MathDB

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)

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

Problem Statement