 156)(157 + 158 + ... + 191) = 475020²,
(1 + 2 + ... + 116)(117 + 118 + ... + 156)(157 + 158 + ... + 191) = 475020²,
(1 + 2 + ... + 119)(120 + 121 + ... + 123)(124 + 125 + ... + 191) = 192780².

191² = 36481 appears in the decimal expression of e:
  e = 2.71828•••36481••• (from the 30933rd digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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192

The smallest squares containing k 192's :
192721 = 439²,
19201924 = 4382²,
19219219264729 = 4383973².

1 / 192 = 0.00520833333333333...,
where 5² + 2² + 0² + 8² + 3² + 3² + 3² + 3² + 3² + 3² + 3² + 3² + 3² + 3² + 3² = 192.

192² = 36864, 36 * 8 / 6 * 4 = 192.

192² = 36864, 3 + 6 + 8 + 64 = 9²,
192² = 36864, 3 + 68 + 6 + 4 = 9²,
192² = 36864, 36 + 864 = 30².

192² is the fourth square which is the sum of 5 fifth powers : (4, 4, 4, 4, 8).

192² is the 9th square which is the sum of 9 sixth powers, and the 2nd square which is the sum of 9 12th powers.

192² = 36864, a square pegged by 6.

192² is the 9th square which is the sum of 2 cubes : 16³ + 32³.

192² is an exchangeable square : 192² = 36864, 86436 = 294².

Komachi equation: 192² = 98⁴ / 7⁴ - 6⁴ - 5⁴ / 4⁴ * 32⁴ / 10⁴.

Cubic polynomials :
(X + 76²)(X + 192²)(X + 513²) = X³ + 553²X² + 106932²X + 7485696²,
(X + 192²)(X + 252²)(X + 301²) = X³ + 437²X~2 + 106932²X + 14563584²,
(X + 192²)(X + 2828²)(X + 9999²) = X³ + 10393²X² + 28347468²X + 5429217024².

192² + 193² + 194² + ... + 279² = 2222²,
192² + 193² + 194² + ... + 2842² = 87483².

(1 + 2)(3 + 4 + ... + 122)(123 + 124 + ... + 192) = 15750²,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 104)(105 + 106 + ... + 192) = 32670²,
(1 + 2 + ... + 6)(7 + 8 + ... + 168)(169 + 170 + ... + 192) = 35910²,
(1 + 2 + ... + 7)(8 + 9 + ... + 167)(168 + 169 + ... + 192) = 42000²,
(1 + 2 + ... + 9)(10 + 11 + ... + 17)(18 + 19 + ... + 192) = 9450²,
(1 + 2 + ... + 9)(10 + 11 + ... + 104)(105 + 106 + ... + 192) = 56430²,
(1 + 2 + ... + 24)(25 + 26 + ... + 38)(39 + 40 + ... + 192) = 48510²,
(1 + 2 + ... + 24)(25 + 26 + ... + 96)(97 + 98 + ... + 192) = 134640²,
(1 + 2 + ... + 24)(25 + 26 + ... + 122)(123 + 124 + ... + 192) = 154350²,
(1 + 2 + ... + 24)(25 + 26 + ... + 150)(151 + 152 + ... + 192) = 154350²,
(1 + 2 + ... + 31)(32 + 33 + ... + 37)(38 + 39 + ... + 192) = 42780²,
(1 + 2 + ... + 90)(91 + 92 + ... + 104)(105 + 106 + ... + 192) = 270270².

(1² + 2² + ... + 12²)(13² + 14² + ... + 71²)(72² + 73² + ... + 192²) = 13330460².

(1³ + 2³ + ... + 77³)(78³ + 79³ + ... + 87³)(88³ + 89³ + ... + 192³) = 129235005900².

192² = 36864 appears in the decimal expression of π:
  π = 3.14159•••36864••• (from the 22668th digit).

Page of Squares : First Upload August 10, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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193

The smallest squares containing k 193's :
1936 = 44²,
12419319364 = 111442²,
193193931934084 = 13899422².

193² = 37249, a square with different digits.

193 is the eighth prime for which the Legendre Symbol (a/193) = 1 for a = 1, 2, 3, 4.

193² = 37249, 3 + 7 + 2 + 4 + 9 = 5².

Komachi equations:
193² = 98 * 76 * 5 + 4 + 3 + 2 */ 1 = 98 * 76 * 5 + 4 + 3 * 2 - 1
= 98 * 76 * 5 + 4 * 3 - 2 - 1 = 98 * 76 * 5 - 4 * 3 + 21
= 98 * 76 * 5 + 4 - 3 - 2 + 10 = - 9 + 8 / 7 * 6 * 5432 + 10.

Two 3-by-3 magic squares consisting of different squares with constant 193²:

(4² - 9)(73² - 9) = (193² - 9).

193² + 194² + 195² + 196² + ... + 288² = 2372².

(1 + 2 + 3 + 4)(5 + 6 + ... + 49)(50 + 51 + ... + 193) = 14580²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 58)(59 + 60 + ... + 193) = 17010²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 103)(104 + 105 + ... + 193) = 26730²,
(1 + 2 + 3 + 4)(5 + 6 + ... + 184)(185 + 186 + ... + 193) = 17010²,
(1 + 2 + ... + 5)(6 + 7 + ... + 21)(22 + 23 + ... + 193) = 7740²,
(1 + 2 + ... + 5)(6 + 7 + ... + 49)(50 + 51 + ... + 193) = 17820²,
(1 + 2 + ... + 6)(7 + 8 + ... + 21)(22 + 23 + ... + 193) = 9030²,
(1 + 2 + ... + 6)(7 + 8 + ... + 58)(59 + 60 + ... + 193) = 24570²,
(1 + 2 + ... + 6)(7 + 8 + ... + 149)(150 + 151 + ... + 193) = 42042²,
(1 + 2 + ... + 8)(9 + 10 + ... + 144)(145 + 146 + ... + 193) = 55692²,
(1 + 2 + ... + 9)(10 + 11 + ... + 13)(14 + 15 + ... + 193) = 6210²,
(1 + 2 + ... + 9)(10 + 11 + ... + 130)(131 + 132 + ... + 193) = 62370²,
(1 + 2 + ... + 11)(12 + 13 + ... + 21)(22 + 23 + ... + 193) = 14190²,
(1 + 2 + ... + 11)(12 + 13 + ... + 103)(104 + 105 + ... + 193) = 68310²,
(1 + 2 + ... + 11)(12 + 13 + ... + 149)(150 + 151 + ... + 193) = 74382²,
(1 + 2 + ... + 12)(13 + 14 + ... + 168)(169 + 170 + ... + 193) = 70590²,
(1 + 2 + ... + 13)(14 + 15 + ... + 130)(131 + 132 + ... + 193) = 88452²,
(1 + 2 + ... + 14)(15 + 16 + ... + 49)(50 + 51 + ... + 193) = 45360²,
(1 + 2 + ... + 18)(19 + 20 + ... + 34)(35 + 36 + ... + 193) = 36252²,
(1 + 2 + ... + 20)(21)(22 + 23 + ... + 193) = 9030²,
(1 + 2 + ... + 20)(21 + 22 + ... + 184)(185 + 186 + ... + 193) = 77490²,
(1 + 2 + ... + 27)(28 + 29 + ... + 49)(50 + 51 + ... + 193) = 74844²,
(1 + 2 + ... + 27)(28 + 29 + ... + 107)(108 + 109 + ... + 193) = 162540²,
(1 + 2 + ... + 27)(28 + 29 + ... + 167)(168 + 169 + ... + 193) = 155610²,
(1 + 2 + ... + 40)(41 + 42 + ... + 175)(176 + 177 + ... + 193) = 199260²,
(1 + 2 + ... + 45)(46)(47 + 48 + ... + 193) = 28980²,
(1 + 2 + ... + 48)(49)(50 + 51 + ... + 193) = 31752²,
(1 + 2 + ... + 59)(60 + 61 + ... + 101)(102 + 103 + ... + 193) = 284970²,
(1 + 2 + ... + 68)(69 + 70 + ... + 78)(79 + 80 + ... + 193) = 164220²,
(1 + 2 + ... + 92)(93 + 94 + ... + 147)(148 + 149 + ... + 193) = 470580²,
(1 + 2 + ... + 97)(98 + 99 + ... + 193) = 8148²,
(1 + 2 + ... + 105)(106 + 107 + ... + 158)(159 + 160 + ... + 193) = 489720²,
(1 + 2 + ... + 114)(115 + 116 + ... + 174)(175 + 176 + ... + 193) = 445740²,
(1 + 2 + ... + 125)(126 + 127 + ... + 130)(131 + 132 + ... + 193) = 226800²,
(1 + 2 + ... + 156)(157)(158 + 159 + ... + 193) = 110214²,
(1 + 2 + ... + 180)(181)(182 + 183 + ... + 193) = 81450².

193² = 37249 appears in the decimal expression of π:
  π = 3.14159•••37249••• (from the 82571st digit).

Page of Squares : First Upload August 10, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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194

The smallest squares containing k 194's :
194481 = 441²,
119450419456 = 345616²,
194011943019489 = 13928817².

1 / 194 = 0.0051546391, 5² + 1² + 5² + 4² + 6² + 3² + 9² + 1² = 194.

194² = 37636, a square with 3 kinds of digits.

194² = 37636, 3 + 7 + 6 + 3 + 6 = 5².

Komachi Fraction : 729/3048516 = (3/194)².

Komachi equations:
194² = - 12² * 3² + 4² + 5² * 6² * 7² - 8² * 9²,
194² = - 1³ + 2³ + 34³ + 5³ - 6³ - 7³ - 8³ - 9³.

(1² + 2)(10² + 2)(11² + 2) = (2² + 2)(7² + 2)(11² + 2) = (194² + 2),
(1² + 2)(112² + 2) = (194² + 2).

1525² = 73² + 74² + 75² + 76² + 77² + ... + 194².

(1 + 2 + ... + 96)(97 + 98 + ... + 194) = 8148².

(1³ + 2³ + ... + 96³)(97³ + 98³ + ... + 194³) = 85358448².

194² = 37636 appears in the decimal expression of e:
  e = 2.71828•••37636••• (from the 59782nd digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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195

The smallest squares containing k 195's :
195364 = 442²,
11951955625 = 109325²,
51957195761956 = 7208134².

The square root of 195 is 13.96424004..., 13² = 9² + 6² + 4² + 2² + 4² + 0² + 0² + 4².

195² = 38025, 380 / 2 + 5 = 195.

195² = 38025, a square with different digits.

195² = (2² + 9)(54² + 9).

195² = 38025, 3 + 8 + 0 + 25 = 6².

4836^k + 6825^k + 8658^k + 17706^k are squares for k = 1,2,3 (195², 21411², 2575053²).

Komachi Square Sum : 195² = 39² + 46² + 52² + 178².

If A = 195², B = 748² and C = 6336², then A + B = 773², B + C = 6380², and C + A = 6339².

195² = 25² + 26² + 27² + 28² + ... + 50².

Three 3-by-3 magic squares consisting of different squares with constant 195²:

(1)(2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13 + 14)(15) = 195².

(1 + 2 + ... + 8)(9 + 10 + ... + 57)(58 + 59 + ... + 195) = 31878².

1³ + 2³ + ... + 195³ = 19110².

(1³ + 2³ + ... + 65³)(66³ + 67³ + ... + 110³)(111³ + 112³ + ... + 195³) = 222017152500².

195² = 38025 appears in the decimal expression of π:
  π = 3.14159•••38025••• (from the 53284th digit).

Page of Squares : First Upload August 10, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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196

The square of 14.

The smallest squares containing k 196's :
196 = 14²,
49196196 = 7014²,
19629206196196 = 4430486².

The squares which begin with 196 and end in 196 are
196392196 = 14014²,   19603920196 = 140014²,   196236596196 = 442986²,
196261404196 = 443014²,   196679832196 = 443486²,...

1 / 196 = 0.005102040816326..., and
  5² + 1² + 0² + 2² + 0² + 4² + 0² + 8² + 1² + 6² + 3² + 2² + 6² = 196.

196 is an exchangeable square : 196 = 14², 961 = 31².

196 = 14² is the first mosaic square : 16 = 4² and 9 = 3².

196² is an exchangeable square : 196² = 38416, 16384 = 128².

196² = 38416, a squre with different digits.

196² = (1² + 3)(2² + 3)(37² + 3) = (5² + 3)(37² + 3).

196² = 38416, 38 + 4 + 1 + 6 = 7²,
196² = 38416, 384 + 16 = 20².

Komachi equation: 196² = 9⁴ * 8⁴ * 7⁴ * 6⁴ / 5⁴ / 432⁴ * 10⁴.

A cubic polynomial : (X + 196²)(X + 528²)(X + 693²) = X³ + 893²X² + 403788²X + 71717184².

196² = 77 x 78 + 78 x 79 + 79 x 80 + 80 x 81 + 81 x 82 + 82 x 83.

(1)(2 + 3 + ... + 97)(98 + 99 + ... + 196) = 8316²,
(1)(2 + 3 + ... + 163)(164 + 165 + ... + 196) = 8910²,
(1 + 2 + ... + 24)(25 + 26 + ... + 75)(76 + 77 + ... + 196) = 112200²,
(1 + 2 + ... + 32)(33 + 34 + ... + 65)(66 + 67 + ... + 196) = 121044²,
(1 + 2 + ... + 32)(33 + 34 + ... + 97)(98 + 99 + ... + 196) = 180180²,
(1 + 2 + ... + 40)(41 + 42 + ... + 49)(50 + 51 + ... + 196) = 77490²,
(1 + 2 + ... + 54)(55 + 56 + ... + 65)(66 + 67 + ... + 196) = 129690²,
(1 + 2 + ... + 54)(55 + 56 + ... + 163)(164 + 165 + ... + 196) = 323730².

196² = 38416 appears in the decimal expression of e:
  e = 2.71828•••38416••• (from the 52109th digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

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197

The smallest squares containing k 197's :
119716 = 346²,
19019719744 = 137912²,
197102197197481 = 14039309².

197² = 38809, 3 + 8 + 80 + 9 = 10²,
197² = 38809, 3 + 88 + 0 + 9 = 10².

42^k + 66^k + 108^k + 145^k are squares for k = 1,2,3 (19², 197², 2161²).

Komachi Fractions : 576/349281 = (8/197)², 7056/349281 = (28/197)².

Komachi Square Sums : 197² = 2² + 4² + 76² + 98² + 153² = 4² + 6² + 73² + 92² + 158².

Two 3-by-3 magic squares consisting of different squares with constant 197²:

(3² - 7)(6² - 7)(26² - 7) = 197² - 7.

197² + 198² + 199² + 200² + ... + 220² = 1022²,
197² + 198² + 199² + 200² + ... + 2204² = 59738²,
197² + 198² + 199² + 200² + ... + 2965² = 93223²,
197² + 198² + 199² + 200² + ... + 3459² = 117468²,
197² + 198² + 199² + 200² + ... + 949² = 16817²,
197² + 198² + 199² + 200² + ... + 119547² = 23864378².

(1 + 2)(3 + 4 + ... + 47)(48 + 49 + ... + 197) = 7875²,
(1 + 2 + ... + 9)(10 + 11 + ... + 39)(40 + 41 + ... + 197) = 24885²,
(1 + 2 + ... + 9)(10 + 11 + ... + 47)(48 + 49 + ... + 197) = 29925²,
(1 + 2 + ... + 10)(11 + 12 + ... + 44)(45 + 46 + ... + 197) = 30855²,
(1 + 2 + ... + 12)(13 + 14 + ... + 117)(118 + 119 + ... + 197) = 81900²,
(1 + 2 + ... + 13)(14 + 15 + ... + 36)(37 + 38 + ... + 197) = 31395²,
(1 + 2 + ... + 125)(126 + 127 + ... + 140)(141 + 142 + ... + 197) = 389025²,
(1 + 2 + ... + 125)(126 + 127 + ... + 159)(160 + 161 + ... + 197) = 508725²,
(1 + 2 + ... + 175)(176 + 177 + ... + 187)(188 + 189 + ... + 197) = 254100².

197² = 38809 appears in the decimal expressions of π and e:
  π = 3.14159•••38809••• (from the 40331st digit),
  e = 2.71828•••38809••• (from the 130095th digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

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198

The smallest squares containing k 198's :
19881 = 141²,
1198198225 = 34615²,
19854198198025 = 4455805².

198² = 39204, a square with different digits.

198² = 39204, 3 - 9 + 204 = 198.

198²± 5 are primes.

198² = (1² + 2)(2² + 2)(3² + 2)(14² + 2) = (1² + 2)(3² + 2)(4² + 2)(8² + 2)
= (1² + 2)(8² + 2)(14² + 2) = (2² + 2)(4² + 2)(19² + 2) = (3² + 2)(4² + 2)(14² + 2).

198² = 39204, 3 + 9 + 20 + 4 = 6².

(3² + 4)(5² + 4)(10² + 4) = (198² + 4).

165^k + 7029^k + 9273^k + 22737^k are squares for k = 1,2,3 (198², 25542², 3591522²).
5390^k + 9482^k + 12034^k + 12298^k are squares for k = 1,2,3 (198², 20372², 2147508²).
6402^k + 9570^k + 9966^k + 13266^k are squares for k = 1,2,3 (198², 20196², 2112660²).

Komachi equations:
198² = 12² + 3² * 4² + 5² * 6² * 7² - 8² * 9² = 9² - 87² + 6² + 5² * 432² / 10².

30² + 31² + 32² + 33² + ... + 198² = 1612².

(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11) = 198²,
(1)(2)(3)(4 + 5 + 6 + 7)(8 + 9 + 10)(11) = 198²,
(1)(2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)(10 + 11 + 12)(13 + 14) = 198²,
(1 + 2)(3)(4)(5 + 6)(7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15) = 198²,
(1 + 2 + 3)(4 + ... + 14)(15 + ... + 18) = 198²,
(1 + 2 + 3)(4 + ... + 7)(8 + ... + 25) = 198²,
(1 + 2 + 3)(4 + 5)(6 + ... + 38) = 198².

(1 + 2 + ... + 9)(10 + 11 + ... + 81)(82 + 83 + ... + 198) = 49140²,
(1 + 2 + ... + 11)(12 + 13 + ... + 164)(165 + 166 + ... + 198) = 74052²,
(1 + 2 + ... + 20)(21 + 22 + ... + 125)(126 + 127 + ... + 198) = 137970²,
(1 + 2 + ... + 23)(24 + 25 + ... + 161)(162 + 163 + ... + 198) = 153180²,
(1 + 2 + ... + 26)(27 + 28 + ... + 90)(91 + 92 + ... + 198) = 143208²,
(1 + 2 + ... + 39)(40 + 41 + ... + 81)(82 + 83 + ... + 198) = 180180²,
(1 + 2 + ... + 78)(79 + 80 + ... + 117)(118 + 119 + ... + 198) = 388206²,
(1 + 2 + ... + 144)(145 + 146 + ... + 149)(150 + 151 + ... + 198) = 255780².

198² = 39204 appears in the decimal expression of π:
  π = 3.14159•••39204••• (from the 92662nd digit).

Page of Squares : First Upload August 10, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

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199

The smallest squares containing k 199's :
199809 = 447²,
19902719929 = 141077²,
1995199403199025 = 44667655².

199² = 39601, a square with different digits.

199² = 39601, 3 + 96 + 0 + 1 = 10²,
199² = 39601, 3 + 9601 = 98²,
199² = 39601, 39³ + 6³ + 0³ + 1³ = 244²,
199² = 39601, 39 + 60 + 1 = 10².

The sum of the concecutive odd primes : 3 + 5 + 7 + 11 + 13 + 17 + ... + 199 = 65².

Komachi equation: 199² = - 12³ + 3³ * 4³ * 5³ + 6³ - 7³ * 8³ + 9³.

Two 3-by-3 magic squares consisting of different squares with constant 199²:

(6² + 5)(31² + 5) = (199² + 5),
(3² + 5)(6² + 5)(8² + 5) = (5² + 9)(34² + 9) = (199² + 9).

199² + 200² + 201² + 202² + ... + 487² = 6001².

(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 199) = 5880²,
(1 + 2 + 3)(4 + 5 + ... + 28)(29 + 30 + ... + 199) = 6840²,
(1 + 2 + 3)(4 + 5 + ... + 52)(53 + 54 + ... + 199) = 12348²,
(1 + 2 + 3)(4 + 5 + ... + 124)(125 + 126 + ... + 199) = 23760²,
(1 + 2 + 3)(4 + 5 + ... + 192)(193 + 194 + ... + 199) = 12348²,
(1 + 2 + ... + 7)(8 + 9 + ... + 28)(29 + 30 + ... + 199) = 14364²,
(1 + 2 + ... + 7)(8 + 9 + ... + 52)(53 + 54 + ... + 199) = 26460²,
(1 + 2 + ... + 10)(11 + 12 + ... + 64)(65 + 66 + ... + 199) = 44550²,
(1 + 2 + ... + 27)(28)(29 + 30 + ... + 199) = 14364²,
(1 + 2 + ... + 27)(28 + 29 + ... + 192)(193 + 194 + ... + 199) = 97020²,
(1 + 2 + ... + 39)(40 + 41 + ... + 160)(161 + 162 + ... + 199) = 257400²,
(1 + 2 + ... + 110)(111 + 112 + ... + 185)(186 + 187 + ... + 199) = 427350²,
(1 + 2 + ... + 120)(121 + 122 + ... + 124)(125 + 126 + ... + 199) = 207900²,
(1 + 2 + ... + 120)(121 + 122 + ... + 195)(196 + 197 + ... + 199) = 260700²,
(1 + 2 + ... + 168)(169 + 170 + ... + 192)(193 + 194 + ... + 199) = 290472².

199² = 39601 appears in the decimal expressions of π and e:
  π = 3.14159•••39601••• (from the 17946th digit),
  e = 2.71828•••39601••• (from the 24250th digit)

Page of Squares : First Upload August 10, 2004 ; Last Revised April 27, 2010
by Yoshio Mimura, Kobe, Japan

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