1. g(955)

g is a function possessing the domain and range of the positive integers satisfying:

(1) g(p+1) > g(p);

(2) g(g(p)) = 3p

Analytically determine all possible values that g(955) can take.

2. Originally Posted by perash
g is a function possessing the domain and range of the positive integers satisfying:

(1) g(p+1) > g(p);

(2) g(g(p)) = 3p

Analytically determine all possible values that g(955) can take.
This probably isn't good enough, but given the two properties we can establish what the function is for all p, so given enough patience we can determine g(955) explicitly.

However, note from property 1 that g is bijective: g has an inverse. (Because if $\displaystyle g(p \prime) = g(p)$ for some $\displaystyle p \prime > p$ that would imply that $\displaystyle g(p + 1) > g(p)$ is false.) And note also from 1 that $\displaystyle g(p) \geq p$. Thus from 2 we have that
$\displaystyle g(g(p)) = 3p$

$\displaystyle g(p) = g^{-1}(3p)$

$\displaystyle g^{-1}(3p) = g(p)$

$\displaystyle 3p \geq g^{-1}(3p) = g(p) \geq p$

So $\displaystyle 955 \leq g(955) \leq 2865$.

(Actually the only p for which g(p) = p is p = 0, so really we have that $\displaystyle 955 < g(955) < 2865$.)

-Dan