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390

The smallest squares containing k 390's :
123904 = 352²,
390339049 = 19757²,
1390383906773904 = 37287852².

390² = (2² + 9)(9² + 9)(11² + 9) = (9² + 9)(41² + 9).

390^k + 2262^k + 4329^k + 6708^k are squares for k = 1,2,3 (117², 8307², 628173²).
6890^k + 43810^k + 46670^k + 54730^k are squares for k = 1,2,3 (390², 84500², 18708300²).

Komachi equations:
390² = 1² * 2² * 3² / 4² * 5² * 6² * 78² / 9² = 1² / 2² / 3² * 4² * 5² / 6² * 78² * 9²
= 1² * 2² * 3² * 45² / 6² * 78² / 9² = 1² / 2² / 3² * 45² * 6² * 78² / 9².

390² + 391² + 392² + 393² + ... + 45126² = 5534606²,
390² + 391² + 392² + 393² + ... + 96101² = 17200244².

(1 + 2 + 3)(4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14 + 15) = 390²,
(1)(2)(3)(4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14 + 15) = 390².

(1 + 2 + 3)(4 + 5 + 6)(7 + 8 + ... + 58) = 390²,
(1 + 2 + 3 + 4 + 5)(6)(7 + 8 + ... + 58) = 390²,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + ... + 123) = 390².

Page of Squares : First Upload January 24, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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391

The smallest squares containing k 391's :
139129 = 373²,
35391391876 = 188126²,
339175391391225 = 18416715².

184¹ + 345¹ = 23², 184² + 345² = 391², 184³ + 345³ = 6877²  (See 23).

Komachi Square Sum : 391² = 2² + 8² + 51² + 94² + 376².

3-by-3 magic squares consisting of different squares with constant 391²:

391² = 152881, 1 + 5 + 2 + 8 + 8 + 1 = 5²,
391² = 152881, 152 + 8 + 8 + 1 = 13².

391² = 152881 appears in the decimal expression of π:
  π = 3.14159•••152881••• (from the 8204th digit),
  (152881 is the third 6-digit square in the expression of π.)

Page of Squares : First Upload January 24, 2005 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

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392

The smallest squares containing k 392's :
13924 = 118²,
83923921 = 9161²,
39205392939241 = 6261421².

392² = 153664, 1 * 5 + 3 + 6 * 64 = 392.

392² = (1² + 3)(5² + 3)(37² + 3).

Komachi equation: 392² = 9² * 8² * 7² * 6² / 54² / 3² * 21².

392² = 14⁴ + 14⁴ + 14⁴ + 14⁴.

392² = 153664, 1 + 5 + 3 + 6 + 6 + 4 = 5²,
392² = 153664, 153 + 6 + 6 + 4 = 13²,
392² = 153664, 1536 + 64 = 40².

Page of Squares : First Upload November 28, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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393

The smallest squares containing k 393's :
393129 = 627²,
39384783936 = 198456²,
29393139343936 = 5421544².

2882^k + 26986^k + 48601^k + 75980^k are squares for k = 1,2,3 (393², 94189², 23939595²).

3-by-3 magic squares consisting of different squares with constant 393²:

393² = 154449, 15 + 4 + 4 + 4 + 9 = 6².

393² = 13³ + 15³ + 53³ = 24³ + 25³ + 50³.

(1³ + 2³ + ... + 87³)(88³ + 89³ + ... + 131³)(132³ + 133³ + ... + 393³) = 2283188818680².

Page of Squares : First Upload January 24, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

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394

The smallest squares containing k 394's :
394384 = 628²,
12239439424 = 110632²,
1394363947839489 = 37341183².

254^k + 362^k + 394^k + 590^k are squares for k = 1,2,3 (40², 836², 18176²).

Komachi Square Sum : 394² = 29² + 41² + 67² + 385².

394² = 155236, 1² + 5² + 5² + 2² + 3² + 6² = 10²,
394² = 155236, 1 + 5 + 5 + 2 + 36 = 7²,
394² = 155236, 15 + 5 + 23 + 6 = 7²,
394² = 155236, 15 + 5 + 236 = 16²,
394² = 155236, 15² + 52² + 36² = 65².

394² = 155236 appears in the decimal expression of π:
  π = 3.14159•••155236••• (from the 84971st digit).

Page of Squares : First Upload January 24, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

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395

The smallest squares containing k 395's :
133956 = 366²,
3952133956 = 62866²,
165139539539569 = 12850663².

3-by-3 magic squares consisting of different squares with constant 395²:

395² + 396² + 397² + ... + 972² = 16915²,
395² + 396² + 397² + ... + 2603² = 76563²,
395² + 396² + 397² + ... + 16236² = 1194469².

Page of Squares : First Upload January 24, 2005 ; Last Revised February 23, 2009
by Yoshio Mimura, Kobe, Japan

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396

The smallest squares containing k 396's :
3969 = 63²,
39639616 = 6296²,
439639662667396 = 20967586².

The squares which begin with 396 and end in 396 are
396567396 = 19914²,   39635235396 = 199086²,   396161983396 = 629414²,
396378531396 = 629586²,   396791647396 = 629914²,...

396² = (1² + 8)(2² + 8)(38² + 8) = (1² + 8)(2² + 8)(5² + 8)(6² + 8) = (10² + 8)(38² + 8)
= (2² + 8)(6² + 8)(17² + 8) = (2² - 1)(10² - 1)(23² - 1) = (5² + 8)(6² + 8)(10² + 8).

396² = 132² + 264² + 264² : 462² + 462² + 231² = 693².

396² = 23³ + 30³ + 49³.

Komachi equations:
396² = 12² * 34² + 5² - 67² - 8² * 9²,
396² = - 9³ - 8³ + 7³ + 6³ + 54³ + 3³ + 2³ - 1³.

Cubic Polynomial : (X + 112²)(X + 147²)(X + 396²) = X³ + 437²X² + 75012²X + 6519744².

396² = 156816, 1 + 5 + 6 + 8 + 16 = 6²,
396² = 156816, 1 + 5 + 68 + 1 + 6 = 9²,
396² = 156816, 1 + 56 + 8 + 16 = 9²,
396² = 156816, 1 + 56 + 81 + 6 = 12²,
396² = 156816, 1 + 568 + 1 + 6 = 24²,
396² = 156816, 15 + 6 + 8 + 1 + 6 = 6².

(1 + 2 + ... + 11)(12)(13 + 14 + ... + 23) = 396²,
(1 + 2)(3 + 4 + 5)(6 + 7 + ... + 93) = 396².

Page of Squares : First Upload January 24, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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397

The smallest squares containing k 397's :
1397124 = 1182²,
8397339769 = 91637²,
397939739774329 = 19948427².

397² = 157609, a square with different digits.

397² = 132² + 253² + 276² : 672² + 352² + 231² = 793².

52^k + 190^k + 233^k + 254^k are squares for k = 1,2,3 (27², 397², 6003²).
173^k + 245^k + 397^k + 629^k are squares for k = 1,2,3 (38², 802², 18202²).

3-by-3 magic squares consisting of different squares with constant 397²:

397² = 157609, 15 + 76 + 0 + 9 = 10²,
397² = 157609, 15 + 760 + 9 = 28².

Page of Squares : First Upload January 24, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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398

The smallest squares containing k 398's :
73984 = 272²,
19398639841 = 139279²,
139825398398841 = 11824779².

398 is the first square which is the sum of a prime and a square in 10 ways :
1² + 397, 3² + 389, 5² + 373, 7² + 349, 9² + 317, 11² + 277, 13² + 229, 15² + 173, 17² + 109, 19² + 37.

398² = 158404, 1³ + 584³ + 0³ + 4³ = 14113².

Page of Squares : First Upload January 24, 2005 ; Last Revised July 10, 2006
by Yoshio Mimura, Kobe, Japan

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399

The smallest squares containing k 399's :
399424 = 632²,
42399339921 = 205911²,
739936399939984 = 27201772².

399² = 133² + 266² + 266² : 662² + 662² + 331² = 993²,
399² = 134² + 254² + 277² : 772² + 452² + 431² = 993²,
399² = 178² + 251² + 254² : 452² + 152² + 871² = 993².

Komachi Fraction : 576 / 2039184 = (8 / 399)².

13034^k + 35644^k + 40033^k + 70490^k are squares for k = 1,2,3 (399², 89509², 21492135²).

3-by-3 magic squares consisting of different squares with constant 399²:

399² = 159201, 1 + 5 + 9 + 20 + 1 = 6²,
399² = 159201, 1 + 59 + 20 + 1 = 9²,
399² = 159201, 15 + 9 + 201 = 15²,
399² = 159201, 15 + 9201 = 96².

Page of Squares : First Upload January 24, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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