The smallest squares containingk 89's :
289 = 172, 6589489 = 25672, 898980289 = 299832,
389896589889 = 6244172, 6276898989895689 = 792268832.
The squares which begin with 89 and end in 89 are
89624089 = 94672, 89927289 = 94832, 890007889 = 298332, 892037689 = 298672,
892993689 = 298832,...
89 is the sum of m squares for m = 2, 3, ..., 75.
892 = 7921, a square with different digits.
892 = 7921, 7 + 92 + 1 = 102,
892 = 7921, 79 + 21 = 102.
892 = 1! + 6! + 6! + 6! + 6! + 7!
Loop of length 8 by the function f(N) = ... + c2 + b2 + a2 for N = ... + 102c + 10b + a:
58 -- 89 -- 145 -- 42 -- 20 -- 4 -- 16 -- 37 -- 58
89624089 = 94672 and 89927289 = 94832 are squares.
The sum of the consecutive odd primes 3 + 5 + 7 + 11 + ... + 89 = 312, a square.
898 = 3936588805702081, 3 + 9 + 3 + 6 + 58 + 8 + 805 + 7020 + 8 + 1 = 892.
Komachi Fractions : (22/89)2 = 4356/71289, (84/89)2 = 63504/71289.
Komachi equations:
892 = 9 * 876 + 54 / 3 * 2 + 1 = 9 * 876 - 5 + 43 - 2 + 1
= 9 * 876 + 5 + 4 * 3 + 2 * 10 = 9 * 876 + 5 * 4 - 3 + 2 * 10
= 9 * 876 + 54 + 3 - 2 * 10,
892 = 12 + 22 - 32 + 42 - 52 - 62 + 72 + 892 = - 12 - 22 + 32 - 42 + 52 + 62 - 72 + 892
= 982 - 72 * 62 + 542 / 32 / 22 */ 12 = - 92 + 82 + 72 / 62 * 542 + 32 * 212
= 92 + 82 - 72 + 652 + 42 * 32 / 22 * 102,
892 = - 93 - 83 + 73 - 63 + 53 * 43 + 33 + 23 + 103,
892 = 94 - 84 - 74 + 64 + 544 / 34 / 24 */ 14.
(892 - 1) = (42 - 1)(232 - 1) = (92 - 1)(102 - 1)
(892 + 9) = (12 + 9)(282 + 9).
12 + 22 + 32 + 42 + ... + 892 = 238965, which consists of different digits.
(1 + 2)(3 + 4 + ... + 29)(30 + 31 + ... + 88) = 21242,
(1 + 2)(3 + 4 + ... + 32)(33 + 34 + ... + 88) = 23102,
(1 + 2 + 3)(4 + 5 + ... + 39)(40 + 41 + ... + 89) = 38702,
(1 + 2 + ... + 8)(9 + 10 + ... + 89) = 3782,
(1 + 2 + ... + 17)(18 + 19 + ... + 38)(39 + 40 + ... + 89) = 171362.
(12 + ... + 202)(212 + ... + 242)(252 + ... + 482)(492 + ... + 892) = 1969221802.
892 = 7921 appears in the decimal expressions of π and e:
π = 3.14159•••7921••• (from the 4810th digit),
e = 2.71828•••7921••• (from the 6733rd digit).
Page of Squares : First Upload March 8, 2004 ; Last Revised April 6, 2010
by Yoshio Mimura, Kobe, Japan