The smallest squares containingk 83's :
8836 = 942, 18378369 = 42872, 18383834569 = 1355872,
67838383796836 = 82364062, 83838308371983936 = 2895484562.
83 is the sum of a square and a prime in 5 ways : 22 + 79, 42 + 67, 62 + 47, 82 + 19, 92 + 2.
838 = 2252292232139041, 2 + 2 + 52 + 2922 + 3 + 2 + 1 + 3904 + 1 = 832,
838 = 2252292232139041, 2 + 2 + 5229 + 223 + 2 + 1390 + 41 = 832,
838 = 2252292232139041, 2 + 25 + 2 + 2922 + 32 + 1 + 3904 + 1 = 832,
838 = 2252292232139041, 22 + 5 + 2 + 2922 + 32 + 1 + 3904 + 1 = 832.
832 = 30 + 31 + 34 + 35 + 38.
Komachi Fraction : 832 = 3782061/549.
Komachi equations:
832 = 1 + 2 * 3456 - 7 - 8 - 9,
832 = 92 + 872 - 62 * 52 + 42 * 32 - 22 - 12 = 92 - 82 - 72 + 62 + 542 + 32 * 212
= - 982 + 72 + 62 + 52 + 42 * 322 - 12.
(832 - 7) = (92 - 7)(102 - 7)
832 + 842 + 852 + ... + 260032 = 24209572,
832 + 842 + 852 + ... + 2762 = 26192.
(1 + 2 + ... + 8)(9 + 10 + ... + 14)(15 + 16 + ... + 83) = 28982,
(1 + 2 + ... + 32)(33 + 34 + ... + 66)(67 + 68 + ... + 83) = 336602,
(1 + 2 + ... + 45)(46 + 47 + ... + 68)(69 + 70 + ... + 83) = 393302.
(12 + 22)(32)(42 + 52 + ... + 282)(292 + 302 + ... + 832) = 2541002,
(12 + 22 + ... + 52)(62)(72 + 72 + ... + 282)(292 + 302 + ... + 832) = 16770602.
832 = 6889 appears in the decimal expressions of π and e:
π = 3.14159•••6889••• (from the 5153rd digit),
e = 2.71828•••6889••• (from the 486th digit).
(6889 is the fifth 4-digit square in the expression of e.)
Page of Squares : First Upload March 1, 2004 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan