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.
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:
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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, 2011by 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).
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:
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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, 2010by 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):
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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.
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:
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3552 = 126025, 1 + 2 + 6 + 0 + 2 + 5 = 42,
3552 = 126025, 12 + 6 + 0 + 2 + 5 = 52.
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, 2014by 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:
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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, 2013by 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, 2006by 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:
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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).
by Yoshio Mimura, Kobe, Japan