MathDB

For every natural

n

let

\phi (n)

be the number of coprime numbers

k \in \{1,2,...,n\}

. (Example:

\phi (12) = 4

, because among the numbers

1, 2, ..., 12

there are only the

4

numbers,

1, 5, 7

and

11

coprime to

12.

) If

k

is a natural number, then one defines \phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)} (Example:

\phi^3 (n)=\phi (\phi (\phi (n)))

) For every whole

n> 2

let

c(n)

be the smallest natural number

k

with

\phi^k (n)= 2

. Prove that

c (ab) = c (a) + c (b)

for odd integers

a

and

b

, both of which are greater than

2

, .

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