In my last formation-of-numbers question I somehow tricked myself, because I wanted to make a puzzle about a number that looks like a year. But the complicated solution I thought of turned out to be outperformed by a much simpler ansatz. Here is a puzzle that (hopefully) has my intended solution as optimum:

Make the number $439204$ by using an arbitrary number of $\Phi$s and

• the operators $+$, $−$, $\cdot$, $/$
• exponentiation
• brackets $($ $)$.

The aim is to use as few $\Phi$s as possible.

You may not use operators or functions other than in this list, so don't even ask for rounding ($\lfloor$ $\rfloor$, $\lceil$ $\rceil$) or logarithm ($\log_a (x)$). If you want to use roots, this is ok as long as you express them as exponent: $\sqrt[n]{x} = x^{\frac{1}{n}}$.

In my last formation-of-numbers question I somehow tricked myself, because I wanted to make a puzzle about a number that looks like a year. But the complicated solution I thought of turned out to be outperformed by a much simpler ansatz. Here is a puzzle that (hopefully) has my intended solution as optimum:

Make the number $439204$ by using an arbitrary number of $\Phi$s and

• the operators $+$, $−$, $\cdot$, $/$ (including unary "$-$")
• exponentiation
• brackets $($ $)$.

The aim is to use as few $\Phi$s as possible.

You may not use operators or functions other than in this list, so don't even ask for rounding ($\lfloor$ $\rfloor$, $\lceil$ $\rceil$) or logarithm ($\log_a (x)$). If you want to use roots, this is ok as long as you express them as exponent: $\sqrt[n]{x} = x^{\frac{1}{n}}$.