Squares:200s

200

The smallest squares containing k 200's :
62001 = 249²,
2002652001 = 44751²,
100200120020001 = 10010001².

200² + 201² + 202² + ... + 17888² = 1381338².

200² = (1² + 4)(6² + 4)(14² + 4) = (2² + 4)(6² + 4)(11² + 4).

A + B, A + C, A + D, B + C, B + D, and C + D are squares
for A = 200, B = 1649, C = 5576, D = 9800.

Komachi equation:
200² = - 1³ + 23³ - 4³ + 5³ * 6³ - 7³ + 8³ + 9³.

200² = 10⁴ + 10⁴ + 10⁴ + 10⁴.

(1 + 2 + 3 + 4)(5 + 6 + ... + 79)(80 + 81 + ... + 200) = 23100²,
(1 + 2 + ... + 12)(13 + 14 + ... + 83)(84 + 85 + ... + 200) = 66456²,
(1 + 2 + ... + 16)(17 + 18 + ... + 47)(48 + 49 + ... + 200) = 50592²,
(1 + 2 + ... + 32)(33 + 34 + ... + 123)(124 + 125 + ... + 200) = 216216²,
(1 + 2 + ... + 175)(176 + 177 + ... + 184)(185 + 186 + ... + 200) = 277200².

(1³ + 2³ + ... + 6³)(7³ + 8³ + ... + 29³)(30³ + 31³ + ... + 200³) = 183356460².

200² = 40000 appears in the decimal expressions of π and e:
π = 3.14159•••40000••• (from the 37321st digit),
e = 2.71828•••40000••• (from the 16765th digit).

Page of Squares : First Upload August 19, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

201

The smallest squares containing k 201's :
10201 = 101²,
2011612201 = 44851²,
2018107201201 = 1420601².

The squares which begin with 201 and end in 201 are
2011612201 = 44851², 2015920201 = 44899², 20121706201 = 141851²,
20135326201 = 141899², 20192694201 = 142101²,...

1 / 201 = 0.0049751243..., 4² + 9² + 7² + 5² + 1² + 2² + 4² + 3² = 201.

201² = 40401 is a reversible square (10404 = 102²).

201² = 40401 is a zigzag square which consists of 3 kinds of digits.

201² is the first square(> 1) that can not be a sum of a power of 2 and a prime.

21^k + 45^k + 201^k + 633^k are squares for k = 1,2,3 ( 30², 666², 16182²).

3-by-3 magic squares consisting of different squares with constant 201²:

1²	104²	172²
136²	127²	76²
148²	116²	71²

4²	71²	188²
97²	164²	64²
176²	92²	31²

7²	44²	196²
116²	161²	32²
164²	112²	31²

201² = 40401, 4 + 0 + 4 + 0 + 1 = 3²,
201² = 40401, 40 + 40 + 1 = 9²,
201² = 40401, 40 + 401 = 21².

940² + 201² = 924001.

(1 + 2 + ... + 6)(7 + 8 + ... + 110)(111 + 112 + ... + 201) = 42588²,
(1 + 2 + ... + 8)(9 + 10 + ... + 15)(16 + 17 + ... + 201) = 7812²,
(1 + 2 + ... + 8)(9 + 10 + ... + 110)(111 + 112 + ... + 201) = 55692²,
(1 + 2 + ... + 11)(12 + 13 + ... + 87)(88 + 89 + ... + 201) = 63954²,
(1 + 2 + ... + 13)(14 + 15 + ... + 77)(78 + 79 + ... + 201) = 67704²,
(1 + 2 + ... + 18)(19 + 20 + ... + 45)(46 + 47 + ... + 201) = 53352²,
(1 + 2 + ... + 20)(21 + 22 + ... + 141)(142 + 143 + ... + 201) = 145530²,
(1 + 2 + ... + 29)(30 + 31 + ... + 173)(174 + 175 + ... + 201) = 182700²,
(1 + 2 + ... + 59)(60 + 61 + ... + 93)(94 + 95 + ... + 201) = 270810²,
(1 + 2 + ... + 80)(81 + 82 + ... + 108)(109 + 110 + ... + 201) = 351540²,
(1 + 2 + ... + 80)(81 + 82 + ... + 149)(150 + 151 + ... + 201) = 484380²,
(1 + 2 + ... + 98)(99 + 100 + ... + 198)(199 + 200 + 201) = 207900²,
(1 + 2 + ... + 111)(112 + 113 + ... + 131)(132 + 133 + ... + 201) = 419580²,
(1 + 2 + ... + 168)(169 + 170 + ... + 183)(184 + 185 + ... + 201) = 360360²,
(1 + 2 + ... + 200)(201) = 2010².

201² = 40401 appears in the decimal expressions of π and e:
π = 3.14159•••40401••• (from the 19623rd digit),
e = 2.71828•••40401••• (from the 39828th digit).

Page of Squares : First Upload April 19, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

202

The smallest squares containing k 202's :
2025 = 45²,
482022025 = 21955²,
202202569842025 = 14219795².

202² = 40804, a palindromic and zigzag square which consists of 3 kinds of even digits.

202² = 40804, 4 + 0 + 8 + 0 + 4 = 4².

46^k + 170^k + 202^k + 258^k are squares for k = 1,2,3 ( 26², 372², 5516²).

Komachi equations:
202² = - 1 - 23 + 4 + 567 * 8 * 9,
202² = 1² + 2² * 34² + 5² * 6² * 7² - 89²,
202² = - 1³ + 2³ + 34³ + 5³ - 6³ + 7³ + 8³ + 9³.

12² + 31² + 50² + 69² + 88² + 107² + 126² + 145² + 164² + 183² + 202² = 407².

(1 + 2 + 3 + 4)(5 + 6 + ... + 94)(95 + 96 + ... + 202) = 26730²,
(1 + 2 + ... + 10)(11 + 12 + ... + 22)(23 + 24 + ... + 202) = 14850²,
(1 + 2 + ... + 10)(11 + 12 + ... + 94)(95 + 96 + ... + 202) = 62370²,
(1 + 2 + ... + 18)(19 + 20 + ... + 31)(32 + 33 + ... + 202) = 33345²,
(1 + 2 + ... + 25)(26 + 27 + ... + 31)(32 + 33 + ... + 202) = 33345²,
(1 + 2 + ... + 35)(36 + 37 + ... + 160)(161 + 162 + ... + 202) = 242550²,
(1 + 2 + ... + 49)(50 + 51 + ... + 112)(113 + 114 + ... + 202) = 297675²,
(1 + 2 + ... + 49)(50 + 51 + ... + 175)(176 + 177 + ... + 202) = 297675²,
(1 + 2 + ... + 110)(111 + 112 + ... + 165)(166 + 167 + ... + 202) = 561660²,
(1 + 2 + ... + 134)(135)(136 + 137 + ... + 202) = 117585².

202² = 40804 appears in the decimal expressions of π and e:
π = 3.14159•••40804••• (from the 59706th digit),
e = 2.71828•••40804••• (from the 96138th digit).

Page of Squares : First Upload April 19, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

203

The smallest squares containing k 203's :
203401 = 451²,
20301120324 = 142482²,
1320305332032036 = 36336006².

203 = (1² + 2² + 3² + ... + 14²) / (1² + 2²).

203² = 41209, a zigzag square with different digits.

203² = 1! + 4! + 4! + 5! + 6! + 8!.

203² = 41209, 4 + 1 + 2 + 0 + 9 = 4²,
203² = 41209, 4 + 12 + 0 + 9 = 5².

203³ = 8365427, 8² + 3² + 6² + 5² + 4² + 2² + 7² = 203.

22^k + 73^k + 130^k + 136^k are squares for k = 1,2,3 ( 19², 203², 2261²).
3190^k + 4524^k + 13137^k + 20358^k are squares for k = 1,2,3 ( 203², 24853², 3290833²).

3-by-3 magic squares consisting of different squares with constant 203²:

18²	86²	183²
103²	162²	66²
174²	87²	58²

23²	78²	186²
102²	166²	57²
174²	87²	58²

(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 93)(94 + 95 + ... + 203) = 32670²,
(1 + 2 + ... + 8)(9 + 10 + ... + 18)(19 + 20 + ... + 203) = 9990²,
(1 + 2 + ... + 8)(9 + 10 + ... + 56)(57 + 58 + ... + 203) = 32760²,
(1 + 2 + ... + 8)(9 + 10 + ... + 93)(94 + 95 + ... + 203) = 50490²,
(1 + 2 + ... + 14)(15 + 16 + ... + 41)(42 + 43 + ... + 203) = 39690²,
(1 + 2 + ... + 27)(28 + 29 + ... + 203) = 2772²,
(1 + 2 + ... + 40)(41)(42 + 43 + ... + 203) = 25830²,
(1 + 2 + ... + 48)(49 + 50 + ... + 75)(76 + 77 + ... + 203) = 187488²,
(1 + 2 + ... + 128)(129)(130 + 131 + ... + 203) = 114552²,
(1 + 2 + ... + 174)(175 + 176 + ... + 203) = 9135².

203² = 41209 appears in the decimal expressions of π and e:
π = 3.14159•••41209••• (from the 29801st digit),
e = 2.71828•••41209••• (from the 53991st digit).

Page of Squares : First Upload April 19, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

204

The smallest squares containing k 204's :
20449 = 143²,
2042041 = 1429²,
20476204204096 = 4525064².

The squares which begin with 204 and end in 204 are
204547204 = 14302², 2042859204 = 45198², 204031083204 = 451698²,
204125047204 = 451802², 204483031204 = 452198²,...

204 = 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8².

204² = 41616, a zigzag square with 3 kinds of digits.

204²± 5 are primes.

204² = (2² + 8)(3² + 8)(14² + 8).

204² = 41616, 4 + 16 + 16 = 6².

204² = 41616, a square pegged by 1.

1 + 2 + 3 + ... + 288 = 204².

204^k + 294^k + 561^k + 966^k are squares for k = 1,2,3 ( 45², 1173², 33345²).
204^k + 3162^k + 7242^k + 12801^k are squares for k = 1,2,3 ( 153², 15045², 1584009²).

The integral triangle of sides 289, 2313, 2600 (or 17, 4904, 4905) has square area 204².

Komachi equation: 204² = - 9² * 8² - 7² + 6² * 5² + 43² + 210².

204² = 1² - 2² + 3² - 4² + ... - 287² + 288².

204² = 41616, (4 and 16 are squares).

204² = 23³ + 24³ + 25³.

204² + 205² + 206² + ... + 925² = 16169².

204² = 32 x 33 + 34 x 35 + 36 x 37 + 38 x 39 + ... + 64 x 65.

(1 + 2 + ... + 16)(17)(18) = 204²,
(1 + 2 + ... + 26)(27 + 28 + 29)(30 + ... + 204) = 24570²,
(1 + 2 + ... + 111)(112 + 113 + ... + 167)(168 + 169 + ... + 204) = 578088².

(1³ + 2³ + ... + 84³)(85³ + 86³ + ... + 203³)(204³) = 212156703360².

204² = 41616 appears in the decimal expressions of π and e:
π = 3.14159•••41616••• (from the 74832nd digit),
e = 2.71828•••41616••• (from the 23679th digit).

Page of Squares : First Upload April 19, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan

205

The smallest squares containing k 205's :
205209 = 453²,
205205625 = 14325²,
420520520520409 = 20506597².

205 = (1² + 2² + 3² + ... + 20²) / (1² + 2² + 3²).

205² = 42025, 420 / 2 - 5 = 205.

33^k + 205^k + 441^k + 477^k are squares for k = 1,2,3 ( 34², 682², 14246²).

3-by-3 magic squares consisting of different squares with constant 205²:

0²	45²	200²
123²	160²	36²
164²	120²	27²

20²	69²	192²
120²	160²	45²
165²	108²	56²

205² = 42025, 4² + 2² + 0² + 2² + 5² = 7²,
205² = 42025, 4 + 20 + 25 = 7²,
205² = 42025, 42 + 0 + 2 + 5 = 7².

205² = 42025, a square pegged by 2.

205² = 42025, (4 = 2², 2025 = 45²)

Komachi Square Sum : 205² = 2² + 4² + 36² + 95² + 178².

205² = 42025, 4³ + 2³ + 0³ + 2³ + 5³ = 205.

205² = 1! + 4! + 5! + 5! + 6! + 6! + 8!.

(1 + 2)(3 + 4 + ... + 118)(119 + 120 + ... + 205) = 17226²,
(1 + 2)(3 + ... + 142)(143 + 144 + ... + 205) = 18270²,
(1 + 2 + ... + 9)(10 + 11 + ... + 130)(131 + 132 + ... + 205) = 69300²,
(1 + 2 + ... + 13)(14 + 15 + ... + 130)(131 + 132 + ... + 205) = 98280²,
(1 + 2 + ... + 15)(16 + 17 + ... + 34)(35 + 36 + ... + 205) = 34200²,
(1 + 2 + ... + 18)(19 + 20 + ... + 37)(38 + 39 + ... + 205) = 43092²,
(1 + 2 + ... + 18)(19 + 20 + ... + 117)(118 + 119 + ... + 205) = 127908²,
(1 + 2 + ... + 22)(23 + 24 + ... + 91)(92 + 93 + ... + 205) = 129789²,
(1 + 2 + ... + 25)(26 + 27 + ... + 169)(170 + 171 + ... + 205) = 175500²,
(1 + 2 + ... + 28)(29 + 30 + ... + 142)(143 + 144 + ... + 205) = 208278²,
(1 + 2 + ... + 30)(31 + 32 + ... + 155)(156 + 157 + ... + 205) = 220875²,
(1 + 2 + ... + 125)(126 + 127 + ... + 130)(131 + 132 + ... + 205) = 252000².

205² = 42025 appears in the decimal expression of π:
π = 3.14159•••42025••• (from the 86855th digit).

Page of Squares : First Upload April 24, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

206

The smallest squares containing k 206's :
206116 = 454²,
20620681 = 4541²,
206227320620689 = 14360617².

206² = 42436, a zigzag square.

206² = 42436, 4² + 2² + 4² + 3² + 6² = 9²,
206² = 42436, 4 + 24 + 36 = 8².

Komachi Fraction : 206² = 6492708/153.

Komachi equation: 206² = 98² / 7² - 6² + 5² - 43² + 210².

(1 + 2)(3 + 4 + ... + 189)(190 + 191 + ... + 206) = 13464²,
(1 + 2 + ... + 6)(7 + 8 + ... + 38)(39 + 40 + ... + 206) = 17640²,
(1 + 2 + ... + 11)(12 + 13 + ... + 24)(25 + 26 + ... + 206) = 18018²,
(1 + 2 + ... + 19)(20 + 21 + ... + 189)(190 + 191 + ... + 206) = 106590²,
(1 + 2 + ... + 30)(31 + 32 + ... + 41)(42 + 43 + ... + 206) = 61380²,
(1 + 2 + ... + 38)(39 + 40 + ... + 78)(79 + 80 + ... + 206) = 177840²,
(1 + 2 + ... + 40)(41 + 42 + ... + 80)(81 + 82 + ... + 206) = 189420²,
(1 + 2 + ... + 54)(55 + 56 + ... + 90)(91 + 92 + ... + 206) = 258390²,
(1 + 2 + ... + 111)(112 + 113 + ... + 147)(148 + 149 + ... + 206) = 550116²,
(1 + 2 + ... + 153)(154 + 155 + ... + 189)(190 + 191 + ... + 206) = 494802².

206² = 42436 appears in the decimal expressions of π and e:
π = 3.14159•••42436••• (from the 68892nd digit),
e = 2.71828•••42436••• (from the 72098th digit).

Page of Squares : First Upload April 24, 2004 ; Last Revised May 7, 2010
by Yoshio Mimura, Kobe, Japan

207

The smallest squares containing k 207's :
20736 = 144²,
20720704 = 4552²,
207937207520784 = 14420028².

207² = 42849, a zigzag square.

207² = 42849, 4 + 28 + 49 = 9²,
207² = 42849, 428 + 4 + 9 = 21².

207² = 42849 is the third square which is the sum of 4 fourth powers in 2 ways :
3⁴ + 4⁴ + 8⁴ + 14⁴, 3⁴ + 6⁴ + 12⁴ + 12⁴.

3450^k + 10833^k + 12972^k + 15594^k are squares for k = 1,2,3 ( 207², 23253², 2699487²).
9177^k + 9660^k + 11454^k + 12558^k are squares for k = 1,2,3 ( 207², 21597², 2270997²).

Komachi Square Sum : 207² = 6² + 8² + 32² + 54² + 197².

3-by-3 magic squares consisting of different squares with constant 207²:

5²	82²	190²
118²	155²	70²
170²	110²	43²

14²	43²	202²
62²	194²	37²
197²	58²	26²

22²	98²	181²
142²	139²	58²
149²	118²	82²

37²	118²	166²
142²	106²	107²
146²	133²	62²

207² = 3⁴ + 6⁴ + 12⁴ + 12⁴ = 3⁴ + 4⁴ + 8⁴ + 14⁴.

(1 + 2 + ... + 100)(101 + 102 + ... + 196)(197 + 198 + ... + 207) = 399960².

207² = 42849 appears in the decimal expressions of π and e:
π = 3.14159•••42849••• (from the 48650th digit),
e = 2.71828•••42849••• (from the 14154th digit).

Page of Squares : First Upload April 24, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

208

The smallest squares containing k 208's :
208849 = 457²,
120802081 = 10991²,
19208208302089 = 4382717².

208 = (1² + 2² + 3² + ... + 32²) / (1² + 2² + 3² + 4² + 5²).

1482^k + 6630^k + 15418^k + 19734^k are squares for k = 1,2,3 (208², 25948², 3412448²).

Komachi equations:
208² = 9 + 8 + 7 + 6 * 5 + 43210 = 9 + 8 + 7 * 6 - 5 + 43210
= 9 + 8 * 7 - 6 - 5 + 43210 = 9 * 8 - 7 - 6 - 5 + 43210
= - 9 - 8 + 76 - 5 + 43210,
208² = 9² * 8² * 7² * 65² * 4² / 3² / 210².

208² = 43264, 4² + 3² + 2² + 6² + 4² = 9²,
208² = 43264, 4 + 32 + 64 = 10².

208² is the 8th square which is the sum of 7 fifth powers : (2, 3, 5, 5, 5, 7, 7).

A + B, A + C, A + D, B + C, B + D, and C + D are squares
for A = 208, B = 576, C = 1728, D = 8073.

(1 + 2 + ... + 31)(32 + 33 + ... + 61)(62 + 63 + ... + 208) = 117180²,
(1 + 2 + ... + 41)(42 + 43 + ... + 201)(202 + 203 + ... + 208) = 154980²,
(1 + 2 + ... + 120)(121 + 122 + ... + 176)(177 + 178 + ... + 208) = 609840².

208² = 43264 appears in the decimal expressions of π and e:
π = 3.14159•••43264••• (from the 74626th digit),
e = 2.71828•••43264••• (from the 121515th digit).

Page of Squares : First Upload April 24, 2004 ; Last Revised February 18, 2011
by Yoshio Mimura, Kobe, Japan

209

The smallest squares containing k 209's :
2209 = 47²,
1020994209 = 31953²,
209720950209 = 457953².

The squares which begin with 209 and end in 209 are
2097365209 = 45797², 20938958209 = 144703², 20966171209 = 144797²,
209034583209 = 457203², 209120546209 = 457297²,...

209 = (1² + 2² + 3² + ... + 38²) / (1² + 2² + 3² + 4² + 5² + 6²).

209² = 43681, a square with different digits.

209² = (3² + 2)(63² + 2).

108¹ + 124¹ + 129¹ = 19², 108² + 124² + 129² = 209², 108³ + 124³ + 129³ = 2305² (See 19).

3-by-3 magic squares consisting of different squares with constant 209²:

9²	72²	196²
124²	156²	63²
168²	119²	36²

12²	129²	164²
144²	124²	87²
151²	108²	96²

209² = 43681, 4 + 36 + 8 + 1 = 7²,
209² = 43681, 4 + 36 + 81 = 11².

209² is the 7th square which is the sum of 6 fifth powers : (3, 4, 4, 6, 7, 7) and 7th square which is the sum of 8 fifth powers: (1, 2, 2, 4, 4, 4, 6, 8).

1² + 2² + 3² + 4² + 5² + ... + 209² = 3064985, different digits.

209² = 43681, 4 + 3 * 68 + 1 = 209.

Cubic polynomial :
(X + 209²)(X + 528²)(X + 684²) = X³ + 889²X² + 403788²X + 75480768².

209² = 43681, (4, 36, and 81 are squares).

67^k + 209^k + 233^k + 275^k are squares for k = 1,2,3 ( 28², 422², 6548²).
152^k + 209^k + 3344^k + 5320^k are squares for k = 1,2,3 ( 95², 6289², 433561²).

Komachi equations:
209² = 1 + 234 * 5 * 6 * 7 * 8 / 9 = 9 * 8 * 7 * 65 * 4 / 3 + 2 - 1
= 9 + 8 * 7 * 65 * 4 * 3 + 2 - 10,
209² = 9² + 8² * 7² - 6² - 5² * 4² * 3² + 210²,
209² = 98⁴ / 7⁴ - 6⁴ + 54⁴ / 3⁴ / 2⁴ * 1⁴ = 98⁴ / 7⁴ - 6⁴ + 54⁴ / 3⁴ / 2⁴ / 1⁴.

(1 + 2 + ... + 10)(11 + 12 + ... + 110)(111 + 112 + ... + 209) = 72600².

(1³ + 2³ + ... + 40³)(41³ + 42³ + ... + 104³)(105³ + 106³ + ... + 209³) = 94083192000²,
(1³ + 2³ + ... + 83³)(84³)(85³ + 86³ + ... + 209³) = 58110922800².

209² = 43681 appears in the decimal expression of π
π = 3.14159•••43681••• (from the 39371st digit).

Page of Squares : First Upload April 24, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan