Reference: Nordic Mathematical Contest 1987, R. Todev:
Question:
Let f be a function, defined for natural numbers, that is strictly increasing, such that values of the function are also natural numbers and which satisfies the conditions and
for all natural numbers m and n. Define the smallest possible value of a.
Solution:
Since, is a function satisfying the conditions of the problem, the smallest possible a is at most 4. Assume that
. It is easy to prove by induction that
for all
. So, taking into account that f is strictly increasing, we get
as well as .
So, we arrive at . But, this is not possible, since
is an integer. So,
.
Cheers,
Nalin Pithwa.