The smallest squares containing k 55's :
27556 = 1662, 5555449 = 23572, 5525543556 = 743342,
15547556555521 = 39430392, 1558655555555344 = 394798122.
The first integer which is the sum of 5 distinct squares: 12 + 22 + 32 + 42 + 52.
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 12 + 22 + 32 + 42 + 52.
The 3rd integer which is the sum of 4 squares in just 3 ways: [1,1,2,7], [1,2,5,5], [1,3,3,6].
The 1st integer which is the sum of 7 squares in just 7 ways:
[1,1,1,1,1,1,7], [1,1,1,1,1,5,5], [1,1,1,2,4,4,4], [1,1,1,3,3,3,5], [1,1,2,2,2,4,5], [1,3,3,3,3,3,3],
[2,2,2,3,3,3,4]
552 is the second square which is the sum of 8 distinct cubes : [1,2,3,4,5,7,9,12].
552 = 3025 is a zigzag square whose digits are distinct.
552 = 3025, 30 * 2 - 5 = 55.
552 = 1! + 4! + 5! + 6! + 6! + 6! + 6!.
552 = 13 + 43 + 63 + 143.
552 + 562 + 572 + 582 + 592 + 602 = 612 + 622 + 632 + 642 + 652.
The fourth Kaprekar numbers : 552 = 3025. 30 + 25 = 55.
(12 + 22 + 32 + ... + 552) = 56980, which consists of different digits.
552 = (12 + 22 + 32 + ... + 152) + (12 + 22 + 32 + ... + 172).
552 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103.
(12 + 22 + 32 + ... + 552) = (12 + 22 + 32 + ... + 422) + (12 + 22 + 32 + ... + 452).
88k + 638k + 1122k + 1177k are squares for k = 1,2,3 (552, 17492, 574752).
(552 - 1) = (72 - 1)(82 - 1).
552 = 30 + 33 + 34 + 36 + 37.
Komachi equations:
552 = 1 + 2 - 3 - 4 + 5 + 6 * 7 * 8 * 9 = 1 - 2 + 3 + 4 - 5 + 6 * 7 * 8 * 9
= 1 / 2 * 3 * 4 - 5 + 6 * 7 * 8 * 9 = 12 - 3 + 45 * 67 - 8 + 9
= - 1 - 2 + 3 - 4 + 5 + 6 * 7 * 8 * 9 = - 1 - 2 * 3 + 45 * 67 + 8 + 9
= 9 * 8 * 7 * 6 + 5 - 4 + 3 - 2 - 1 = 9 * 8 * 7 * 6 + 5 - 4 - 3 + 2 + 1
= 9 * 8 * 7 * 6 - 5 + 4 + 3 - 2 + 1 = 9 * 8 * 7 * 6 - 5 + 4 * 3 / 2 */ 1
= 9 + 8 * 76 * 5 - 4 * 3 * 2 */ 1 = 9 * 8 * 7 / 6 * 54 / 3 * 2 + 1
= 98 / 7 * 6 * 54 / 3 * 2 + 1 = - 9 + 8 * 76 * 5 - 4 - 3 + 2 - 1
= - 9 + 8 * 76 * 5 - 4 * 3 / 2 */ 1 = - 9 - 8 + 765 * 4 + 3 - 21
= - 98 + 765 * 4 + 3 * 21 = 9 * 8 * 7 * 6 + 5 + 4 * 3 / 2 - 10
= 9 * 8 * 7 * 6 + 5 - 4 * 3 - 2 + 10 = 9 + 8 * 76 * 5 - 4 * 3 - 2 - 10
= 9 * 8 * 7 * 6 - 54 / 3 / 2 + 10 = 9 * 8 * 7 + 6 - 5 + 4 * 3 * 210
= - 9 + 8 * 7 * 6 * 54 / 3 / 2 + 10 = - 9 + 8 * 7 / 6 * 54 * 3 * 2 + 10
= - 9 - 8 + 7 * 654 / 3 * 2 - 10 = - 9 - 8 * 7 - 6 * 5 * 4 + 3210
= - 9 * 8 + 7 - 6 * 5 * 4 + 3210,
552 = 12 - 22 * 32 + 42 + 52 - 62 + 72 * 82 - 92 = 12 * 22 - 342 + 562 / 72 * 82 + 92
= 12 * 22 - 32 * 42 * 52 / 62 * 72 + 892 = - 12 * 22 + 32 - 42 * 52 - 62 * 72 + 82 * 92
= - 12 + 22 + 32 - 42 - 562 + 782 + 92 = 92 * 82 - 72 * 62 - 52 * 42 + 32 - 22 */ 12
= - 92 + 82 * 72 - 62 + 52 + 42 - 32 * 22 + 12 = 92 + 82 * 72 + 62 - 542 / 32 - 22 + 102
= 92 - 82 - 72 + 62 + 542 + 32 - 22 + 102 = - 92 + 82 + 72 - 62 + 542 + 32 + 22 + 102,
552 = 14 * 24 * 34 - 44 + 54 + 64 - 74 - 84 + 94 = 94 - 84 - 74 + 64 + 54 - 44 + 34 * 24 */ 14.
12 + 22 + 32 + 42 + ... + 552 = 56980, which consists of different digits.
552 + 562 + 572 + ... + 35332 = 1212682.
(12 + 22 + ... + 242)(252 + 262 + ... + 482)(492 + 502 + ... + 522)(532 + 542 + 552)= 1203930002.
(13 + 23 + ... + 173)(183 + 193 + 203)(213 + 223 + ... + 553) = 335758502.
552 = 2305 appears in the decimal expressions of π and e:
π = 3.14159•••2305••• (from the 8751st digit),
e = 2.71828•••2305••• (from the 10583rd digit).
Page of Squares : First Upload January 26, 2004 ; Last Revised September 5, 2011
by Yoshio Mimura, Kobe, Japan