Functions — “s’wat” Math is about !! :-)

Posted by Nalin Pithwa

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

${f(3)}^{4}=f(3^{4})=f(81)>f(64)=f(2^{6})={f(2)}^{6}=3^{6}=27^{2}>25^{2}=5^{4}$

as well as ${f(3)}^{8}=f(3^{8})=f(6561)<f(8192)=f(2^{13})={f(2)}^{13}=3^{13}<6^{8}$ .

So, we arrive at $5<f(3)<6$ . But, this is not possible, since $f(3)$ is an integer. So, $a=4$ .

Cheers,

Nalin Pithwa.

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