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, 2008by Yoshio Mimura, Kobe, Japan
3101
31012 = 9616201, a reversible square (1026169 = 10132).
Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007by 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, 2007by 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, 2008by 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, 2007by 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, 2013by 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.
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.
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, 2007by 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.
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, 2011by 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, 2007by Yoshio Mimura, Kobe, Japan
3119
31192 = 15 + 85 + 95 + 205 + 235.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2013by Yoshio Mimura, Kobe, Japan
3121
1 / 3121 = 0.00032041, and 32041 = 1792.
Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007by 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, 2013by 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, 2007by 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, 2013by 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, 2007by Yoshio Mimura, Kobe, Japan
3126
31262 = 15 + 55 + 55 + 255.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
3129
31292 = 983 + 1213 + 1923.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by Yoshio Mimura, Kobe, Japan
3132
31322 = 363 + 1263 + 1983.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2007by Yoshio Mimura, Kobe, Japan
3135
31352 = 573 + 1183 + 2003.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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.
by Yoshio Mimura, Kobe, Japan
3137
31372 = 34 + 64 + 344 + 544.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2011by 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, 2007by 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, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
3144
31442 = 523 + 603 + 2123 = 963 + 1003 + 2003.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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)
by Yoshio Mimura, Kobe, Japan
3146
31462± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by 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, 2007by 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, 2007by Yoshio Mimura, Kobe, Japan
3155
31552 = 263 + 1223 + 2013 = 483 + 1523 + 1853.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by Yoshio Mimura, Kobe, Japan
3156
31562 = 1303 + 1523 + 1623.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2008by 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, 2007by 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, 2007by 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, 2014by Yoshio Mimura, Kobe, Japan
3163
31632 = 1293 + 1343 + 1763.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by Yoshio Mimura, Kobe, Japan
3165
31652 = 93 + 1453 + 1913.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by 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, 2007by 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, 2013by 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, 2008by Yoshio Mimura, Kobe, Japan
3172
317231721 = 178112.
Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007by 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, 2013by 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, 2008by Yoshio Mimura, Kobe, Japan
3179
31792 = 10106041, a zigzag square.
Page of Squares : First Upload June 4, 2007 ; Last Revised June 4, 2007by Yoshio Mimura, Kobe, Japan
3180
31802 = 173 + 313 + 2163.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by Yoshio Mimura, Kobe, Japan
3181
A cubic polynomial :
(X + 5312)(X + 8322)(X + 30242) = X3 + 31812X2 + 30172322X + 13359790082.
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, 2008by 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.)
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, 2011by 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, 2008by 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, 2007by 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, 2013by 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, 2013by 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, 2008by 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, 2008by 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, 2007by 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.
by Yoshio Mimura, Kobe, Japan
3198
31982 = 114 + 134 + 434 + 514.
Page of Squares : First Upload August 4, 2008 ; Last Revised August 4, 2008by Yoshio Mimura, Kobe, Japan