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120 - 129

120

The smallest squares containing k 120's :
41209 = 2032,
120012025 = 109552,
91201201204624 = 95499322.

1202 = 14400 (1, 4 and 400 are squares).

1202 = 43 + 83 + 243.

1202 = 6 x 6! + 2 x 7!.

1202 = 120,(32 - 1)(42 - 1)(112 - 1) = (42 - 1)(312 - 1).

132 + 120 = 172, 132 - 120 = 72.

Komachi equations:
1202 = 1 - 2 + 345 * 6 * 7 - 89 = 9 * 8 * 765 / 4 + 3 * 210,
1202 = 12 - 22 * 32 + 452 + 672 + 892 = 982 - 72 * 62 + 542 * 32 / 22 - 12
  = 982 + 72 / 62 * 52 * 42 * 32 - 22 - 102 = - 92 - 82 - 72 * 62 + 52 + 42 * 322 - 102,
1202 = - 14 * 24 - 34 - 44 + 564 / 74 + 84 + 94.

1692 = 1192 + 1202.

120 and 121 are consecutive integers having square factors (the 9th case).

(1)(2 + 3)(4)(5)(6)(7 + 8 + 9) = 1202,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9)(10 + 11 + ... + 18) = 2102,
(1 + 2 + 3 + 4)(5 + 6 + ... + 16)(17 + 18) = 2102,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 38) = 1202,
(1 + 2 + ... + 12)(13 + 14 + ... + 68)(69 + 70 + ... + 120) = 294842.

(13 + 23 + ... + 1043)(1053 + 1063 + ... + 1193)(1203) = 330220800002.

13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 + 113 + 123 + 133 + 143 + 153 = 1202.

1202 = 14400 appears in the decimal expression of e:
  e = 2.71828•••14400••• (from the 142273rd digit)

Page of Squares : First Upload April 26, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

121

the square of 11.

The smallest squares containing k 121's :
121 = 112,
12173121 = 34892,
12131212374121 = 34829892.

The squares which begin with 121 and end in 121 are
12173121 = 34892,   121242121 = 110112,   12102420121 = 1100112,
12152637121 = 1102392,   12157488121 = 1102612,...

1212 = 14641, a palindromic square.

1212 = 14641, 1 + 4 + 6 + 4 + 1 = 42,
1212 = 14641, 14 + 6 + 4 + 1 = 52,
1212 = 14641, 143 + 63 + 43 + 13 = 552.

1212 = 14641 (1, 4 and 64 are squares).

1212 = 123 + 173 + 203.

1212 + 1222 + 1232 + ... + 1442 = 6502.

Komachi equations:
1212 = 12 + 232 * 42 - 52 + 62 + 782 + 92 = 92 * 82 + 762 * 52 / 42 - 32 + 212.

A 3-by-3 magic square consisting of different squares with constant 1212:

424821112
842812322
872762362

(1)(2 + 3 + ... + 13)(14 + 15 + ... + 121) = 8102,
(1)(2 + 3 + ... + 25)(26 + 27 + ... + 121) = 15122,
(1 + 2 + 3 + 4)(5 + 6 + ... + 13)(14 + 15 + ... + 121) = 24302,
(1 + 2 + 3 + 4)(5 + 6 + ... + 40)(41 + 42 + ... + 121) = 72902,
(1 + 2 + ... + 7)(8 + 9 + ... + 49)(50 + 51 + ... + 121) = 143642,
(1 + 2 + ... + 8)(9 + 10 + ... + 25)(26 + 27 + ... + 121) = 85682,
(1 + 2 + ... + 8)(9 + 10 + ... + 40)(41 + 42 + ... + 121) = 136082,
(1 + 2 + ... + 8)(9 + 10 + ... + 112)(113 + 114 + ... + 121) = 154442,
(1 + 2 + ... + 16)(17 + 18 + ... + 31)(32 + 33 + ... + 121) = 183602,
(1 + 2 + ... + 16)(17 + 18 + ... + 67)(68 + 69 + ... + 121) = 385562,
(1 + 2 + ... + 16)(17 + 18 + ... + 85)(86 + 87 + ... + 121) = 422282,
(1 + 2 + ... + 16)(17 + 18 + ... + 118)(119 + 120 + 121) = 183602,
(1 + 2 + ... + 28)(29 + 30 + ... + 58)(59 + 60 + ... + 121) = 548102.

(12 + 22 + 32 + ... + 1212) = 597861, which consists of different digits.

1212 = 14641 appears in the decimal expression of e:
  e = 2.71828•••14641••• (from the 51919th digit)

Page of Squares : First Upload April 26, 2004 ; Last Revised April 16, 2010
by Yoshio Mimura, Kobe, Japan

122

The smallest squares containing k 122's :
1225 = 352,
1221222916 = 349462,
122029912291225 = 110467152.

1222 = 14884, a reversible square (48841 = 2212).

1222 = 14884, 1 + 4 + 8 + 8 + 4 = 52.

(12 + 22 + 32 + ... + 1222) = 612745, which consists of different digits.

1222 = 14884, 1 * 488 / 4 = 122.

14k + 58k + 122k + 130k are squares for k = 1,2,3 (182, 1882, 20522).

Komachi equations:
1222 = 92 + 82 * 72 + 62 * 542 / 32 + 22 - 12 = - 92 * 82 - 762 + 542 * 32 - 22 * 102.

1222 = 34 + 34 + 34 + 114.

(42 - 7)(62 - 7)(82 - 7) = (1222 - 7).

(1 + 2)(3 + 4 + ... + 122) = 1502 (1 + 2 + ... + 5)(6 + 7 + ... + 59)(60 + 61 + ... + 122) = 122852 (1 + 2 + ... + 24)(25 + 26 + ... + 122) = 14702.

(13 + 23 + ... + 243)(253 + 263 + ... + 1223) = 22491002,
(13 + 23 + ... + 1173)(1183 + 1193 + ... + 1223) = 202948202.

1222 = 14884 appears in the decimal expressions of π and e:
  π = 3.14159•••14884••• (from the 33985th digit),
  e = 2.71828•••14884••• (from the 138537th digit)

Page of Squares : First Upload April 26, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

123

The smallest squares containing k 123's :
11236 = 1062,
1231237921 = 350892,
123212353211236 = 111001062.

1232 = 15129, 1 + 5 + 1 + 29 = 62,
1232 = 15129, 1 + 51 + 29 = 92,
1232 = 15129, 15 + 12 + 9 = 62,
1232 = 15129, 15 + 129 = 122,
1232 = 15129, 1512 + 9 = 392.

1232 = 15129, 15 + 12 * 9 = 123.

1232 = 1! + 2! + 3! + 7! + 7! + 7!.

1232 = (12 + 2)(712 + 2).

32 + 112 + 192 + 272 + ... + 1232 = 2922.

Komachi equations:
1232 = 1 * 2 + 3 + 4 + 5 * 6 * 7 * 8 * 9 = 12 * 3 / 4 + 5 * 6 * 7 * 8 * 9
  = 12 * 3 / 4 * 5 * 6 * 7 * 8 + 9 = 1 + 2 - 3 + 45 * 6 * 7 * 8 + 9
  = 12 * 3 * 45 / 6 * 7 * 8 + 9 = - 1 - 2 + 3 * 4 + 5 * 6 * 7 * 8 * 9
  = - 1 + 2 * 3 + 4 + 5 * 6 * 7 * 8 * 9 = - 1 - 2 + 3 + 45 * 6 * 7 * 8 + 9
  = 9 * 8 * 7 * 6 * 5 + 4 + 3 + 2 */ 1 = 9 * 8 * 7 * 6 * 5 + 4 + 3 * 2 - 1
  = 9 * 8 * 7 * 6 * 5 + 4 * 3 - 2 - 1 = 9 * 8 * 7 * 6 * 5 - 4 * 3 + 21
  = 9 * 8 * 7 * 6 * 5 + 4 - 3 - 2 + 10,
1232 = - 12 - 22 + 32 * 452 - 62 - 72 * 82 + 92,
1232 = 123 * 33 - 43 * 563 / 73 + 83 + 93 = 93 - 83 * 73 - 63 + 543 + 323 */ 13.

A 3-by-3 magic square consisting of different squares with constant 1232:

 723421182
6221032262
1062582232

(22 + 1)(552 + 1) = (1232 + 1),
(22 + 7)(372 + 7) = (1232 + 7),
(22 + 7 )(52 + 7 )(62 + 7 ) = (1232 + 7),
(112 - 9)(122 - 9) = (1232 - 9),
(42 - 9)(52 - 9)(122 - 9) = (1232 - 9).

282 + 292 + 302 + ... + 1232 = 7882.

(1)(2 + 3 + ... + 6)(7 + 8 + ... + 123) = 3902,
(1)(2 + 3 + ... + 28)(29 + 30 + ... + 123) = 17102,
(1 + 2 + ... + 9)(10 + 11 + ... + 28)(29 + 30 + ... + 123) = 108302,
(1 + 2 + ... + 32)(33 + 34 + ... + 110)(111 + 112 + ... + 123) = 669242,
(1 + 2 + ... + 39)(40 + 41 + ... + 104)(105 + 106 + ... + 123) = 889202.

1232 = 15129 appears in the decimal expressions of π and e:
  π = 3.14159•••15129••• (from the 102355th digit)
  e = 2.71828•••15129••• (from the 14011st digit)

Page of Squares : First Upload April 26, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan

124

The smallest squares containing k 124's :
33124 = 1822,
124791241 = 111712,
21245812425124 = 46093182.

The squares which begin with 124 and end in 124 are
1247361124 = 353182,   12472869124 = 1116822,   124032161124 = 3521822,
124127973124 = 3523182,   124384593124 = 3526822,...

1242 = 15376, a zigzag square with different digits.

1242 = 15376, a square with odd digits except the last digit 6.

1242 = 15376, 1 + 5 + 37 + 6 = 72.

(112- 8)(122 - 8) = (1242 - 8),
(42 - 8)(52 - 8)(112 - 8) = (1242 - 8).

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

124, 125 and 126 are three consecutive integers having square factors (the third case).

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

113k + 124k + 262k + 590k are squares for k = 1,2,3 (332, 6672, 150572).
403k + 961k + 5363k + 8649k are squares for k = 1,2,3 (1242, 102302, 8956522).

Komachi equations:
1242 = 982 + 762 + 52 - 42 - 32 - 22 */ 12 = 982 + 762 - 52 + 42 + 32 - 22 */ 12
  = 982 + 762 - 52 - 42 + 32 * 22 + 12 = 92 + 82 * 72 - 652 + 42 * 322 */ 12
  = - 92 * 82 - 72 + 652 + 42 * 322 */ 12 = - 92 * 82 - 72 + 652 + 42 * 322 / 12.

(1 + 2)(3)(4 + 5 + ... + 124) = 2642,
(1 + 2 + ... + 17)(18 + 19 + ... + 22)(23 + 24 + ... + 124) = 107102,
(1 + 2 + ... + 42)(43)(44 + 45 + ... + 124) = 162542.

(12 + 22 + 32 + ... + 1242) = 643250, which consists of different digits.

(12 + 22 + ... + 72)(82 + 92 + ... + 122)(132 + 142 + ... + 1242) = 2142002.

1242 = 15376 appears in the decimal expressions of π and e:
  π = 3.14159•••15376••• (from the 33526th digit),
  e = 2.71828•••15376••• (from the 64498th digit)

Page of Squares : First Upload April 26, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

125

The smallest squares containing k 125's :
12544 = 1122,
12501252481 = 1118092,
12512512513809 = 35373032.

1252 = 15625, 1 + 56 + 2 + 5 = 82.

1252 = 15625, (1 and 5625 are squares).

1252 = 15625, 1 / 5 * 625 = 125.

A cubic polynomial (X + 352)(X + 722)(X + 962) = X3 + 1252X2 + 80882X + 2419202.

1252 = 43 + 173 + 223.

1252 is the third square which is the sum of 5 fifth powers [5,5,5,5,5].

1252 = (13 + 14 + 15 + 16 + 17)2 + (18 + 19 + 20 + 21 + 22)2.

125k + 241k + 305k + 485k are squares for k = 1,2,3 (342, 6342, 125862).

Komachi Fractions : 1252 = 9421875/603, (13/125)2 = 9126/843750.

Komachi equations:
1252 = 982 - 72 + 652 + 432 - 22 */ 12.

A 3-by-3 magic square consisting of different squares with constant 1252:

 023521202
752962282
1002722212

(112 - 7)(122 - 7) = (1252 - 7),
(32 - 7)(82 - 7)(122 - 7) = (1252 - 7).

(1 + 2 + 3 + 4)(5 + 6 + ... + 70)(71 + 72 + ... + 125) = 115502,
(1 + 2 + ... + 6)(7 + 8 + ... + 27)(28 + 29 + ... + 125) = 74972,
(1 + 2 + ... + 16)(17 + 18 + ... + 27)(28 + 29 + ... + 125) = 157082,
(1 + 2 + ... + 17)(18 + 19 + ... + 27)(28 + 29 + ... + 125) = 160652,
(1 + 2 + ... + 17)(18 + 19 + ... + 50)(51 + 52 + ... + 125) = 336602,
(1 + 2 + ... + 17)(18 + 19 + ... + 108)(109 + 110 + ... + 125) = 417692,
(1 + 2 + ... + 48)(49 + 50)(51 + 52 + ... + 125) = 277202,
(1 + 2 + ... + 48)(49 + 50 + ... + 77)(78 + 79 + ... + 125) = 1023122,
(1 + 2 + ... + 68)(69 + 70 + ... + 102)(103 + 104 + ... + 125) = 1337222.

(12 + 22)(32 + 42)(52 + 62 + ... + 1252) = 90752.

(13 + 23 + ... + 233)(243 + 253 + ... + 453)(463 + 473 + ... + 1253) = 21493058402.

1252 = 15625 appears in the decimal expressions of π and e:
  π = 3.14159•••15625••• (from the 6927th digit),
  e = 2.71828•••12100••• (from the 110272nd digit)

Page of Squares : First Upload May 10, 2004 ; Last Revised February 8, 2011
by Yoshio Mimura, Kobe, Japan

126

The smallest squares containing k 126's :
126025 = 3552,
12612637636 = 1123062,
112612612624281 = 106119092.

126 = (12 + 22 + 32 + ... + 272) / (12 + 22 + 32 + 42 + 52).

1262 = 15876 with different digits.

1262 = 15876, 15 + 8 + 7 + 6 = 62.

1262 = 3! + 3! + 4! + 6! + 7! + 7! + 7!.

1262 = (12 + 5)(22 + 5)(172 + 5) = (12 + 5)(22 + 5)(32 + 5)(42 + 5) = (12 + 5)(42 + 5)(112 + 5)
= (22 + 5)(32 + 5)(112 + 5) = (32 + 5)(42 + 5)(72 + 5) = (72 + 5)(172 + 5).

126k + 538k + 1050k + 1422k are squares for k = 1,2,3 (562, 18522, 647362).
154k + 3038k + 3626k + 9058k are squares for k = 1,2,3 (1262, 102202, 9049322).
2282k + 3010k + 3626k + 6958k are squares for k = 1,2,3 (1262, 87082, 6509162).
3045k + 3381k + 4305k + 5145k are squares for k = 1,2,3 (1262, 81062, 5318462).

Komachi Fraction : (13/126)2 = 9126/857304.

Komachi equation:
1262 = - 9 * 8 * 7 + 65 * 4 * 3 * 21,
1262 = 122 / 32 * 42 * 5672 / 82 / 92 = - 92 * 82 * 72 * 62 / 542 * 32 + 2102.

(62 - 3)(222 - 3) = (1262 - 3),
(112 - 6)(122 - 6) = (1262 - 6).

(1)(2 + 3 + 4 + 5)(6)(7)(8 + 9 + 10) = 1262,
(1)(2)(3 + 4)(5 + 6 + 7)(8 + 9 + 10 + 11 + 12 + 13) = 1262,
(1)(2)(3 + 4 + 5 + 6)(7)(8 + 9 + 10 + 11 + 12 + 13) = 1262,
(1 + 2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + 13) = 1262,
(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13)(14) = 1262,
(1 + 2 + ... + 6)(7)(8 + 9 + ... + 16) = 1262,
(1 + 2)(3 + 4 + ... + 9)(10 + 11 + ... + 18) = 1262,
(1 + 2 + 3)(4 + 5 + ... + 17)(18) = 1262,
(1 + 2 + ... + 7)(8 + 9 + ... + 34) = 1262,
(1 + 2)(3)(4 + 5 + ... + 59) = 1262.

(1 + 2)(3 + 4 + ... + 45)(46 + 47 + ... + 126) = 46442,
(1 + 2 + ... + 6)(7 + 8 + ... + 42)(43 + 44 + ... + 126) = 114662,
(1 + 2 + ... + 6)(7 + 8 + ... + 69)(70 + 71 + ... + 126) = 167582,
(1 + 2 + ... + 9)(10 + 11 + ... + 94)(95 + 96 + ... + 126) = 265202,
(1 + 2 + ... + 48)(49 + 50 + ... + 113)(114 + 115 + ... + 126) = 982802,
(1 + 2 + ... + 63)(64 + 65 + ... + 69)(70 + 71 + ... + 126) = 670322.

1262 = 15876 appears in the decimal expression of e:
  e = 2.71828•••15876••• (from the 127499th digit)

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

127

The smallest squares containing k 127's :
12769 = 1132,
1271279025 = 356552,
1271277812730289 = 356549832.

The square root of 127 is 11.269..., 112 = 22+62+92.

Komachi equations:
1272 = - 9 + 8 * 7 / 6 * 54 * 32 + 10 = - 9 + 8 + 76 * 5 * 43 - 210,
1272 = - 92 - 82 - 72 - 62 - 52 + 42 * 322 */ 12,
1272 = 93 - 83 * 73 - 63 + 543 + 323 + 103.

(22 - 1)(52 - 1)(152 - 1) = (1272 - 1),
(112 - 5)(122 - 5) = (1272 - 5).

A 3-by-3 magic square consisting of different squares with constant 1272:

1825321142
6621022372
1072542422

(1 + 2)(3 + 4 + ... + 122)(123 + 124 + ... + 127) = 37502,
(1 + 2 + 3 + 4)(5 + 6 + ... + 19)(20 + 21 + ... + 127) = 37802,
(1 + 2 + ... + 14)(15 + 16 + ... + 110)(111 + 112 + ... + 127) = 357002,
(1 + 2 + ... + 24)(25)(26 + 27 + ... + 127) = 76502,
(1 + 2 + ... + 24)(25 + 26 + ... + 32)(33 + 34 + ... + 127) = 228002,
(1 + 2 + ... + 24)(25 + 26 + ... + 78)(79 + 80 + ... + 127) = 648902,
(1 + 2 + ... + 24)(25 + 26 + ... + 122)(123 + 124 + ... + 127) = 367502,
(1 + 2 + ... + 54)(55 + 56 + ... + 110)(111 + 112 + ... + 127) = 1178102,
(1 + 2 + ... + 69)(70 + 71 + ... + 104)(105 + 106 + ... + 127) = 1400702,
(1 + 2 + ... + 86)(87)(88 + 89 + ... + 127) = 374102.

(13 + 23 + ... + 173)(183 + 193 + ... + 1223)(1233 + 1243 + ... + 1273) = 35873145002.

1272 = 16129 appears in the decimal expressions of π and e:
  π = 3.14159•••16129••• (from the 34763rd digit),
  e = 2.71828•••16129••• (from the 45749th digit)

Page of Squares : First Upload May 10, 2004 ; Last Revised April 16, 2010
by Yoshio Mimura, Kobe, Japan

128

The smallest squares containing k 128's :
71289 = 2672,
811281289 = 284832,
128012886661284 = 113142782.

1282 = 16384, 1 + 6 + 38 + 4 = 72,
1282 = 16384, 16 + 384 = 202.

1282 = 46 + 46 + 46 + 46 = 84 + 84 + 84 + 84.

128 is the second integer which is written as the sum of a square and a prime in 5 ways:
    12 + 127, 52 + 103, 72 + 79, 92 + 47, 112 + 7.

(42 - 4)(372 - 4) = (52 - 4)(282 - 4) = (112 - 4)(122 - 4) = (1282 - 4).

1282 = (12 + 7)(32 + 7)(112 + 7).

1282 = 16384, a zigzag square with different digits.

1282 = 16384, 38416 = 1962.

1282 is the fifth square which is the sum of 4 sixth powers and the second square which is the sum of 4 twelfth powers.

Komachi equations:
1282 = - 92 * 82 * 72 + 652 / 42 * 322 */ 12,
1282 = 984 / 74 - 64 - 54 * 44 * 34 * 24 / 104.

1282 + 1292 + 1302 + ... + 33062 = 1097692.

(1 + 2 + 3 + 4)(5 + 6 + ... + 88)(89 + 90 + ... + 128) = 130202,
(1 + 2 + ... + 8)(9 + 10 + ... + 116)(117 + 118 + ... + 128) = 189002,
(1 + 2 + ... + 14)(15 + 16 + ... + 23)(24 + 25 + ... + 128) = 119702,
(1 + 2 + ... + 14)(15 + 16 + ... + 95)(96 + 97 + ... + 128) = 415802,
(1 + 2 + ... + 18)(19 + 20 + ... + 23)(24 + 25 + ... + 128) = 119702,
(1 + 2 + ... + 18)(19 + 20 + ... + 95)(96 + 97 + ... + 128) = 526682,
(1 + 2 + ... + 18)(19 + 20 + ... + 114)(115 + 116 + ... + 128) = 430922,
(1 + 2 + ... + 19)(20 + 21 + ... + 56)(57 + 58 + ... + 128) = 421802,
(1 + 2 + ... + 23)(24 + 25 + ... + 32)(33 + 34 + ... + 128) = 231842,
(1 + 2 + ... + 32)(33 + 34 + ... + 47)(48 + 49 + ... + 128) = 475202,
(1 + 2 + ... + 32)(33 + 34 + ... + 95)(96 + 97 + ... + 128) = 887042,
(1 + 2 + ... + 36)(37 + 38 + ... + 91)(92 + 93 + ... + 128) = 976802,
(1 + 2 + ... + 40)(41 + 42 + ... + 76)(77 + 78 + ... + 128) = 959402,
(1 + 2 + ... + 69)(70 + 71 + ... + 114)(115 + 116 + ... + 128) = 1304102,
(1 + 2 + ... + 81)(82 + 83 + ... + 87)(88 + 89 + ... + 128) = 863462.

1282 = 16384 appears in the decimal expressions of π and e:
  π = 3.14159•••16384••• (from the 37619th digit),
  e = 2.71828•••16384••• (from the 9318th digit)

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

129

The smallest squares containing k 129's :
1296 = 362,
112933129 = 106272,
12932129400129 = 35961272.

The squares which begin with 129 and end in 129 are
129345129 = 113732,   129436129 = 113772,   12910186129 = 1136232,
12911095129 = 1136272,   12967060129 = 1138732,...

1292 = 16641 with 3 kinds of digits.

1292 = 16641, 16 + 64 + 1 = 92.

1292 = 16641, (16, 64 and 1 are squares).

1292 is the second square which is the sum of 4 seventh powers : 17 + 27 + 27 + 47.
1292 is the third square which is the sum of 6 fifth powers : 15 + 25 + 25 + 45 + 65 + 65,
1292 is the 7th square which is the sum of 9 sixth powers.

(12 + 22 + 32 + ... + 1292) = 723905, which consists of different digits.

6k + 12k + 36k + 129k + 258k are squares (212, 2912, 44012, 686252) for k = 1,2,3,4.

1292 = 16641, 11664 = 1082.

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

42k + 129k + 660k + 1194k are squares for k = 1,2,3 (452, 13712, 446312).
18k + 84k + 129k + 210k are squares for k = 1,2,3 (212, 2612, 34652).
344k + 1505k + 4558k + 10234k are squares for k = 1,2,3 (1292, 113092, 10816652).
1505k + 4042k + 4988k + 6106k are squares for k = 1,2,3 (1292, 89872, 6489992).

1292 = 40 + 44 + 47.

Komachi equations:
1292 = 982 / 72 + 62 + 52 + 42 * 322 */ 12 = 982 / 72 + 62 + 52 + 42 * 322 / 12
1292 = 92 + 82 - 72 + 62 + 52 + 42 * 322 + 102.

Two 3-by-3 magic squares consisting of different squares with constant 1192:

 725621162
762922492
1042712282
      
1627621032
892682642
922792442

(112- 3)(122 - 3) = (1292 - 3),
(62 + 9)(192 + 9) = (1292 + 9),
(62 - 9)(252 - 9) = (1292 - 9).

1292 + 1302 + 1312 + ... + 45222 = 1755912.

(1 + 2 + ... + 24)(25 + 26 + ... + 30)(31 + 32 + ... + 129) = 198002,
(1 + 2 + ... + 27)(28 + 29 + ... + 126)(127 + 128 + 129) = 332642,
(1 + 2 + ... + 44)(45 + 46 + ... + 55)(56 + 57 + ... + 129) = 610502,
(1 + 2 + ... + 121)(122)(123 + 124 + ... + 129) = 281822,
(1 + 2 + ... + 128)(129) = 10322.

1292 = 16641 appears in the decimal expressions of π and e:
  π = 3.14159•••16641••• (from the 4742nd digit),
  (16641 is the eighth 5-digit square in the expression of π.)
  e = 2.71828•••16641••• (from the 44037th digit)

Page of Squares : First Upload May 10, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan