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Re: PNT look-alikes
Posted:
May 18, 2011 2:16 PM
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On May 18, 9:07 am, marcus_b <marcus_bruck...@yahoo.com> wrote:
> On May 1, 7:48 am, Han de Bruijn <umum...@gmail.com> wrote:
> > I would like to have many _more_ of these PNT look-alikes. Any takers?
> > Please do me a favour and leave out the _very_ trivial ones.
> OK, here's one I don't know the answer to. Let m be a positive
> integer.
> Define m to be 'decimal-maximal' if the decimal expansion of 1/m
> repeats
> after m - 1 digits. For example, 1/7 = .142857142... , so 7 is
> decimal-
> maximal. The number 11 is not decimal-maximal. Let M be the set of
> decimal-maximal numbers. What is the asymptotic density of M ?
By Fermat's Little Theorem (Carmichael generalization), the number
of digits after which 1/m repeats must divide lambda(m), which in
turn divides phi(m). It follows that if m is "decimal-maximal" (or
"base-b-maximal to any base b), it must be _prime_.
http://en.wikipedia.org/wiki/Repeating_decimal
http://en.wikipedia.org/wiki/Carmichael_function
So M is a subset of the set P of prime numbers. Therefore the
density of M is at most that of P, which is _zero_. QED
Related question: what is rho(M,P) (Katz notation), the density of
M _in_P_? I don't know the answer to this one.
> More general: let T3 = the set of numbers m for which the decimal
> expansion
> of 1/m repeats after 3 digits. For example, 37 is in T3. What is
> the
> density of T3? Is the set T3 infinite? (Probably easy to prove).
It took me a while to figure this one out, but then I realized
that T3 contains not only 37, but 37*10^n for every natural n.
1/37 = .027027027...
1/370 = .0027027027...
1/3700 = .00027027027...
etc.
Therefore T3 is infinite. QED
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