Your intuition isn't completely off base. If the two periods M & N are relatively prime, then the product period should be a divisor of 9MN. This is because 10^M - 1 and 10^N - 1 both divide 10^MN - 1, and if gcd(M,N)=1, then gcd(10^M-1,10^N-1) is 9. The 9 in 9MN provides the required factor of 9, so that (10^M-1)*(10^N-1) divides 10^9MN - 1. This suggests the general period formula is (a divisor of) lcm(M,N) * (10^gcd(M,N) - 1). Rich ---- Quoting James Propp <jamespropp@gmail.com>:
I should have said "cannot be bounded by any subexponential function of m and n".
Or perhaps I should have said "I had assumed that if you multiply two eventually repeating decimals of period less than or equal to n, you get an eventually repeating decimal whose period is bounded by some polynomial function of n."
In fact, my intuition told me that there'd be a quadratic bound. Wrong!
Jim
On Mon, Sep 29, 2014 at 6:03 PM, Dan Asimov <dasimov@earthlink.net> wrote:
To make a short story long, what does "the period length *can be* exponentially large in n and m" mean?
Maybe it means sup L(rs) over all r, s in Q with L(r) = m, (s) = n, is >= exp(Cm + Dn), where L(t) is the period of the repeating decimal of t in Q, for some C and D in R+. Is that it?
--Dan
On Sep 29, 2014, at 2:53 PM, James Propp <jamespropp@gmail.com> wrote:
I had assumed that if you multiply an eventually repeating decimal of period m and an eventually repeating decimal of period n, you get an eventually repeating decimal whose period is bounded by some polynomial function of m and n. But today I learned from Henry Cohn that that's not true: the period length can be exponentially large in m and n.
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