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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).

12586^k + 19530^k + 29698^k + 34286^k are squares for k = 1,2,3 (310², 50964², 8714348²).
13485^k + 15345^k + 22165^k + 45105^k 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).

Page of Squares : First Upload November 29, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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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:

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).

Page of Squares : First Upload November 29, 2004 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

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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).

Page of Squares : First Upload November 29, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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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.

240^k + 306^k + 313^k + 366^k are squares for k = 1,2,3 (35², 619², 11053²).

3-by-3 magic squares consisting of different squares with constant 313²:

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).

Page of Squares : First Upload November 29, 2004 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

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314

The smallest squares containing k 314's :
314721 = 561²,
53141314576 = 230524²,
17231431431491209 = 131268547².

314² = 15³ + 16³ + 45³.

190^k + 314^k + 410^k + 530^k 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).

Page of Squares : First Upload November 29, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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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:

(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).

Page of Squares : First Upload November 29, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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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.)

Page of Squares : First Upload November 29, 2004 ; Last Revised May 28, 2010
by Yoshio Mimura, Kobe, Japan

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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:

317² = 100489, 100 + 4 + 8 + 9 = 11².

Page of Squares : First Upload November 29, 2004 ; Last Revised January 27, 2009
by Yoshio Mimura, Kobe, Japan

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318

The smallest squares containing k 318's :
183184 = 428²,
5318493184 = 72928²,
31831893184576 = 5641976².

318² = 101124, 1 + 0 + 1 + 1 + 2 + 4 = 3².

16218^k + 17490^k + 23214^k + 44202^k are squares for k = 1,2,3 (318², 55332², 10415772²).
14522^k + 19186^k + 22366^k + 45050^k 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.

Page of Squares : First Upload November 29, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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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:

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).

Page of Squares : First Upload November 29, 2004 ; Last Revised January 27, 2009
by Yoshio Mimura, Kobe, Japan

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