Denote by S the set of all primes such the decimal representation of
has the fundamental period divisible by 3. For every
such that
has the fundamental period
one may write
where
; for every
and every integer
define
by
a) Prove that
is infinite.
b) Find the highest value of
for
and
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Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
$$\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots ,$$
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by $$f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}$$
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
Školjka