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100 - 109

100

The square of 10.

The smallest squares containing k 100's :
100 = 102,
25100100 = 50102,
100100100996 = 3163862.

The squares which begin with 100 and end in 100 are
100200100 = 100102,   1004256100 = 316902,   1005524100 = 317102,
10002000100 = 1000102,   10018008100 = 1000902,...

53 + 243 + 433 + 623 + 813 + 1003 = 13652.

1002 = 103 + 103 + 203.

The sum of the consecutive primes : 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100.

53 + 100 = 152, 53 - 100 = 52.

Komachi equations:
1002 = 9876 - 5 + 4 * 32 + 1,
1002 = 92 * 82 * 72 / 62 * 52 / 42 - 322 - 12 = 982 / 72 * 62 * 52 / 42 - 322 - 12
  = 982 - 72 + 62 - 52 - 42 + 32 + 212 = - 92 - 82 + 762 + 52 * 42 + 32 * 212
  = 92 * 82 + 72 * 62 + 542 + 32 * 22 + 102,
1002 = - 93 - 83 + 73 - 63 + 53 + 43 * 33 + 213,
1002 = 14 * 2344 * 54 * 64 / 784 / 94.

(1)(2 + 3 + ... + 97)(98 + 99 + 100) = 11882,
(1 + 2 + ... + 14)(15 + 16 + ... + 60)(61 + 62 + ... + 100) = 241502,
(1 + 2 + ... + 32)(33)(34 + 35 + ... + 100) = 88442,
(1 + 2 + ... + 32)(33 + 34 + ... + 43)(44 + 45 + ... + 100) = 300962,
(1 + 2 + ... + 32)(33 + 34 + ... + 86)(87 + 88 + ... + 100) = 471242,
(1 + 2 + ... + 32)(33 + 34 + ... + 97)(98 + 99 + 100) = 257402,
(1 + 2 + ... + 44)(45 + 46 + ... + 99)(100) = 198002,
(1 + 2 + ... + 51)(52)(53 + 54 + ... + 100) = 159122,
(1 + 2 + ... + 98)(99)(100) = 69302.

(13 + 23 + ... + 683)(693 + 703 + ... + 1003) = 104913122,
(13 + 23 + ... + 953)(963 + 973 + ... + 1003) = 98952002.

1002 = 10000 appears in the decimal expression of e:
  e = 2.71828•••10000••• (from the 26683rd digit)

Page of Squares : First Upload March29, 2004 ; Last Revised July 22, 2011
by Yoshio Mimura, Kobe, Japan

101

The smallest squares containing k 101's :
101124 = 3182,
9810110116 = 990462,
101510199101529 = 100752272.

101 is the sum of m squares for m = 2, 3, 4, ..., 87.

101 = 102 + 12.

1012 = 10201 is a palindromic and zigzag square.

1012 = 10201, 1 + 0 + 2 + 0 + 1 = 22, a square.

1012 = 10201, 102 + 0 - 1 = 101.

1012 = 1! + 5! + 7! + 7!

Komachi equations:
1012 = 9876 + 54 * 3 * 2 + 1,
1012 = 982 - 72 + 62 + 52 + 42 * 32 + 212 = 92 * 82 + 762 + 52 - 42 / 32 * 212.

(22 + 5)(42 + 5)(72 + 5) = (1012 + 5),
(102 - 9)(112 - 9) = (1012 - 9),
(42 - 9)(52 - 9)(102 - 9) = (1012 - 9).

(1 + 2)(3 + 4 + ... + 24)(25 + 26 + ... + 101) = 20792,
(1 + 2 + 3)(4 + 5 + ... + 38)(39 + 40 + ... + 101) = 44102,
(1 + 2 + ... + 5)(6 + 7 + ... + 14)(15 + 16 + ... + 101) = 26102,
(1 + 2 + ... + 5)(6 + 7 + ... + 38)(39 + 40 + ... + 101) = 69302,
(1 + 2 + ... + 10)(11 + 12 + ... + 24)(25 + 26 + ... + 101) = 80852,
(1 + 2 + ... + 10)(11 + 12 + ... + 43)(44 + 45 + ... + 101) = 143552,
(1 + 2 + ... + 13)(14 + 15 + ... + 52)(53 + 54 + ... + 101) = 210212,
(1 + 2 + ... + 26)(27 + 28 + ... + 38)(39 + 40 + ... + 101) = 245702,
(1 + 2 + ... + 44)(45 + 46 + ... + 63)(64 + 65 + ... + 101) = 564302,
(1 + 2 + ... + 45)(46 + 47 + ... + 59)(60 + 61 + ... + 101) = 507152,
(1 + 2 + ... + 63)(64 + 65 + ... + 90)(91 + 92 + ... + 101) = 665282.

1012 = 10201 appears in the decimal expressions of π and e:
  π = 3.14159•••10201••• (from the 52994th digit),
  e = 2.71828•••10201••• (from the 19675th digit)

Page of Squares : First Upload March 29, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

102

The smallest squares containing k 102's :
1024 = 322,
102111025 = 101052,
810261021025 = 9001452.

1022 = 10404 is a zigzag square, and a reversible square : 40401 = 2012.

1022 = 10404, 1 + 0 + 4 + 0 + 1 = 32.

57k + 102k + 222k + 348k are squares for k = 1,2,3 (272, 4292, 73712).
578k + 2074k + 2890k + 4862k are squares for k = 1,2,3 (1022, 60522, 3849482).
85k + 1649k + 4097k + 4573k are squares for k = 1,2,3 (1022, 63582, 4109582).
786k + 1614k + 3882k + 4122k are squares for k = 1,2,3 (1022, 59402, 3650042).

Komachi equations:
1022 = - 122 + 32 * 42 * 52 + 62 * 72 + 82 * 92 = 982 + 72 + 62 * 52 - 42 * 32 - 22 - 12
  = 982 - 72 + 62 * 52 - 42 - 32 * 22 + 12 = 92 + 872 + 62 * 52 + 432 + 22 + 12
  = 92 * 82 * 72 / 62 + 542 - 32 + 212 = 982 / 72 * 62 + 542 - 32 + 212
  = - 92 + 872 - 62 + 542 + 32 * 22 */ 12 = - 92 + 872 + 62 + 542 - 32 * 22 */ 12
  = - 92 + 872 + 62 * 542 / 32 / 22 */ 12 = 92 - 872 + 62 - 542 * 32 + 2102.

(102 - 8)(112 - 8) = (1022 - 8),
(42 - 9)(62 - 9)(82 - 9) = (1022 - 9).

(1 + 2 + ... + 8)(9 + 10 + ... + 25) = 1022,
(1 + 2)(3 + 4 + ... + 77)(78 + 79 + ... + 102) = 45002,
(1 + 2 + ... + 11)(12 + 13 + ... + 87)(88 + 89 + ... + 102) = 188102,
(1 + 2 + ... + 15)(16 + 17 + ... + 20)(21 + 22 + ... + 102) = 73802,
(1 + 2 + ... + 15)(16 + 17 + ... + 77)(78 + 79 + ... + 102) = 279002,
(1 + 2 + ... + 27)(28 + 29 + ... + 72)(73 + 74 + ... + 102) = 472502,
(1 + 2 + ... + 27)(28 + 29 + ... + 77)(78 + 79 + ... + 102) = 472502,
(1 + 2 + ... + 65)(66 + 67 + ... + 77)(78 + 79 + ... + 102) = 643502.

1022 = 10404 appears in the decimal expressions of π and e:
  π = 3.14159•••10401••• (from the 1270th digit),
  (10404 is the third 5-digit square in the expression of π.)
  e = 2.71828•••10404••• (from the 97919th digit)

Page of Squares : First Upload March 29, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

103

The smallest squares containing k 103's :
103041 = 3212,
103103716 = 101542,
103451031103744 = 101710882.

1032 = 10609 is a zigzag square, and a reversible square : 90601 = 3012.

1032 = 10609, 1 + 0 + 6 + 0 + 9 = 42,
1032 = 10609, 10 + 6 + 0 + 9 = 52.

1032 = 24 + 74 + 84 + 84.

132 + 432 + 732 + 1032 = 1342.

Komachi Fractions : 381924/576 = (103/4)2, 7056/381924 = (14/103)2,
  20736/95481 = (48/103)2, 30276/95481 = (58/103)2.

Komachi equations:
1032 = 982 + 72 * 62 + 52 - 42 / 32 * 212 = 92 * 82 + 762 * 52 / 42 - 32 * 22 * 102,
1032 = - 13 + 233 - 43 - 53 + 63 - 73 - 83 - 93.

(12 + 5)(422 + 5) = (1032 + 5),
(102 - 7)(112 - 7) = (1032 - 7),
(32 - 7)(82 - 7)(102 - 7) = (1032 - 7).

1032 + 1042 + 1052 + ... + 1422312 = 309694192.

(1)(2 + 3 + ... + 22)(23 + 24 + ... + 103) = 11342,
(1 + 2 + 3)(4 + 5 + ... + 11)(12 + 13 + ... + 103) = 13802,
(1 + 2 + 3)(4 + 5 + ... + 31)(32 + 33 + ... + 103) = 37802,
(1 + 2 + ... + 4)(5 + 6 + ... + 31)(32 + 33 + ... + 103) = 48602,
(1 + 2 + ... + 7)(8 + 9 + ... + 22)(23 + 24 + ... + 103) = 56702,
(1 + 2 + ... + 7)(8 + 9 + ... + 34)(35 + 36 + ... + 103) = 86942,
(1 + 2 + ... + 30)(31)(32 + 33 + ... + 103) = 83702,
(1 + 2 + ... + 32)(33 + 34 + ... + 92)(93 + 94 + ... + 103) = 462002,
(1 + 2 + ... + 38)(39 + 40 + ... + 65)(66 + 67 + ... + 103) = 577982,
(1 + 2 + ... + 42)(43 + 44 + ... + 85)(86 + 87 + ... + 103) = 650162,
(1 + 2 + ... + 56)(57)(58 + 59 + ... + 103) = 183542,
(1 + 2 + ... + 56)(57 + 58 + ... + 76)(77 + 78 + ... + 103) = 718202,
(1 + 2 + ... + 56)(57 + 58 + ... + 84)(85 + 86 + ... + 103) = 750122,
(1 + 2 + ... + 80)(81 + 82 + ... + 88)(89 + 90 + ... + 103) = 561602.

1032 = 10609 appears in the decimal expressions of π and e:
  π = 3.14159•••10609••• (from the 64384th digit),
  e = 2.71828•••10609••• (from the 6728th digit)

Page of Squares : First Upload March 29, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

104

The smallest squares containing k 104's :
10404 = 1022,
1046393104 = 323482,
104104313104 = 3226522.

The squares which begin with 104 and end in 104 are
1046393104 = 323482,   10435031104 = 1021522,   10475113104 = 1023482,
104104313104 = 3226522,   104230831104 = 3228482,...

104 is the sum of m squares for m = 2, 3, 4, 5, ..., 90.

1 / 104 = 0.00961..., 961 = 312.

1042 = 10816 is a zigzag square.

1042 = 2! + 2! + 3! + 3! + 6! + 7! + 7!.

(12 + 22 + 32 + ... + 1042) = 380380, which consists of 3 kinds of digits.

1042 = 10816, 1 + 0 + 8 + 1 + 6 = 42,
1042 = 10816, 1 + 0 + 8 + 16 = 52,
1042 = 10816, 10 + 8 + 1 + 6 = 52.

Komachi equations:
1042 = 12 + 22 * 342 + 562 + 72 * 82 - 92 = - 12 * 232 + 42 * 52 * 62 - 72 * 82 + 92
  = 982 + 72 * 62 + 52 - 42 * 32 * 22 - 12 = 982 - 72 - 62 + 542 / 32 * 22 + 12
  = 92 * 82 + 762 - 52 * 42 * 32 * 22 / 102,
1042 = 14 * 24 - 34 - 44 - 54 - 64 + 74 + 84 + 94 = 94 + 84 + 74 - 64 - 54 - 44 - 34 + 24 */ 14.

(102 - 6)(112 - 6) = 1042 - 6.

1042 + 1052 + 1062 + ... + 101852 = 5934892,
1042 + 1052 + 1062 + ... + 9672 = 173642.

(1 + 2 + ... + 20)(21 + 22 + ... + 104) = 10502,
(1 + 2 + ... + 27)(28 + 29 + ... + 104) = 13862,
(1 + 2 + ... + 44)(45 + 46 + ... + 55)(56 + 57 + ... + 104) = 462002.

(13 + 23 + ... + 143)(153 + 163 + ... + 203)(213 + 223 + ... + 1043) = 1041862502.

1042 = 10816 appears in the decimal expression of e:
  e = 2.71828•••10201••• (from the 13677th digit)

Page of Squares : First Upload March 29, 2004 ; Last Revised April 13, 2010
by Yoshio Mimura, Kobe, Japan

105

The smallest squares containing k 105's :
105625 = 3252,
10510564 = 32422,
1051050110529409 = 324199032.

105 = (12 + 22 + 32 + ... + 492) / (12 + 22 + 32 + ... + 102).

1052 = 11025, 1 + 1 + 0 + 2 + 5 = 32,
1052 = 11025, 11 + 0 + 25 = 62.

1052 + 1062 + 1072 + ... + 1122 = 1132 + 1142 + 1152 + ... + 1192.

Komachi Fraction : 1052 = 9327150/846.

Komachi equations:
1052 = 122 / 32 * 42 * 52 / 62 * 72 / 82 * 92 = 122 / 32 * 452 * 62 * 72 / 82 / 92
  = 1232 + 452 + 62 - 782 - 92 = - 122 * 32 / 42 + 562 + 72 + 892
  = 92 * 82 + 762 + 52 + 42 * 32 - 22 - 102.

(12 + 22 + 32 + ... + 1052) = 391405, which consists of different digits,
(the first 6-digit integer which is the sum of the consecutuve squares : 12 + 22 + 32 + ... + n2).

(12 + 22 + 32 + ... + 212) + (12 + 22 + 32 + ... + 282) = 1052,
(12 + 22 + 32 + ... + 192) + (12 + 22 + 32 + ... + 292) = 1052.

Two 3-by-3 magic squares consisting of different squares with constant 1052:

 123221002
682762252
802652202
      
 823121002
442922252
952402202

Cubic Polynomial : (X + 1052)(X + 1402)(X + 2882) = X3 + 3372X2 + 525002X + 42336002.

(102 - 5)(112 - 5) = (1052 - 5).

(1 + 2)(3 + 4)(5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15) = 1052,
(1 + 2 + ... + 14)(15 + 16 + ... + 20) = 1052,
(1)(2 + 3 + ... + 8)(9 + 10 + ... + 26) = 1052,
(1)(2 + 3)(4 + 5 + ... + 66) = 1052,
(1 + 2 + ... + 50)(51 + 52 + ... + 102)(103 + 104 + 105) = 397802,
(1 + 2 + ... + 57)(58 + 59 + ... + 86)(87 + 88... + 105) = 793442,
(1)(2 + 3 + ... + 148) = 1052.

(13 + 23 + ... + 103)(113 + 123 + ... + 293)(303 + 313 + ... + 1053) = 1316700002,
(13 + 23 + ... + 633)(643 + 653 + ... + 1053) = 104569922.

13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 + 113 + 123 + 133 + 143 = 1052.

1052 = 11025 appears in the decimal expression of e:
  e = 2.71828•••10201••• (from the 5989th digit)

Page of Squares : First Upload April 5, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

106

The smallest squares containing k 106's :
10609 = 1032,
106110601 = 103012,
1061068603106304 = 325740482.

1 / 106 = 0.00943, 92 + 42 + 32 = 106.

1062 = 11236, 1 + 12 + 36 = 72,
1062 = 11236, 11 + 2 + 36 = 72,
1062 = 11236, 112 + 3 + 6 = 112.

1062 = 11236, the digits of which is a non-decreasing sequence.

1062 is the first square which is the sum of 8 sixth powers:
    26 + 26 + 36 + 36 + 36 + 36 + 46 + 46 = 1062.

106k + 266k + 814k + 1314k are squares for k = 1,2,3 (502, 15722, 531802).

Komachi Square Sums : 1062 = 12 + 22 + 52 + 392 + 462 + 872
  = 12 + 22 + 62 + 352 + 492 + 872 = 12 + 22 + 32 + 42 + 62 + 572 + 892.

1062 = (7 x 8)2 + (9 x 10)2.

1062 = 92 + 102 + 112 + ... + 322.

(102 - 4)(112 - 4) = (1062 - 4),
(12 + 6)(402 + 6) = (1062 + 6).

1062 + 1072 + 1082 + ... + 13792 = 295752,
1062 + 1072 + 1082 + ... + 4022 = 46202.

(1 + 2)(3 + 4 + ... + 10)(11 + 12 + ... + 106) = 9362,
(1 + 2 + 3)(4 + 5 + ... + 73)(74 + 75 + ... + 106) = 69302,
(1 + 2 + ... + 7)(8 + 9 + ... + 64)(65 + 66 + ... + 106) = 143642,
(1 + 2 + ... + 10)(11 + 12 + ... + 25)(26 + 27 + ... + 106) = 89102,
(1 + 2 + ... + 10)(11 + 12 + ... + 73)(74 + 75 + ... + 106) = 207902,
(1 + 2 + ... + 10)(11 + 12 + ... + 88)(89 + 90 + ... + 106) = 193052,
(1 + 2 + ... + 13)(14 + 15 + ... + 28)(29 + 30 + ... + 106) = 122852,
(1 + 2 + ... + 13)(14 + 15 + ... + 103)(104 + 105 + 106) = 122852,
(1 + 2 + ... + 19)(20 + 21 + ... + 64)(65 + 66 + ... + 106) = 359102,
(1 + 2 + ... + 21)(22 + 23 + ... + 29)(30 + 31 + ... + 106) = 157082,
(1 + 2 + ... + 21)(22 + 23 + ... + 55)(56 + 57 + ... + 106) = 353432,
(1 + 2 + ... + 26)(27 + 28 + ... + 64)(65 + 66 + ... + 106) = 466832,
(1 + 2 + ... + 34)(35 + 36 + ... + 55)(56 + 57 + ... + 106) = 481952,
(1 + 2 + ... + 39)(40 + 41 + ... + 88)(89 + 90 + ... + 106) = 655202,
(1 + 2 + ... + 42)(43)(44 + 45 + ... + 106) = 135452,
(1 + 2 + ... + 56)(57 + 58 + ... + 64)(65 + 66 + ... + 106) = 526682,
(1 + 2 + ... + 69)(70 + 71 + ... + 91)(92 + 93 + ... + 106) = 796952,
(1 + 2 + ... + 80)(81 + 82 + ... + 89)(90 + 91 + ... + 106) = 642602.

(12 + 22 + ... + 112)(122 + 132 + ... + 602)(612 + 622 + ... + 1062) = 34924122.

1062 = 11236 appears in the decimal expressions of π and e:
  π = 3.14159•••11236••• (from the 56398th digit),
  e = 2.71828•••11236••• (from the 135844th digit)

Page of Squares : First Upload April 5, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

107

The smallest squares containing k 107's :
51076 = 2262,
410710756 = 202662,
10710735471076 = 32727262.

1072 = 11449, the digits of which is a non-decreasing sequence.

1072 = 11449, 1 + 14 + 49 = 82,
1072 = 11449, 11 + 4 + 49 = 82,
1072 = 11449, 11 + 44 + 9 = 82.

1072 = 11449 = 11449 = 11449, where 1, 4, 49 and 144 are squares.

112 + 432 + 752 + 1072 = 1382.

The sum of the consecutive odd primes : 3 + 5 + 7 + 11 + 13 + 17 + ... + 107 = 372.

1072 = 213 + 15 + 37.

Komachi equations:
1072 = 982 + 72 * 62 + 542 / 32 / 22 */ 12 = 982 - 72 * 62 + 52 - 42 + 32 * 22 * 102.

A 3-by-3 magic square consisting of different squares with constant 1072:

 62622872
732662422
782572462

(102 - 3)(112 - 3) = (1072 - 3).

(1 + 2 + ... + 32)(33 + 34 + ... + 57)(58 + 59 + ... + 107) = 495002.

1072 = 11449 appears in the decimal expressions of π and e:
  π = 3.14159•••11449••• (from the 33503rd digit),
  e = 2.71828•••11449••• (from the 147898th digit)

Page of Squares : First Upload April 5, 2004 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

108

The smallest squares containing k 108's :
1089 = 332,
1080108225 = 328652,
11421081081081 = 33795092.

1082 = 11664, 1 + 16 + 64 = 92,
1082 = 11664, 11 + 6 + 64 = 92,
1082 = 11664, 11 + 66 + 4 = 92.

1082 = 183 + 183, the 7th square which is the sum of two cubes.

1082 = 15 + 15 + 25 + 35 + 35 + 35 + 55 + 65, the 5th square which is the sum of 5 fifth powers.

1082 = 11664, where 1, 16 and 64 are squares.

Cubic Polynomial : (X+72)(X + 1082)(X + 3362) = X3 + 3532X2 + 363722X + 2540162.

1082 = 11664 is an exchangeable square, 16641 = 1292.

1082 = 4! + 5! + 6! + 6! + 7! + 7!.

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

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

Komachi Fraction : (108/5)2 = 314928/675.

Komachi equations:
1082 = 9 * 876 + 54 / 3 * 210,
1082 = 92 + 872 + 62 + 52 - 42 + 32 * 212 = 92 + 82 - 72 + 62 * 542 / 32 + 22 - 102
  = - 92 - 82 + 72 + 62 * 542 / 32 - 22 + 102,
1082 = 94 - 84 - 74 + 64 + 54 - 44 - 34 + 24 + 104.

(102 - 2)(112 - 2) = (1082 - 2),
(32 - 2)(42 - 2)(112 - 2) = (1082 - 2),
(42 - 3)(302 - 3) = (1082 - 3).

6522 = 132 + 142 + 152 + ... + 1082.

(1 + 2)(3)(4 + 5)(6)(7 + 8 + 9) = 1082,
(1)(2 + 3 + 4)(5 + 6 + 7)(8)(9) = 1082,
(1 + 2 + 3)(4)(5 + 6 + 7)(8 + 9 + 10) = 1082,
(1)(2)(3)(4)(5 + 6 + 7)(8 + 9 + 10) = 1082,
(1 + 2 + 3)(4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)(13 + 14) = 1082,
(1)(2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)(13 + 14) = 1082,
(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 21) = 1082,
(1 + 2 + 3)(4)(5 + 6 + ... + 31) = 1082.

(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 108) = 15842,
(1 + 2 + ... + 6)(7 + 8 + ... + 27)(28 + 29 + ... + 108) = 64262,
(1 + 2 + ... + 6)(7 + 8 + ... + 38)(39 + 40 + ... + 108) = 88202,
(1 + 2 + ... + 8)(9 + 10 + ... + 17)(18 + 19 + ... + 108) = 49142,
(1 + 2 + ... + 10)(11 + 12 + ... + 45)(46 + 47 + ... + 108) = 161702,
(1 + 2 + ... + 16)(17 + 18 + ... + 27)(28 + 29 + ... + 108) = 134642,
(1 + 2 + ... + 16)(17 + 18 + ... + 91)(92 + 93 + ... + 108) = 306002,
(1 + 2 + ... + 17)(18 + 19 + ... + 27)(28 + 29 + ... + 108) = 137702,
(1 + 2 + ... + 21)(22 + 23 + ... + 77)(78 + 79 + ... + 108) = 429662,
(1 + 2 + ... + 23)(24 + 25 + ... + 45)(46 + 47 + ... + 108) = 318782,
(1 + 2 + ... + 32)(33 + 34 + ... + 45)(46 + 47 + ... + 108) = 360362,
(1 + 2 + ... + 32)(33 + 34 + ... + 77)(78 + 79 + ... + 108) = 613802,
(1 + 2 + ... + 38)(39 + 40 + ... + 51)(52 + 53 + ... + 108) = 444602,
(1 + 2 + ... + 71)(72 + 73 + ... + 104)(105 + 106 + ... + 108) = 562322,
(13 + 23 + + ... + 803)(813 + 823 + ... + 1083) = 159213602.

1082 = 11664 appears in the decimal expressions of π and e:
  π = 3.14159•••11664••• (from the 51861st digit),
  e = 2.71828•••11664••• (from the 109641st digit)

Page of Squares : First Upload April 5, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

109

The smallest squares containing k 109's :
109561 = 3312,
10971096049 = 1047432,
109109264109444 = 104455382.

1092 = 11881, a square which consists of 2 kind of digits : 1 and 8.

1092 = 11881, 1 + 18 + 81 = 102,
1092 = 11881, 11 + 8 + 81 = 102,
1092 = 11881, 11 + 88 + 1 = 102.

1092 = 11881, 118 - 8 - 1 = 109.

1092 = 272 + 442 + 962 = 692 + 442 + 722.

9k + 10k + 60k + 90k are squares for k = 1,2,3 (132, 1092, 9732).
29k + 109k + 301k + 461k are squares for k = 1,2,3 (302, 5622, 112502).

Komachi Fraction : (2/109)2 = 936/2780154.

Komachi equations:
1092 = 1232 - 42 - 562 + 72 - 82 - 92 = 982 - 72 * 62 + 52 * 42 * 32 + 212
  = 982 + 72 * 62 + 52 * 42 + 32 + 22 + 102.

A 3-by-3 magic square consisting of different squares with constant 1092:

 82512962
692722442
842642272

(102 - 1)(112 - 1) = (1092 - 1),
(32 - 1)(42 - 1)(102 - 1) = (1092 - 1),
(72 + 9)(142 + 9) = (1092 + 9).

(12 + ... + 542)(552 + 562 + ... + 1092) = 1438802.

(1 + 2 + ... + 4)(5 + 6 + ... + 79)(80 + 81 + ... + 109) = 94502,
(1 + 2 + ... + 6)(7 + 8 + ... + 93)(94 + 95 + ... + 109) = 121802,
(1 + 2 + ... + 16)(17 + 18 + ... + 34)(35 + 36 + ... + 109) = 183602,
(1 + 2 + ... + 20)(21 + 22 + ... + 70)(71 + 72 + ... + 109) = 409502,
(1 + 2 + ... + 32)(33 + 34 + ... + 88)(89 + 90 + ... + 109) = 609842,
(1 + 2 + ... + 38)(39 + 40 + ... + 52)(53 + 54 + ... + 109) = 466832,
(1 + 2 + ... + 48)(49 + 50 + ... + 73)(74 + 75 + ... + 109) = 768602,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + ... + 109) = 831602,
(1 + 2 + ... + 81)(82)(83 + 84 + ... + 109) = 265682.

1092 = 11881 appears in the decimal expressions of π and e:
  π = 3.14159•••11881••• (from the 846th digit),
  (11881 is the first 5-digit square in the expression of π.)
  e = 2.71828•••11881••• (from the 68539th digit)

Page of Squares : First Upload April 5, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan