Squares:320s

320

The smallest squares containing k 320's :
32041 = 179²,
3203673201 = 56601²,
332032063732009 = 18221747².

10^k + 50^k + 190^k + 320^k + 330^k is a square (30², 500², 8700², 153800²) for k = 1, 2, 3, 4.

Komachi Square Sums : 320² = 5² + 6² + 7² + 149² + 283² = 5² + 7² + 9² + 143² + 286².

Page of Squares : First Upload December 6, 2004 ; Last Revised June 29, 2006
by Yoshio Mimura, Kobe, Japan

321

The smallest squares containing k 321's :
12321 = 111²,
3321332161 = 57631²,
27724321321321 = 5265389².

The squares which begin with 321 and end in 321 are
32170368321 = 179361², 32180413321 = 179389², 321048025321 = 566611²,
321079756321 = 566639², 321331393321 = 566861²,...

321² = 103041, a zigzag square.

3-by-3 magic squares consisting of different squares with constant 321²:

4²	103²	304²
172²	256²	89²
271²	164²	52²

4²	184²	263²
212²	199²	136²
241²	172²	124²

7²	56²	316²
124²	292²	49²
296²	121²	28²

16²	89²	308²
217²	224²	76²
236²	212²	49²

18²	186²	261²
219²	198²	126²
234²	171²	138²

44²	143²	284²
199²	236²	88²
248²	164²	121²

321² = (1² + 2² + ... + 50²) + (1² + 2² + ... + 56²).

321² = 103041, 1 + 0 + 3 + 0 + 4 + 1 = 3²,
321² = 103041, 1 + 0 + 30 + 4 + 1 = 6²,
321² = 103041, 10 + 30 + 41 = 9²,
321² = 103041, 103 + 0 + 41 = 12².

321² + 322² + 323² + 324² + 325² + ... + 1856² = 46064².

Page of Squares : First Upload December 6, 2004 ; Last Revised February 2, 2009
by Yoshio Mimura, Kobe, Japan

322

The smallest squares containing k 322's :
13225 = 115²,
1113223225 = 33365²,
13221332293225 = 3636115².

1 / 322 = 0.003105590062111801242236..., where the sum of their digits is 322.

322²± 3 are primes.

322 = (1² + 2² + 3² + ... + 1127²) / (1² + 2² + 3² + ... + 164²).

322² = 103684, a square with different digits.

322² = 103684, 1 + 0 + 36 + 8 + 4 = 7²,
322² = 103684, 1 + 0 + 36 + 84 = 11²,
322² = 103684, 10² + 3² + 6² + 8² + 4² = 15²,
322² = 103684, 103 + 6 + 8 + 4 = 11².

322² + 323² + 324² + 325² + 326² + ... + 1019² = 18497²,
322² + 323² + 324² + 325² + 326² + ... + 100810² = 18479832².

Page of Squares : First Upload December 6, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

323

The smallest squares containing k 323's :
232324 = 482²,
32303232361 = 179731²,
1963237323532356 = 44308434².

323² = 104329, a square with different digits.

323² = 104329, 104 * 3 + 2 + 9 = 323.

3-by-3 magic squares consisting of different squares with constant 323²:

10²	177²	270²
198²	210²	145²
255²	170²	102²

17²	54²	318²
102²	303²	46²
306²	98²	33²

17²	102²	306²
222²	226²	63²
234²	207²	82²

33²	154²	282²
186²	222²	143²
262²	177²	66²

323² = 104329, 10 + 43 + 2 + 9 = 8².

323² = 9³ + 10³ + 11³ + 12³ + 13³ + ... + 25³.

323² + 324² + 325² + 326² + 327² + ... + 2531² = 73461².

323² = 104329 appears in the decimal expression of e:
e = 2.71828•••104329••• (from the 105621st digit).

Page of Squares : First Upload December 6, 2004 ; Last Revised February 2, 2009
by Yoshio Mimura, Kobe, Japan

324

the square of 18.

The smallest squares containing k 324's :
324 = 18²,
81324324 = 9018²,
32450317324324 = 5696518².

The squares which begin with 324 and end in 324 are
324648324 = 18018², 3246948324 = 56982²32406480324 = 180018²,
324309748324 = 569482², 324350752324 = 569518²,...

a zigzag square ( = 18²).

324² = 104976 a square with different digits.

Komachi equation: 324² = 98⁴ / 7⁴ / 6⁴ * 54⁴ * 3⁴ / 21⁴.

324² = 3³ + 24³ + 45³.

324² = 104976, 10 + 4 + 9 + 7 + 6 = 6².

324² + 325² + 326² + 327² + 328² + ... + 44725² = 5460999².

(1)(2 + 3 + ... + 7)(8 + 9 + ... + 88) = 324².

(1³ + 2³ + ... + 28³)(29³ + 30³ + ... + 35³)(36³ + 37³ + ... + 324³) = 10296679680²,
(1³ + 2³ + ... + 28³)(29³ + 30³ + ... + 260³)(261³ + 262³ + ... + 324³) = 554549748480².

Page of Squares : First Upload December 6, 2004 ; Last Revised May 28, 2010
by Yoshio Mimura, Kobe, Japan

325

The smallest squares containing k 325's :
2325625 = 1525²,
8325832516 = 91246²,
2325832533256996 = 48226886².

325² = 105625, 10 * 5 * 6 + 25 = 10 + 5 + 62 * 5 = 10 + 5 * 62 + 5 = 325.

325² = (11² + 4)(29² + 4) = (2² + 1)(8² + 1)(18² + 1).

325² = 105625 with 1 = 1² and 5625 = 75².

Komachi equation: 325² = - 9³ + 8³ + 7³ + 6³ * 5³ + 43³ - 2³ - 10³.

325 is the smallest integer which is the sum of 2 squares in 3 ways.

3-by-3 magic squares consisting of different squares with constant 325²:

0²	80²	315²
195²	252²	64²
260²	189²	48²

0²	91²	312²
125²	288²	84²
300²	120²	35²

0²	125²	300²
195²	240²	100²
260²	180²	75²

0²	165²	280²
195²	224²	132²
260²	168²	99²

8²	156²	285²
219²	208²	120²
240²	195²	100²

44²	117²	300²
180²	260²	75²
267²	156²	100²

325² = 105625, 1 + 0 + 56 + 2 + 5 = 8².

Page of Squares : First Upload December 6, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

326

The smallest squares containing k 326's :
43264 = 208²,
3263265625 = 57125²,
32635032693264 = 5712708².

326² = 106276, a zigzag square.

326²± 3 are primes.

326² = 124² + 204² + 222² : 222² + 402² + 421² = 623².

Komachi equation: 326² = 1³ - 23³ + 4³ + 56³ * 7³ / 8³ + 9³.

326² = 106276, 1³ + 0³ + 6³ + 2³ + 7³ + 6³ = 28².

326² = 106276, 10² + 6² + 2² + 7² + 6² = 15²,
326² = 106276, 10 + 6 + 27 + 6 = 7²,
326² = 106276, 106 + 2 + 7 + 6 = 11².

1599² = 301² + 302² + 303² + 304² + 305² + ... + 326²,
3406² = 15² + 16² + 17² + 18² + 19² + ... + 326².

(1² + 2² + ... + 7²)(8² + 9² + ... + 14²)(15² + 16² + ... + 326²) = 1192100².

Page of Squares : First Upload December 6, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

327

The smallest squares containing k 327's :
32761 = 181²,
3273327369 = 57213²,
70327327327321 = 8386139².

Komachi equation: 327² = 9³ * 8³ + 7³ - 65³ - 4³ + 3³ + 2³ * 10³.

(2² + 3)(3² + 3)(4² + 3)(8² + 3) = 327² + 3.

3-by-3 magic squares consisting of different squares with constant 327²:

13²	134²	298²
178²	253²	106²
274²	158²	83²

24²	153²	288²
207²	216²	132²
252²	192²	81²

35²	98²	310²
130²	290²	77²
298²	115²	70²

38²	83²	314²
106²	302²	67²
307²	94²	62²

38²	122²	301²
182²	259²	82²
269²	158²	98²

67²	178²	266²
214²	227²	98²
238²	154²	163²

327² = 106929, 1 + 0 + 69 + 2 + 9 = 9²,
327² = 106929, 10 + 6 + 9 + 2 + 9 = 6²,
327² = 106929, 106 + 9 + 29 = 12².

1546² = 304² + 305² + 306² + 307³ + 308³ + ... + 327².

Page of Squares : First Upload December 6, 2004 ; Last Revised May 28, 2010
by Yoshio Mimura, Kobe, Japan

328

The smallest squares containing k 328's :
233289 = 483²,
13285328644 = 115262²,
283287913283281 = 16831159².

328² = 107584, a zigzag square with different digits.

Komachi equations:
328² = 987 * 654 / 3 / 2 + 1,
328² = - 1² * 2² * 34² + 5² * 67² + 8² - 9².

328² = 107584, 1 + 0 + 7 + 5 + 8 + 4 = 5².

328329 = 573².

328² = 2⁴ + 6⁴ + 6⁴ + 18⁴.

328² = 107584, 10 + 75 + 84 = 13²,
328² = 107584, 107 + 5 + 84 = 14²,
328² = 107584, 107 + 58 + 4 = 13².

(3² - 8)(5² - 8)(8² - 8)(11² - 8) = 328² - 8.

Page of Squares : First Upload December 6, 2004 ; Last Revised May 28, 2010
by Yoshio Mimura, Kobe, Japan

329

The smallest squares containing k 329's :
5329 = 73²,
732947329 = 27073²,
17329095329329 = 4162823².

The squares which begin with 329 and end in 329 are
3297860329 = 57427², 32915756329 = 181427², 32968754329 = 181573²,
329105300329 = 573677², 329272835329 = 573823²,...

1 / 329 = 0.0030395136..., 3² + 0² + 3² + 9² + 5² + 13² + 6² = 329.

329² = 108241, a zigzag square.

329² = 121² + 204² + 228² : 822² + 402² + 121² = 923².

Komachi Square Sum : 329² = 2² + 6² + 58² + 79² + 314².

3-by-3 magic squares consisting of different squares with constant 329²:

4²	57²	324²
159²	284²	48²
288²	156²	31²

4²	135²	300²
225²	220²	96²
240²	204²	95²

33²	176²	276²
204²	228²	121²
256²	159²	132²

36²	193²	264²
212²	216²	129²
249²	156²	148²

329² = 108241, 1 + 0 + 82 * 4 * 1 = 1 * 0 + 82 * 4 + 1 = 329.

329² = 108241, 1 + 0 + 8 + 2 + 4 + 1 = 4²,
329² = 108241, 10 + 8 + 2 + 4 + 1 = 5²,
329² = 108241, 10² + 8² + 2² + 41² = 43².

(1³ + 2³ + ... + 119³)(120³ + 121³ + ... + 140³)(141³ + 142³ + ... + 329³) = 2597238189000².

Page of Squares : First Upload December 6, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan