The intriguing number 1/998001: exploration and visualization

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Very nice! The next assignment could be: ポテト一郎🥔 on X: "1/109の中にフィボナッチ数列を見つけました！" / X

E69a1t7UcAEsABS.jpg
E69a1t7UcAEsABS.jpg1536×2048 390 KB

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1/109, Fibonacci, and Fractional Part
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Uau, this is interesting! :slight_smile: Adding it to my postits for future investigation…

Thanks for the hint!

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This post was interesting enough to make me delurk and offer a few comments.

First, using N is not really necessary here; RealDigits is smart enough to handle rational numbers with repeating decimal representations:

Short[rd = RealDigits[1/998001]]
   {{{1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 9,
       0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 6, 0, 1, 7, 0,
       1, 8, 0, 1, 9, 0, 2, 0, 0, 2, 1, <<2876>>, 0, 9, 8, 1, 9, 8, 2, 9, 8, 3,
       9, 8, 4, 9, 8, 5, 9, 8, 6, 9, 8, 7, 9, 8, 8, 9, 8, 9, 9, 9, 0, 9, 9, 1, 9, 9,
       2, 9, 9, 3, 9, 9, 4, 9, 9, 5, 9, 9, 6, 9, 9, 7, 9, 9, 9, 0, 0, 0, 0, 0}},
    <<1>>}

rd[[2]]
   -5

which signifies that the decimal form starts out as 0.00000100200… (five zero digits after the decimal point, and before the first non-zero digit).

For convenience, I’ll rotate the digits:

rdrot = RotateRight[rd[[1, 1]], 4];

for a more convenient representation:

FromDigits @ {{rdrot}, -1}
   1/998001

Note that

Length[rdrot]
   2997

which is a multiple of 3,

Divisible[%, 3]
   True

and this length can be computed like so:

MultiplicativeOrder[10, 998001]
   2997
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Thanks for your hints J. M.