The smallest squares containing k 56's :
256 = 162, 2056356 = 14342, 565678656 = 237842,
565565665681 = 7520412, 630565635656356 = 251110662.
The squares which begin with 56 and end in 56 are
5683456 = 23842, 56010256 = 74842, 56490256 = 75162, 56761156 = 75342,
560079556 = 236662,...
The 2nd integer which is the sum of 5 squares in just 5 ways.
The 1st integer which is the sum of 8 squares in just 8 ways:
[1,1,1,1,1,1,1,7], [1,1,1,1,1,1,5,5], [1,1,1,1,2,4,4,4], [1,1,1,1,3,3,3,5], [1,1,1,2,2,2,4,5],
[1,1,3,3,3,3,3,3], [1,2,2,2,3,3,3,4], [2,2,2,2,2,2,4,4]
562 is the second square which is the sum of 5 fifth powers : [2,2,4,4,4].
562 = 3136, a square with odd digits except the last digit.
563 + 653 = 6712, a square.
5683456 = 23842.
126k + 538k + 1050k + 1422k are squares for k = 1,2,3 (562, 18522, 647362).
226k + 610k + 822k + 1478k are squares for k = 1,2,3 (562, 18122, 634242).
Komachi equations:
562 = 1 + 2 - 3 * 4 + 56 * 7 * 8 + 9 = 1 - 2 * 3 - 4 + 56 * 7 * 8 + 9
= 1 * 2 + 3 + 4 + 56 * 7 * 8 - 9 = 12 * 3 / 4 + 56 * 7 * 8 - 9
= 12 * 3 / 4 * 56 * 7 * 8 / 9 = - 1 - 2 + 3 * 4 + 56 * 7 * 8 - 9
= - 1 + 2 * 3 + 4 + 56 * 7 * 8 - 9 = - 1 * 2 - 3 - 4 + 56 * 7 * 8 + 9
= - 12 * 3 / 4 + 56 * 7 * 8 + 9 = 9 * 8 + 765 * 4 + 3 + 2 - 1
= 9 + 8 * 76 * 5 + 43 * 2 + 1 = 9 * 8 + 765 * 4 - 3 * 2 + 10
= 98 + 765 * 4 - 32 + 10 = 9 - 8 - 76 + 5 - 4 + 3210
= - 9 * 8 - 7 + 6 - 5 + 4 + 3210 = - 98 / 7 - 6 - 54 + 3210,
562 = 12 + 22 - 32 - 42 - 52 - 62 + 72 * 82 + 92 = 12 * 22 * 32 - 42 + 52 + 62 + 72 * 82 - 92
= 122 / 32 + 42 + 562 - 72 - 82 + 92 = 12 * 22 * 342 * 52 / 62 - 782 / 92
= - 12 - 22 + 32 + 42 + 52 + 62 + 72 * 82 - 92 = - 12 * 22 * 32 + 42 - 52 - 62 + 72 * 82 + 92
= - 12 * 22 * 32 - 452 - 62 + 72 + 82 * 92 = - 122 / 32 - 42 + 562 + 72 + 82 - 92
= 92 + 82 * 72 - 62 - 52 + 42 - 32 * 22 */ 12 = 92 + 82 * 72 - 62 - 52 - 42 - 32 + 22 + 12
= 982 / 72 + 62 + 542 - 32 - 22 + 12 = 92 + 82 * 72 + 62 + 542 / 32 - 212
= - 92 + 82 * 72 + 62 + 52 - 42 + 32 * 22 */ 12 = - 92 + 82 * 72 + 62 + 52 + 42 + 32 - 22 - 12
= - 92 - 82 - 72 - 62 + 542 + 32 + 212 = - 92 + 82 * 72 - 62 - 542 / 32 + 212,
562 = 14 * 24 - 34 + 44 - 54 - 64 + 74 - 84 + 94 = 94 - 84 + 74 - 64 - 54 + 44 - 34 + 24 * 14
= 94 - 84 + 74 - 64 - 54 + 44 - 34 + 24 / 14.
2452 = 72 + 82 + 92 + ... + 562.
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 56) = 15122,
(1 + 2 + ... + 5)(6 + 7 + ... + 39)(40 + 41 + ... + 56) = 30602.
(12 + 22 + 32 + ... + 562) = 60116, which consists of 3 kinds of digits.
(12 + 22)(32 + 42 + 52)(62 + 72)(82 + 92 + ... + 562) = 357002.
562 = 3136 appears in the decimal expressions of π and e:
π = 3.14159•••3136••• (from the 1303rd digit),
(3136 is the fifth 4-digit square in the expression of π,)
e = 2.71828•••3136••• (from the 1862nd digit).
Page of Squares : First Upload January 26, 2004 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan