Functions — “s’wat” Math is about !! :-) | For the love of Maths: the Queen of all Sciences

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 $f(2)=a>2$ and $f(mn)=f(m)f(n)$ for all natural numbers m and n. Define the smallest possible value of a.

Solution:

Since, $f(n)=n^{2}$ is a function satisfying the conditions of the problem, the smallest possible a is at most 4. Assume that $a=3$ . It is easy to prove by induction that $f(n^{k})={f(n)}^{k}$ for all $k \geq 1$ . So, taking into account that f is strictly increasing, we get