The smallest squares containing k 53's :
5329 = 732, 553536 = 7442, 5320535364 = 729422,
1753653953536 = 13242562, 1531531553535369 = 391347872.
53 is the sum of m squares for m = 2, 3, ..., 39.
The first integer which is the sum of 5 squares in just 5 ways:
[1,1,1,1,7], [1,1,1,5,5], [1,2,4,4,4], [1,3,3,3,5], [2,2,2,4,5].
532 is the first square which is the sum of 6 fifth powers : 25 + 35 + 35 + 35 + 45 + 45.
532 is the 5th square which is the sum of distinct 3 cubes : 13 + 43 + 143.
532 = 2809 is a zigzag square whose digits are distinct.
532 = 2809, 280 + 9 = 172.
536 = 22164361129, 2 + 2164 + 3 + 611 + 29 = 532,
538 = 62259690411361, 6 + 2 + 2596 + 90 + 41 + 13 + 61 = 532,
6 + 2259 + 6 + 90 + 411 + 36 + 1 = 6 + 2259 + 69 + 0411 + 3 + 61 = 532,
62 + 2596 + 9 + 04 + 1 + 136 + 1 = 62 + 2596 + 90 + 41 + 13 + 6 + 1 = 532.
Komachi Fractions : (6/53)2 = 3564/278091 = 4860/379215, (8/53)2 = 8640/379215,
(53/16)2 = 75843/6912.
Komachi equations:
532 = 9 * 8 + 76 * 54 / 3 * 2 + 1 = 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 + 65 * 43 + 2 * 10 = - 9 + 8 + 7 + 65 * 43 - 2 + 10
= - 9 + 8 * 76 * 5 - 4 * 3 - 210 = - 9 - 8 - 76 * 5 - 4 + 3210,
532 = 12 / 22 * 32 * 42 * 52 + 62 * 72 + 82 + 92 = 12 * 22 / 32 * 452 + 62 * 72 + 82 + 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 * 12 = - 92 + 82 - 72 - 62 + 542 - 32 + 22 / 12.
(532 - 3) = (72 - 3)(82 - 3), (532 + 5) = (32 + 5)(142 + 5)
(532 + 7) = (32 + 7)(132 + 7) = (52 + 7)(92 + 7) = (12 + 7)(22 + 7)(52 + 7).
(1 + 2 + 3)(4 + 5 + ... + 41)(42 + 43 + ... + 53) = 17102,
(1 + 2 + ... + 9)(10 + 11 + ... + 23)(24 + 25 + ... + 53) = 34652,
(1 + 2 + ... + 10)(11 + 12 + ... + 45)(46 + 47 + ... + 53) = 46202,
(1 + 2 + ... + 14)(15 + 16 + ... + 48)(49 + 50 + ... + 53) = 53552,
(1 + 2 + ... + 21)(22 + 23)(24 + 25 + ... + 53) = 34652,
(1 + 2 + ... + 23)(24 + 25 + ... + 45)(46 + 47 + ... + 53) = 91082,
(1 + 2 + ... + 26)(27)(28 + 29 + ... + 53) = 31592,
(1 + 2 + ... + 30)(31 + 32 + ... + 39)(40 + 41 + ... + 53) = 97652,
(1 + 2 + ... + 32)(33 + 34 + ... + 45)(46 + 47 + ... + 53) = 102962.
12 + 22 + 32 + 42 + ... + 532 = 51039, which consists of different digits.
(12 + 22 + ... + 142)(152 + 162 + ... + 272)(282 + 292 + ... + 532) = 5146052.
532 = 2809 appears in the decimal expressions of π and e:
π = 3.14159•••2809••• (from the 10842nd digit),
e = 2.71828•••2809••• (from the 12166th digit).
Page of Squares : First Upload January 19, 2004 ; Last Revised March 19, 2010
by Yoshio Mimura, Kobe, Japan