310
The smallest squares containing k 310's :
23104 = 1522,
10993103104 = 1048482,
31093103102161 = 55761192.
3102 = 96100 is an exchangeable square (61009 = 2472).
3102 = (22 + 6)(982 + 6).
12586k + 19530k + 29698k + 34286k are squares for k = 1,2,3 (3102, 509642, 87143482).
13485k + 15345k + 22165k + 45105k are squares for k = 1,2,3 (3102, 542502, 104268502).
Komachi Square Sum : 3102 = 42 + 52 + 672 + 832 + 2912 = 52 + 72 + 432 + 812 + 2962.
3102 = 96100 appears in the decimal expression of e:
e = 2.71828•••96100••• (from the 32516th digit).
by Yoshio Mimura, Kobe, Japan
311
The smallest squares containing k 311's :
311364 = 5582,
3311311936 = 575442,
83115331131121 = 91167612.
3112 = 96721, a square with different digits.
3112 = 96721, a reversible square (12769 = 1132).
3112 = 96721, 9 + 6 + 7 + 2 + 1 = 52,
3112 = 96721, 96 + 72 + 1 = 132.
311 is the first prime for which the Legendre Symbol (a/311) = 1 for a = 1, 2,..., 10.
3112 is the third square(> 1) that can not be a sum of a power of 2 and a prime.
3-by-3 magic squares consisting of different squares with constant 3112:
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25962 = 2162 + 2172 + 2182 + 2192 + 2202 + ... + 3112.
3112 + 3122 + 3132 + 3142 + 3152 + ... + 103922 = 6116652.
3112 = 96721 appears in the decimal expressions of π and e:
π = 3.14159•••96721••• (from the 17360th digit),
e = 2.71828•••96721••• (from the 138254th digit).
by Yoshio Mimura, Kobe, Japan
312
The smallest squares containing k 312's :
33124 = 1822,
312653124 = 176822,
31231209303121 = 55884892.
3122 = 97344, 97 + 3 + 44 = 122,
3122 = 97344, 97 + 344 = 212.
3122 = 23 + 463.
3122 = (12 + 3)(32 + 3)(452 + 3) = (12 + 3)(32 + 3)(62 + 3)(72 + 3).
3122 = 143 + 153 + 163 + 173 + ... + 253.
25302 = 2252 + 2262 + 2272 + 2282 + 2292 + 2302 + ... + 3122.
3122 + 3132 + 3142 + 3152 + 3162 + ... + 147362 = 10328302,
3122 + 3132 + 3142 + 3152 + 3162 + ... + 245272 = 22177822.
(1 + 2 + ... + 12)(13 + 14 + ... + 51) = 3122.
(13 + 23 + ... + 63)(73 + 83 + ... + 123)(133 + 143 + ... + 3123) = 770269502.
3122 = 97344 appears in the decimal expressions of π and e:
π = 3.14159•••97344••• (from the 101596th digit),
e = 2.71828•••97344••• (from the 75271st digit).
by Yoshio Mimura, Kobe, Japan
313
The smallest squares containing k 313's :
3136 = 562,
9131331364 = 955582,
31331379333136 = 55974442.
3132 = 97969, a square every digit of which is greater than 5.
240k + 306k + 313k + 366k are squares for k = 1,2,3 (352, 6192, 110532).
3-by-3 magic squares consisting of different squares with constant 3132:
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3132 = 97969, a zigzag square with 3 kinds of digits.
3132 = 97969, 9 + 7 + 96 + 9 = 112,
3132 = 97969, 97 + 9 + 6 + 9 = 112.
3133 - 3123 + 3113 - 3103 + 3093 - .. + 13 = 39252.
(13 + 23 + ... + 2153)(2163 + 2173 + ... + 3133) = 10056349802.
3132 = 97969 appears in the decimal expression of e:
e = 2.71828•••97969••• (from the 35279th digit).
by Yoshio Mimura, Kobe, Japan
314
The smallest squares containing k 314's :
314721 = 5612,
53141314576 = 2305242,
17231431431491209 = 1312685472.
3142 = 153 + 163 + 453.
190k + 314k + 410k + 530k are squares for k = 1,2,3 (382, 7642, 159882).
Komachi equation: 3142 = 92 + 82 + 762 * 52 - 432 - 2102.
3142 = 98596 appears in the decimal expression of e:
e = 2.71828•••98596••• (from the 11855th digit).
by Yoshio Mimura, Kobe, Japan
315
The smallest squares containing k 315's :
315844 = 5622,
3153159409 = 561532,
207315823153156 = 143984662.
3152 = 99225, a square with 3 kinds of digits.
3152 = (22 - 1)(42 - 1)(62 - 1)(82 - 1).
3152 = 99225 with 9 = 32 and 225 = 152.
Komachi Fraction : 4761 / 893025 = (23 / 315)2.
3152 = 99225, 99 + 225 = 182.
3152 = 253 + 263 + 273 + 283 + 293.
3-by-3 magic squares consisting of different squares with constant 3152:
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(1 + 2 + ... + 6)(7 + 8 + ... + 11)(12 + 13 + ... + 18) = 3152,
(1 + 2 + ... + 9)(10 + 11)(12 + 13 + ... + 18) = 3152,
(1 + 2 + ... + 5)(6 + 7 + 8)(9 + 10 + ... + 26) = 3152,
(1 + 2 + ... + 6)(7 + 8)(9 + 10 + ... + 26) = 3152,
(1 + 2 + ... + 6)(7)(8 + 9 + ... + 37) = 3152.
3152 = 99225 appears in the decimal expression of e:
e = 2.71828•••99225••• (from the 103673rd digit).
by Yoshio Mimura, Kobe, Japan
316
The smallest squares containing k 316's :
21316 = 1462,
2316593161 = 481312,
83167943165316 = 91196462.
The squares which begin with 316 and end in 316 are
31632045316 = 1778542, 316008125316 = 5621462, 316242021316 = 5623542,
316570521316 = 5626462, 316804625316 = 5628542,...
1 / 316 = 0.00316...
Cubic Polynomial : (X + 2882)(X + 3162)(X + 3572) = X3 + 5572X2 + 1777082X + 32489856.
Komachi equation: 3162 = - 92 * 82 - 72 + 62 * 542 + 32 + 22 + 102.
3162 + 3172 + 3182 + 3192 + 3202 + ... + 21642 = 580502.
3162 = (12 + 22 + ... + 312) + (12 + 22 + ... + 642).
3162 = 99856 appears in the decimal expression of e:
e = 2.71828•••99856••• (from the 3227th digit),
(99856 is the tenth 5-digit square in the expression of e.)
by Yoshio Mimura, Kobe, Japan
317
The smallest squares containing k 317's :
131769 = 3632,
317231721 = 178112,
1463179317831729 = 382515272.
3172 is the 10th square which is the sum of 6 fifth powers (1, 1, 1, 3, 3, 10).
(161 / 317)2 = 0.257948631... (Komachic).
3-by-3 magic squares consisting of different squares with constant 3172:
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3172 = 100489, 100 + 4 + 8 + 9 = 112.
Page of Squares : First Upload November 29, 2004 ; Last Revised January 27, 2009by Yoshio Mimura, Kobe, Japan
318
The smallest squares containing k 318's :
183184 = 4282,
5318493184 = 729282,
31831893184576 = 56419762.
3182 = 101124, 1 + 0 + 1 + 1 + 2 + 4 = 32.
16218k + 17490k + 23214k + 44202k are squares for k = 1,2,3 (3182, 553322, 104157722).
14522k + 19186k + 22366k + 45050k are squares for k = 1,2,3 (3182, 557562, 106180202).
3182 = 101124, 1 + 0 + 11 + 24 = 62,
3182 = 101124, 10 + 1 + 1 + 24 = 62,
3182 = 101124, 101 + 124 = 152.
(13 + 23 + ... + 1433)(1443 + 1453 + ... + 3183) = 5113508402,
(13 + 23 + ... + 2313)(2323 + 2333 + ... + 3183) = 11539697402,
(13 + 23 + ... + 2643)(2653 + 2663 + ... + 3183) = 12847804202.
by Yoshio Mimura, Kobe, Japan
319
The smallest squares containing k 319's :
319225 = 5652,
331931961 = 182192,
319031931946225 = 178614652.
3-by-3 magic squares consisting of different squares with constant 3192:
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3192 = 101761, 1 + 0 + 1 + 7 + 6 + 1 = 42,
3192 = 101761, 1 + 0 + 17 + 6 + 1 = 52,
3192 = 101761, 10 + 1 + 7 + 6 + 1 = 52,
3192 = 101761, 101 + 7 + 61 = 132,
3192 = 101761, 1017 + 6 + 1 = 322.
3192 + 3202 + 3212 + 3222 + 3232 + ... + 615682 = 88201752.
3192 = 101761 appears in the decimal expression of π:
π = 3.14159•••101761••• (from the 49435th digit).
by Yoshio Mimura, Kobe, Japan