# Dedicated to my beloved

**Number Theory**Level 5

let \(f\) be a function on real numbers and whole \(x\) such that ,

\(f (\varphi + x)+3 = f (ix)\)

also \(f(2)=7\)

Let us define another function \(P\) on \(i \) and \(x\) such that ,

\[\left.\begin{matrix} &P_{x} (i)\geq P_{x}(x+i) & \\ &P_{x-1}(i)+3 \geq P_{x}(x+i)& \\ &P_2 (i)=10 & \end{matrix}\right\}\] holds true , where \(i=\phi (x)\)

Then evaluate :

\[\left(P_4 (i) + f \left(\frac {\sqrt5 (\sqrt5 +1)}{2}\right)\right)\]

**Notations**:

\(\varphi\) denotes the Golden ratio, \(\varphi = \dfrac{1+\sqrt5}{2}\).

\(\phi(\cdot) \) denotes the Euler's totient function.

\(x\) and \(i\) are whole numbers.