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3100 - 3199

3100

31002 = 303 + 383 + 2123.

31002 = 264 x 265 + 266 x 267 + 268 x 269 + 270 x 271 + ... + 422 x 423.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3101

31012 = 9616201, a reversible square (1026169 = 10132).

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3102

31022 = 11 x 12 + 13 x 14 + 15 x 16 + 17 x 18 + ... + 385 x 386.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3105

31052 = 9641025, a square with different digits.

31053 = 29935382625, and 292 + 92 + 32 + 52 + 382 + 262 + 22 + 52 = 3105.

Page of Squares : First Upload June 4, 2007 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

3106

1 / 3106 = 0.000321957501, 32 + 212 + 92 + 52 + 72 + 502 + 12 = 3106.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3108

31082 = (22 + 3)(32 + 3)(162 + 3)(212 + 3) = (92 + 3)(162 + 3)(212 + 3).

3108k + 10212k + 39072k + 58497k are squares for k = 1,2,3 (3332, 711512, 161528312).

Page of Squares : First Upload May 27, 2011 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3111

31112 = 9678321, a square with different digits.

31112 = 9678321, a reversible square (1238769 = 11132).

1 / 3111 = 0.00032144005143, 322 + 142 + 42 + 02 + 02 + 52 + 12 + 432 = 3111,
1 / 3111 = 0.00032144005143, 322 + 142 + 42 + 02 + 052 + 12 + 432 = 3111,
1 / 3111 = 0.00032144005143, 322 + 142 + 42 + 002 + 52 + 12 + 432 = 3111,
1 / 3111 = 0.00032144005143, 322 + 142 + 42 + 0052 + 12 + 432 = 3111.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3113

31132 = 9690769, a zigzag square.

1 / 3113 = 0.0003212335367, 32 + 22 + 12 + 232 + 352 + 362 + 72 = 3113.

Komachi equations:
31132 = 94 - 84 - 74 * 64 + 54 * 44 * 34 - 24 * 104 = 94 - 84 - 74 * 64 - 54 * 44 + 34 * 24 * 104.

Page of Squares : First Upload June 4, 2007 ; Last Revised September 28, 2010
by Yoshio Mimura, Kobe, Japan

3114

31142 = 9696996, a square with just 2 kinds of digits.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3115

31152 = 303 + 1293 + 1963.

Komachi equations:
31152 = 12 * 22 / 32 / 42 * 52 * 62 * 72 * 892 = 12 / 22 * 32 * 42 * 52 / 62 * 72 * 892
 = 12 * 22 / 32 * 452 / 62 * 72 * 892 = 12 * 23452 / 672 * 892.

Page of Squares : First Upload August 4, 2008 ; Last Revised September 28, 2010
by Yoshio Mimura, Kobe, Japan

3116

55 + 3116 = 792, 55 - 3116 = 32.

Komachi equation: 31162 = 14 * 234 * 44 + 54 * 64 + 74 - 894.

Page of Squares : First Upload September 28, 2010 ; Last Revised July 27, 2011
by Yoshio Mimura, Kobe, Japan

3117

31172 = 9715689, 9 * 71 * 5 - 6 - 8 * 9 = 3117.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3119

31192 = 15 + 85 + 95 + 205 + 235.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3120

31202 = (22 - 1)(142 - 1)(1292 - 1) = (32 - 1)(142 - 1)(792 - 1) = (92 - 1)(142 - 1)(252 - 1).

Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3121

1 / 3121 = 0.00032041, and 32041 = 1792.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3122

31222 = (12 + 3)(22 + 3)(5902 + 3) = (52 + 3)(5902 + 3).

31222 = 174 + 414 + 434 + 434.

Page of Squares : First Upload August 4, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3123

31232 = 9753129, 9 * 7 * 5 + 312 * 9 = 3123.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3124

31242 = 9759376, a square with odd digits except the last digit 6.

31242 = 823 + 1243 + 1943.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

3125

31252 = 9765625, and 9 = 32, 765625 = 8752.

31252 = 9765625, 92 + 72 + 62 + 52 + 62 + 22 + 52 = 162.

31252 = (63 + 64 + 65 + ... + 87)2 + (88 + 89 + 90 + ... + 112)2.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3126

31262 = 15 + 55 + 55 + 255.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3128

31282 = 573 + 953 + 2063.

31282 = 202 + 212 + 222 + 232 + 242 + 252 + 262 + ... + 3082.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3129

31292 = 983 + 1213 + 1923.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3132

31322 = 363 + 1263 + 1983.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3133

31332 = 9815689, 9 * 8 + 1 + 5 * 68 * 9 = 3133.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3135

31352 = 573 + 1183 + 2003.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3136

The square of 56.

31362 = 9834496, 98 * 3 * 4 * 4 / 9 * 6 = 98 / 3 * 4 / 4 * 96 = 98 / 3 / 4 * 4 * 96 = 3136.

Komachi equations:
31362 = 122 * 32 / 42 + 562 * 72 * 82 - 92 = 122 * 32 / 42 * 562 * 72 * 82 / 92
 = - 122 * 32 / 42 + 562 * 72 * 82 + 92.

Page of Squares : First Upload June 4, 2007 ; Last Revised September 28, 2010
by Yoshio Mimura, Kobe, Japan

3137

31372 = 34 + 64 + 344 + 544.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3138

31385 = 304273896638043168 : 302 + 422 + 72 + 32 + 82 + 92 + 62 + 62 + 32 + 82 + 02 + 42 + 32 + 12 + 62 + 82 = 3138.

498k + 942k + 3138k + 3522k are squares for k = 1,2,3 (902, 48362, 2748602).

Page of Squares : First Upload December 8, 2008 ; Last Revised May 27, 2011
by Yoshio Mimura, Kobe, Japan

3139

31392 = 9853321, a square with non-increasing digits.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3142

31422 = 9872164, a square with different digits.

31422 = 313 + 1303 + 1973.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3143

31432 = 24 + 314 + 464 + 464 = 44 + 64 + 424 + 514.

The square root of 3143 is 56.0624..., and 56 = 02 + 62 + 22 + 42.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3144

31442 = 523 + 603 + 2123 = 963 + 1003 + 2003.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3145

31452 = 9891025, 92 + 82 + 92 + 12 + 02 + 22 + 52 = 162.

31452 = 9891025 appears in the decimal expressions of π:
  π = 3.14159•••9891025••• (from the 41308th digit)

Page of Squares : First Upload June 4, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

3146

31462± 3 are primes.

Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

3148

31482 = 9909904, a square with just 3 kinds of digits.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3154

31542 = 9947716, 9 + 9 + 4 * 7 * 7 * 16 = 3154.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3155

31552 = 263 + 1223 + 2013 = 483 + 1523 + 1853.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3156

31562 = 1303 + 1523 + 1623.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3157

31572 = 9966649, a square with just 3 kinds of digits.

31572 = 45 + 105 + 105 + 255.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3159

31592 = 9979281, 92 + 92 + 72 + 92 + 22 + 82 + 12 = 192.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3161

31612 = 9991921, a square with just 3 kinds of digits.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3162

31622± 5 are primes.

1 / 3162 = 0.0003162....

Page of Squares : First Upload June 4, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

3163

31632 = 1293 + 1343 + 1763.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3165

31652 = 93 + 1453 + 1913.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3166

31662 = 10023556, 12 + 02 + 02 + 22 + 32 + 52 + 52 + 62 = 102.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3168

31682 = 10036224, a reversible square (42263001 = 65012).

31682 = 10036224, and 1 * 0036 * 22 * 4 = 3168.

31682 = 10036224, and
1 * 0 + 0 + 36 * 22 * 4 = 1 * 0 * 0 + 36 * 22 * 4 = 10 * 0 + 36 * 22 * 4 = 3168.

31682 = (22 + 8)(42 + 8)(62 + 8)(282 + 8) = (22 + 8)(62 + 8)(82 + 8)(162 + 8)
= (22 - 1)(32 - 1)(102 - 1)(652 - 1) = (32 - 1)(52 - 1)(102 - 1)(232 - 1) = (52 - 1)(102 - 1)(652 - 1).

31682 = 923 + 1203 + 1963.

Page of Squares : First Upload June 4, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3171

31712 = 733 + 743 + 2103.

The square root of 3171 is 56.31163...., and 56 = 32 + 12 + 12 + 62 + 32.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3172

317231721 = 178112.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3174

3174, 3175, 3176 and 3177 are four consecutive integers having square factors (the second case).

Page of Squares : First Upload November 10, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

3177

31772 = 23 + 253 + 2163 = 274 + 344 + 444 + 464.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3179

31792 = 10106041, a zigzag square.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3180

31802 = 173 + 313 + 2163.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3181

A cubic polynomial :
(X + 5312)(X + 8322)(X + 30242) = X3 + 31812X2 + 30172322X + 13359790082.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3182

31825 = 326212708800926432 : 322 + 62 + 22 + 122 + 72 + 02 + 82 + 82 + 02 + 02 + 92 + 262 + 42 + 322 = 3182.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3183

31832 = 10131489 appears in the decimal expressions of e:
  e = 2.71828•••10131489••• (from the 18067th digit)
  (10131489 is the first 8-digit square in the expression of e.)

Page of Squares : First Upload November 4, 2008 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

3184

31842 = 25 + 145 + 205 + 205 + 205.

The square root of 3184 is 56.426...., and 56 = 42 + 22 + 62.

31842 = 1973 + 195 + 47.

31844 = 102776124276736, and 102 + 272 + 72 + 62 + 122 + 42 + 272 + 62 + 72 + 362 = 3184.

Page of Squares : First Upload June 4, 2007 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

3186

31862 = 393 + 1173 + 2043.

31862 = 10150596, 12 + 02 + 12 + 52 + 02 + 52 + 92 + 62 = 132.

Page of Squares : First Upload June 4, 2007 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan

3188

1 / 3188 = 0.0003136...., and 3136 = 562.

The square root of 3188 is 56.462...., and 56 = 42 + 62 + 22.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3190

31902 = (22 + 6)(72 + 6)(1362 + 6).

31902 = (68 + 69 + 70 + 71)2 + (72 + 73 + 74 + 75)2 + (76 + 77 + 78 + 79)2 + ... + (196 + 197 + 198 + 199)2.

31904 = 103553011210000, and 12 + 02 + 32 + 552 + 32 + 02 + 12 + 122 + 12 + 02 + 02 + 02 + 02 = 3190.

3190k + 4524k + 13137k + 20358k are squares for k = 1,2,3 (2032, 248532, 32908332).

Page of Squares : First Upload June 4, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3192

31922 = (12 + 3)(22 + 3)(42 + 3)(92 + 3)(152 + 3) = (22 + 3)(32 + 3)(152 + 3)(232 + 3)
= (22 + 3)(32 + 3)(42 + 3)(52 + 3)(152 + 3) = (32 + 3)(152 + 3)(612 + 3)
= (32 + 3)(42 + 3)(92 + 3)(232 + 3) = (42 + 3)(52 + 3)(92 + 3)(152 + 3) = (92 + 3)(152 + 3)(232 + 3).

31922 = 423 + 683 + 2143 = 125 + 145 + 165 + 205 + 225.

Page of Squares : First Upload August 4, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

3193

3193 = (12 + 22 + 32 + ... + 1542) / (12 + 22 + 32 + ... + 102).

31934 = 103943102172001, and 12 + 02 + 392 + 42 + 312 + 02 + 22 + 172 + 202 + 02 + 12 = 3193.

Page of Squares : First Upload November 25, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

3194

31945 = 332410366382328224 : 32 + 32 + 22 + 412 + 02 + 32 + 62 + 62 + 32 + 82 + 232 + 282 + 22 + 22 + 42 = 3194.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

3195

31952 = 10208025, 1 * 0 + 20 * 80 * 2 - 5 = 3195.

Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007
by Yoshio Mimura, Kobe, Japan

3196

31964 = 104334294221056,
and 12 + 02 + 42 + 32 + 342 + 22 + 92 + 422 + 22 + 102 + 52 + 62 = 3196,
31964 = 104334294221056,
and 102 + 42 + 32 + 342 + 22 + 92 + 422 + 22 + 12 + 02 + 52 + 62 = 3196.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

3198

31982 = 114 + 134 + 434 + 514.

Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008
by Yoshio Mimura, Kobe, Japan