MathDB

S 18

Source:

May 25, 2007

Miscellaneous Problems

Problem Statement

Denote by

\frac{1}{p}

the set of all primes

p

such that the decimal representation of

\frac{1}{p}

has the fundamental period of 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}\cdots a_{3r}a_{1}a_{2}\cdots a_{3r}\cdots,

where

r=r(p)

. For every

p \in S

and every integer

k \ge 1

define

f(k, p)=a_{k}+a_{k+r(p)}+a_{k+2r(p)}.

[*] Prove that

S

is finite. [*] Find the highest value of

f(k, p)

for

k \ge 1

and

p \in S

.

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