The smallest squares containingk 99's :
12996 = 1142, 9909904 = 31482, 6999999556 = 836662,
219992999929921 = 148321612, 102799699999909929 = 3206239232.
992 = 9801, a reversible square (1089 = 332).
992 = 9801, 98 + 0 + 1 = 99.
992 = (52 + 2)(192 + 2) = (52 + 8)(172 + 8).
330k + 1353k + 4026k + 4092k are squares for k = 1,2,3 (992, 59072, 3691712).
704k + 1034k + 1870k + 6193k are squares for k = 1,2,3 (992, 65892, 4954952).
1122k + 1353k + 2508k + 4818k are squares for k = 1,2,3 (992, 57092, 3626372).
1276k + 1529k + 1738k + 5258k are squares for k = 1,2,3 (992, 58852, 3953072).
Komachi equations:
992 = 9876 - 5 / 4 * 3 * 2 * 10 = - 9 + 87 * 6 * 5 * 4 - 3 * 210,
992 = 12 - 22 + 32 * 42 - 52 + 62 * 72 + 892 = - 12 + 232 - 42 * 52 + 672 + 82 * 92
= 982 + 72 - 62 - 52 - 42 + 32 / 22 * 102.
Two 3-by-3 magic squares consisting of different squares with constant 992:
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(992 + 1) = (12 + 1)(702 + 1), (992 + 9) = (92 + 9)(102 + 9).
992 = (1)(2 + 3 + 4)(5 + 6)(7 + 8 + 9 + ... + 15) = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 16).
4302 = 762 + 772 + 782 + ... + 992.
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 75)(76 + 77 + ... + 99) = 94502,
(1 + 2 + ... + 44)(45 + 46 + ... + 99) = 19802,
(1 + 2 + ... + 98)(99) = 6932.
(12 + 22 + 32 + 42)(52 + 62 + ... + 182)(192 + 202 + 212 + 222)(232 + 242 + ... + 992) = 58419902.
993 = 970299, 9702 + 99 = 992.
998 = 9227446944279201, 9 + 2 + 2 + 7 + 44 + 6944 + 2792 + 01 = 992,
9 + 2 + 27 + 44 + 69 + 442 + 7 + 9201 = 9 + 2 + 2744 + 6944 + 2 + 7 + 92 + 01
= 9 + 2 + 2744 + 6944 + 2 + 79 + 20 + 1 = 9 + 22 + 7 + 44 + 69 + 442 + 7 + 9201
= 92 + 2744 + 6944 + 2 + 7 + 9 + 2 + 01 = 9227 + 4 + 469 + 44 + 27 + 9 + 20 + 1
= 9227 + 44 + 69 + 4 + 427 + 9 + 20 + 1 = 9227 + 44 + 69 + 442 + 7 + 9 + 2 + 01 = 992.
992 = 9801 appears in the decimal expressions of π and e:
π = 3.14159•••9801••• (from the 25320th digit),
e = 2.71828•••9801••• (from the 1319th digit).
Page of Squares : First Upload March 22, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan