Almost Integer

[Dean Hickerson]

Yasutoshi Kohmoto wrote:

> Recently Mathworld's description about "Almost Integer" was updated.
> I expected my example which I had mailed to Eric Weisstein was described
> on it.
>          Pi^14/(9103887*Zeta(9))=1.00000000004539
> But he didn't do so.
>
>     Is it not interesting?

Not very.  Suppose you compute all numbers of the form

    pi^ab / (cdefghi * zeta(j))

where a,b,...,j are decimal digits.  There are 8999999100 choices for the
digits, so it's not surprising that one of them has 10 zeroes after the
decimal point.  If you found one that had a lot more than 10 zeroes, that
might be interesting.  (E.g., pi^8/(9450 * zeta(8)) = 1.000000000000..., but
here we already know that it's exactly 1.)

Another way to write your equation is:  pi^14/zeta(9) = 9103887.0004132...
There are 900 choices for the digits in  pi^ab/zeta(c),  so again it's not
surprising that there's one with 3 zeroes after the decimal point.

Dean Hickerson
dean at math.ucdavis.edu