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350 - 359

350

The smallest squares containing k 350's :
63504 = 2522,
3505350436 = 592062,
8350643503504 = 28897482.

350 = (12 + 22 + 32 + ... + 242) / (12 + 22 + 32).

3502 = (22 + 6)(82 + 6)(132 + 6).

3502 = 122500 with 1 = 12 and 22500 = 1502.

3502 = 253 + 353 + 403.

3502± 3 are primes.

3502 = 152 x 153 + 154 x 155 + 156 x 157 + 158 x 159 + 160 x 161,
3502 = 44 x 45 + 46 x 47 + 48 x 49 + 50 x 51 + ... + 92 x 93.

350, 351 and 352 are three consecutive integers having square factors (the 5th case).

10290k + 21210k + 39690k + 51310k are squares for k = 1,2,3 (3502, 690202, 144305002).

Komachi equation: 3502 = 92 * 82 * 72 * 62 * 52 / 4322 * 102.

(12 + 22 + 32 + 42 + 52 + 62 + 72)(82 + 92 + 102 + 112 + 122 + 132 + 142) = 3502.

(13 + 23 + ... + 3123)(3133 + 3143)(3153 + 3163 + ... + 3503) = 139641849547202,
(13 + 23 + ... + 2403)(2413 + 2433 + ... + 3143)(3153 + 3163 + ... + 3503) = 422676500400002.

Page of Squares : First Upload December 27, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

351

The smallest squares containing k 351's :
351649 = 5932,
35113512996 = 1873862,
35135102135169 = 59274872.

(12 + 22 + ... + 3512) = (12 + 22 + ... + 1362) + (12 + 22 + ... + 3442).

3512 + 3522 + 3532 + ... + 3642 = 3652 + 3662 + 3672 + ... + 3772.

3-by-3 magic squares consisting of different squares with constant 3512:

225923462
22922622462
26622262372
   
1125823462
9423342532
3382912262
   
1129423382
21422772262
27821942912
   
11221422782
242220621492
254218721542
   
1428223412
10723262742
33421012382
14213923222
203226621062
28621822912
   
2629123382
20622782592
28321942742
   
26213023252
245222621102
25022352742
   
26218222992
214222921582
27721942942
   
39215623122
22822492962
264219221292

3512 = 123201, 1 + 2 + 3 + 2 + 0 + 1 = 32,
3512 = 123201, 1 + 2 + 32 + 0 + 1 = 62,
3512 = 123201, 1 + 2 + 320 + 1 = 182,
3512 = 123201, 1 + 23 + 201 = 152,
3512 = 123201, 123 + 33 + 23 + 03 + 13 = 422,
3512 = 123201, 12 + 3 + 20 + 1 = 62,
3512 = 123201, 123 + 20 + 1 = 122,
3512 = 123201, 123 + 201 = 182,
3512 = 123201, 12320 + 1 = 1112.

(1 + 2)(3)(4 + 5 + 6 + 7 + 8 + ... + 165) = 3512.

Page of Squares : First Upload December 27, 2004 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

352

The smallest squares containing k 352's :
83521 = 2892,
352763524 = 187822,
3523526935236 = 18771062.

3522 = 123904, a square with different digits.

3522 = (12 + 7)(22 + 7)(32 + 7)(92 + 7) = (12 + 7)(32 + 7)(312 + 7) = (12 + 7)(92 + 7)(132 + 7)
= (112 + 7)(312 + 7) = (22 + 7)(92 + 7)(112 + 7).

Komachi Square Sum : 3522 = 12 + 22 + 72 + 82 + 692 + 3452.

3522 = 123904, 1 + 2 + 3 + 90 + 4 = 102,
3522 = 123904, 122 + 392 + 02 + 42 = 412.

3522 = 123904 appears in the decimal expression of π:
  π = 3.14159•••123904••• (from the 75742nd digit).

Page of Squares : First Upload December 27, 2004 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

353

The smallest squares containing k 353's :
35344 = 1882,
35335352529 = 1879772,
3536935317235321 = 594721392.

3532 = 124609, a square with different digits.

Komachi equation: 3532 = 122 + 32 * 42 * 52 * 62 + 72 - 82 * 92.

Cubic Polynomial : (X + 72)(X + 1082)(X + 3362) = X3 + 3532X2 + 363722X + 2540162.

3-by-3 magic squares consisting of different squares with constant 3532:

7210823362
20422732922
28821962572
     
1625723482
13223242472
32721282362
     
48212423272
15623032922
31321322962

3532 = 124609, 1 + 246 + 0 + 9 = 162,
3532 = 124609, 12 + 4 + 609 = 252.

3532 + 3542 + 3552 + 3562 + 3572 + ... + 3762 = 17862.

(13 + 23 + ... + 813)(823 + 833 + ... + 1223)(1233 + 1243 + ... + 3533) = 13859534082242.

Page of Squares : First Upload December 27, 2004 ; Last Revised June 1, 2010
by Yoshio Mimura, Kobe, Japan

354

The smallest squares containing k 354's :
135424 = 3682,
13545235456 = 1163842,
3542354113547041 = 595176792.

The square root of 354 is 18.8148..., 182 = 82 + 142 + 82.

222k + 354k + 370k + 498k are squares for k = 1,2,3 (382, 7482, 151482).
7198k + 16874k + 20650k + 80594k are squares for k = 1,2,3 (3542, 851962, 231834602).
11682k + 27966k + 30090k + 55578k are squares for k = 1,2,3 (3542, 700922, 149126042).

The 4-by-4 magic square consisting of different squares with constant 354(the least value):

 02 42 72172
 82162 32 52
112 12142 62
132 92102 22

3542 = 125316, 1 + 2 + 5 + 316 = 182,
3542 = 125316, 1 + 25 + 3 + 1 + 6 = 62,
3542 = 125316, 12 + 5 + 3 + 16 = 62,
3542 = 125316, 12 + 53 + 16 = 92,
3542 = 125316, 125 + 3 + 16 = 122,
3542 = 125316, 125 + 316 = 212.

Page of Squares : First Upload December 27, 2004 ; Last Revised March 2, 2011
by Yoshio Mimura, Kobe, Japan

355

The smallest squares containing k 355's :
355216 = 5962,
2355355024 = 485322,
2355953558643556 = 485381662.

3552 = 126025, 12 * 60 / 2 - 5 = 355.

3552 -1 = 4! + 7! + 8! + 8! + 8!

3552 = 503 + 45 + 17.

3-by-3 magic squares consisting of different squares with constant 3552:

1529423422
23022582812
27022252502
     
15213023302
23022552902
27022102952
     
15218623022
230222521502
270220221112
     
22215023212
246222521222
25522302902
     
49215023182
21022702952
282217521262

3552 = 126025, 1 + 2 + 6 + 0 + 2 + 5 = 42,
3552 = 126025, 12 + 6 + 0 + 2 + 5 = 52.

Page of Squares : First Upload December 27, 2004 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

356

The smallest squares containing k 356's :
4356 = 662,
555356356 = 235662,
4235635660356 = 20580662.

The squares which begin with 356 and end in 356 are
35696056356 = 1889342,   356330200356 = 5969342,   356487808356 = 5970662,
356927384356 = 5974342,   3560519920356 = 18869342,...

3562± 3 are primes.

3562 = 126736, 1 + 2 + 6 + 7 + 3 + 6 = 52,
3562 = 126736, 122 + 672 + 362 = 772,
3562 = 126736, 126 + 7 + 36 = 132.

3562 = (12 + 22 + ... + 242) + (12 + 22 + ... + 712).

3562 = 23 + 123 + 503.

(13 + 23 + ... + 1633)(1643 + 1653 + ... + 2043)(2053 + 2063 + ... + 3563) = 128973606664322.

Page of Squares : First Upload December 27, 2004 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

357

The smallest squares containing k 357's :
35721 = 1892,
35735743521 = 1890392,
3570357073576225 = 597524652.

357 = (12 + 22 + 32 + ... + 172) / (12 + 22).

3572 = 252 + 262 + 272 + 282 + 292 + ... + 732.

3572 = 1362 + 2122 + 2532 : 3522 + 2122 + 6312 = 7532.

Cubic Polynomial : (X + 2882)(X + 3162)(X + 3572) = X3 + 5572X2 + 1777082X + 324898562.

3-by-3 magic squares consisting of different squares with constant 3572:

825923522
13123282522
33221282292
     
8216423172
187227221362
30421632922
     
9210223422
16223062872
31821532542
     
18218323062
222223421532
279219821022
     
20210723402
23522602682
26822202852
32216323162
197225621522
29621882672
     
3728823442
13623232682
32821242672
     
43212423322
21222772762
284218821072
     
44222722722
248217621872
253221221362
     
64220822832
227224421282
268215721762

3572 = 127449, 1 + 27 + 4 + 49 = 92,
3572 = 127449, 1 + 27 + 44 + 9 = 92,
3572 = 127449, 1 + 274 + 49 = 182,
3572 = 127449, 12 + 7 + 4 + 4 + 9 = 62,
3572 = 127449, 127 + 4 + 4 + 9 = 122,
3572 = 127449, 127 + 449 = 242.

(13 + 23 + 33 + ... + 833)(843)(853 + 863 + 873 + ... + 3573) = 1712335191842.

Page of Squares : First Upload December 27, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

358

The smallest squares containing k 358's :
358801 = 5992,
93358358116 = 3055462,
3588676358535844 = 599055622.

3582 = 128164, 1 + 28 + 16 + 4 = 72,
3582 = 128164, 123 + 83 + 163 + 43 = 802.

(12 + 22 + ... + 172)(182 + 192 + ... + 2372)(2382 + 2392 + ... + 3582) = 2946035402.

3582 + 3592 + 3602 + 3612 + 3622 + ... + 7182 = 104122.

Page of Squares : First Upload December 27, 2004 ; Last Revised July 3, 2006
by Yoshio Mimura, Kobe, Japan

359

The smallest squares containing k 359's :
1359556 = 11662,
12359435929 = 1111732,
235965635935929 = 153611732.

359 is the 6th prime for which the Legendre Symbol (a/359) = 1 for a =1, 2,..., 6.

3592 = 128881, a square with 3 kinds of digits.

3592 = 1342 + 2342 + 2372 : 7322 + 4322 + 4312 = 9532.

12 + 22 + ... + 3592 = 15487260, which consistes of different digits (the second 8-digit sum).

3-by-3 magic squares consisting of different squares with constant 3592:

 3217423142
206225821412
294217921022
     
 6210923422
13423182992
33321262462
     
 6211823392
18622912982
30721742662
     
10214123302
210227021092
29121902902
42220622912
234223721342
269217421622
     
66218922982
24622422992
253218621742

3592 = 128881, 1 + 2 + 8 + 8 + 81 = 102,
3592 = 128881, 1 + 2 + 8 + 88 + 1 = 102,
3592 = 128881, 1 + 2 + 88 + 8 + 1 = 102,
3592 = 128881, 1288 + 81 = 372.

3592 = 128881 appears in the decimal expression of π:
  π = 3.14159•••128881••• (from the 99034th digit).

Page of Squares : First Upload December 27, 2004 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan