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340

The smallest squares containing k 340's :
23409 = 153²,
12434034064 = 111508²,
334073403405625 = 18277675².

340² = 69 x 70 + 71 x 72 + 73 x 74 + 75 x 76 + ... + 99 x 100.

340² = (4² + 4)(76² + 4) = (4² + 4)(8² + 4)(9² + 4).

340²± 3 are primes.

Komachi equation: 340² = - 1² - 2² + 345² + 6² * 7² - 8² * 9².

(1³ + 2³ + ... + 75³)(76³ + 77³ + ... + 124³)(125³ + 126³ + ... + 340³) = 1180002600000².

Page of Squares : First Upload December 20, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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341

The smallest squares containing k 341's :
341056 = 584²,
63417341584 = 251828²,
3341982341234161 = 57809881².

253^k + 295^k + 341^k + 407^k are squares for k = 1,2,3 (36², 658², 12204²).

341² = 116281, 1 + 1 + 6 + 281 = 17²,
341² = 116281, 1 + 16 + 2 + 81 = 10²,
341² = 116281, 11 + 6 + 2 + 81 = 10².

3-by-3 magic squares consisting of different squares with constant 341²:

341² = 1³ - 2³ + 3³ - 4³ + 5³ - 6³ + ... - 60³ + 61³.

(1³ + 2³ + ... + 133³)(134³ + 135³ + ... + 209³)(210³ + 211³ + ... + 341³) = 9654318835392².

341² = 116281 appears in the decimal expression of π
  π = 3.14159•••116281••• (from the 29230th digit).

Page of Squares : First Upload December 20, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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342

The smallest squares containing k 342's :
34225 = 185²,
34234201 = 5851²,
903423429342009 = 30057003².

342²± 5 are primes.

342² = 116964 = 116964  (1, 9, 16, 64, 169 are squares.)

342² = (1² + 2)(2² + 2)(6² + 2)(13² + 2) = (1² + 2)(6² + 2)(32² + 2) = (4² + 2)(6² + 2)(13² + 2).

342² = 1³ + 35³ + 42³.

342, 343 and 344 are consecutive integers having square factors (the 4th case).

1482^k + 2850^k + 32718^k + 79914^k are squares for k = 1,2,3 (342², 86412², 23353812²).

Komachi equations:
342² = 1 / 2 * 345 * 678 + 9,
342² = 9² * 8² * 76² * 5² * 4² / 32² / 10² = 9² / 8² * 76² * 5² / 4² * 32² / 10²,
342² = - 1³ + 2³ - 3³ + 4³ + 56³ * 7³ / 8³ - 9³.

342² = 116964, 1 + 1 + 6 + 9 + 64 = 9²,
342² = 116964, 1 + 1 + 69 + 6 + 4 = 9²,
342² = 116964, 1 + 16 + 9 + 6 + 4 = 6²,
342² = 116964, 11 + 6 + 9 + 6 + 4 = 6²,
342² = 116964, 11 + 69 + 64 = 12².

(1 + 2 + 3 + ... + 8)(9)(10 + 11 + 12 + ... + 28) = 342².

Page of Squares : First Upload December 20, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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343

The smallest squares containing k 343's :
343396 = 586²,
23431343329 = 153073²,
343343741434369 = 18529537².

Komachi equations:
343² = 98² * 7² * 6² * 5² / 4² / 3² * 2² / 10² = 98² * 7² * 6² / 5² / 4² / 3² / 2² * 10²
= 98² * 7² / 6² * 5² * 4² * 3² / 2² / 10² = 98² * 7² / 6² / 5² / 4² * 3² * 2² * 10²,
343² = 12³ * 3³ / 4³ + 56³ * 7³ / 8³ - 9³ = 12³ * 3³ / 4³ * 56³ * 7³ / 8³ / 9³
= - 12³ * 3³ / 4³ + 56³ * 7³ / 8³ + 9³,
343² = 9⁶ * 8⁶ + 7⁶ - 6⁶ * 5⁶ * 4⁶ * 3⁶ * 2⁶ / 10⁶ = 9⁶ * 8⁶ + 7⁶ - 6⁶ / 5⁶ * 4⁶ * 3⁶ / 2⁶ * 10⁶
= 9⁶ * 8⁶ * 7⁶ / 6⁶ * 5⁶ / 4⁶ / 3⁶ * 2⁶ / 10⁶ = 9⁶ * 8⁶ * 7⁶ / 6⁶ / 5⁶ / 4⁶ / 3⁶ / 2⁶ * 10⁶
= 9⁶ / 8⁶ + 7⁶ - 6⁶ * 5⁶ / 4⁶ * 3⁶ / 2⁶ / 10⁶ = 9⁶ / 8⁶ * 7⁶ / 6⁶ / 5⁶ * 4⁶ / 3⁶ * 2⁶ * 10⁶
= 98⁶ / 7⁶ * 6⁶ * 5⁶ / 4⁶ / 3⁶ * 2⁶ / 10⁶ = 98⁶ / 7⁶ * 6⁶ / 5⁶ / 4⁶ / 3⁶ / 2⁶ * 10⁶
= 98⁶ / 7⁶ / 6⁶ * 5⁶ * 4⁶ * 3⁶ / 2⁶ / 10⁶ = 98⁶ / 7⁶ / 6⁶ / 5⁶ / 4⁶ * 3⁶ * 2⁶ * 10⁶
= - 9⁶ * 8⁶ + 7⁶ + 6⁶ * 5⁶ * 4⁶ * 3⁶ * 2⁶ / 10⁶ = - 9⁶ * 8⁶ + 7⁶ + 6⁶ / 5⁶ * 4⁶ * 3⁶ / 2⁶ * 10⁶
= - 9⁶ / 8⁶ + 7⁶ + 6⁶ * 5⁶ / 4⁶ * 3⁶ / 2⁶ / 10⁶.

3-by-3 magic squares consisting of different squares with constant 343²:

343² = 117649 with 1 = 1², 1764 = 42², and 9 = 3².

343² = 117649, 1 + 1 + 7 + 6 + 49 = 8²,
343² = 117649, 11 + 76 + 4 + 9 = 10²,
343² = 117649, 11 + 764 + 9 = 28²,
343² = 117649, 1176 + 49 = 35².

2908² = 248² + 249² + 250² + 251² + 252² + ... + 343².

Page of Squares : First Upload December 20, 2004 ; Last Revised June 1, 2010
by Yoshio Mimura, Kobe, Japan

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344

The smallest squares containing k 344's :
35344 = 188²,
234457344 = 15312²,
13440934451344 = 3666188².

The squares which begin with 344 and end in 344 are
3444281344 = 58688²,   34480033344 = 185688²,   344202809344 = 586688²,
344348323344 = 586812²,   344789747344 = 587188²,...

344² = 118336, 1 - 1 + 8 + 336 = 1 * 1 * 8 + 336 = 344.

344^k + 1505^k + 4558^k + 10234^k are squares for k = 1,2,3 (129², 11309², 1081665²).

344² = 118336, 1³ + 1³ + 8³ + 3³ + 3³ + 6³ = 28²,
344² = 118336, 1 + 1 + 8 + 3 + 36 = 7²,
344² = 118336, 1 + 1 + 8 + 33 + 6 = 7²,
344² = 118336, 1 + 1 + 83 + 36 = 11²,
344² = 118336, 1 + 1 + 833 + 6 = 29².

344² + 345² + 346² + 346² + 347² + ... + 1065² = 19741².

Page of Squares : First Upload December 20, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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345

The smallest squares containing k 345's :
13456 = 116²,
26345834596 = 162314²,
234534553459441 = 15314521².

1 / 345 = 0.00289..., 289 = 17².

345 = (1² + 2² + 3² + ... + 45²) / (1² + 2² + 3² + 4² + 5² + 6²).

The sum of (3k - 2)³ + (3k - 1)³ + (3k)⁵ for k = 1,2,...,115 is 9805245².

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

345² = 119025 with 1 = 1², 9025 = 95².

The sum of the cubes of the divisors of 345 is 6552².

Komachi equations:
345² = 1² * 2² + 345² - 6² + 7² + 8² - 9² = - 1² * 2² + 345² + 6² - 7² - 8² + 9².

3-by-3 magic squares consisting of different squares with constant 345²:

345² = 119025, 1 + 1 + 9 + 0 + 25 = 6²,
345² = 119025, 119 + 0 + 25 = 12².

(1² + 2²)(3² + 4² + 5² + ... + 169²)(170² + 171² + 172² + ... + 345²) = 9919800²,
(1² + 2² + ... + 12²)(13² + 14² + ... + 59²)(60² + 61² + ... + 345²) = 24867700²,
(1³ + 2³ + ... + 104³)(105³ + 106³ + ... + 299³)(300³ + 301³ + ... + 345³) = 9571663647000².

345² = 119025 appears in the decimal expression of e:
  e = 2.71828•••119025••• (from the 138644th digit).

Page of Squares : First Upload December 20, 2004 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

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346

The smallest squares containing k 346's :
134689 = 367²,
13467834601 = 116051²,
2346773467346064 = 48443508².

1 / 346 = 0.00289..., 289 = 17².

346² = 119716, a square with odd digits except the last digit 6.

98^k + 212^k + 305^k + 346^k are squares for k = 1,2,3 (31², 517², 8959²).

Komachi equation: 346² = - 1² * 2² + 345² - 67² + 8² * 9².

346² = 119716, 1 + 1 + 9 + 7 + 1 + 6 = 5²,
346² = 119716, 1² + 1² + 9² + 7² + 1² + 6² = 13²,
346² = 119716, 119 + 71 + 6 = 14².

346² + 347² + 348² + 349² + 350² + ... + 5922² = 263120².

Page of Squares : First Upload December 20, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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347

The smallest squares containing k 347's :
1153476 = 1074²,
3472273476 = 58926²,
4934733473476 = 2221426².

347² = 120409, a zigzag square.

347² = 103² + 222² + 246² : 642² + 222² + 301² = 743².

3-by-3 magic squares consisting of different squares with constant 347²:

347² = 120409, 1 + 2 + 0 + 4 + 0 + 9 = 4²,
347² = 120409, 12 + 0 + 4 + 0 + 9 = 5²,
347² = 120409, 120 + 40 + 9 = 13²,
347² = 120409, 120 + 409 = 23².

Page of Squares : First Upload December 20, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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348

The smallest squares containing k 348's :
3481 = 59²,
19583483481 = 139941²,
348703486348816 = 18673604².

348² = 4! + 5! + 8! + 8! + 8!

348² = 6 x 7 x 8 x 9 + 9 x 10 x 11 x 12 + 12 x 13 x 14 x 15 + 15 x 16 x 17 x 18.

Cubic Polynomial :
(X + 348²)(X + 539²)(X + 1584²) = (X³ + 1709²X² + 1033428²X + 297114048²).

57^k + 102^k + 222^k + 348^k are squares for k = 1,2,3 (27², 429², 7371²).

348² = 121104, 1 + 2 + 1 + 1 + 0 + 4 = 3²,
348² = 121104, 121 + 104 = 15².

Page of Squares : First Upload December 20, 2004 ; Last Revised February 28, 2011
by Yoshio Mimura, Kobe, Japan

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349

The smallest squares containing k 349's :
34969 = 187²,
23534934921 = 153411²,
3499533493491025 = 59156855².

349² = 121801, a zigzag square.

349² = 1! + 5! + 6! + 8! + 8! + 8!

349² = 11² + 228² + 264² : 462² + 822² + 11² = 943².

Komachi equation: 349² = 1² * 2² * 34² * 5² + 6² + 78² + 9².

3-by-3 magic squares consisting of different squares with constant 349²:

349² + 350² + 351² + 352² + 353²... + 5677² = 246959².

Page of Squares : First Upload December 20, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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